Model Building
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ﺟﺎﻣﻌﺔ ﺍﻟﻤﻠﻚ ﺳﻌﻮﺩ ﻗﺴﻢ ﺍﻹﺣﺼﺎﺀ ﻭﺑﺤﻮﺙ ﺍﻟﻌﻤﻠﻴﺎﺕ
ﺑﻨﺎﺀﺍﻟﻨﻤﺎﺫﺝ ﺒﺈﺴﺘﺨﺩﺍﻡ
Excel and Vensim
ﺗﺄﻟﻴﻒ ﺩ .ﻋﺪﻧﺎﻥ ﻣﺎﺟﺪ ﻋﺒﺪﺍﻟﺮﺣﻤﻦ ﺑﺮﻱ ﺃﺳﺘﺎﺫ ﺍﻹﺣﺼﺎﺀ ﻭﺑﺤﻮﺙ ﺍﻟﻌﻤﻠﻴﺎﺕ ﺍﻟﻤﺸﺎﺭﻙ
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ا ا ا " ا!ف ا و# ا رب ا واة وا م . و$% و$& " #و . أ Excel and Vensim *)ام+ *ب ء ا ذج/ "و0دة ا2 ا3 ه56ه .س28ر2/ ا9 ب: "ء أول – " وGH – 3@A اB*ق اD " 9; ا3* آ9; 3= وآ ذآت Excel, SIMAN, Arena and General *)ام+ وا آة9H6 ا $; *ب آ/ا ا6" أن هPurpose Simulation System (GPSS WORLD) 58JK و582:* م2Lف أ2 "K ذن ا+ و30 دة إ" !ء ا2 NO 9;*م وا2 *; ان ا# أ30 ا6 ه3*و/A ا$ /P NO * وN/P $ Kو Q R *8S 3/K* H آ*ب3= TU ان وV * رعN/P و28 ر2:*K .X*Aت وا2 رة ا2Y # 3= 9% ع و2U2 ا9/8د .Model Building ع ء ا ذج2U2 9و0 ت ا0*ب ا/ا ا6 ه3:Z8 Dynamic Stochastic 3@ا2P# 9 آO ذج3 ه هT ي6وا ذج ا \ 9 آ9 O0 ]6 وآ9;; ا9 O0 اO N/PK 3* وواSystems اعA ا3 # *K 3 وه# = و9H6 * اK .اH درة3 ه3* وا9@ا2P# N= 3= D 6 أ2 ^ = ّقN # أيN^ T ^ 98رات ا`دT وا)ل وا أدواتQ H *)ام0 9T وا9 ا9` ) ا$8 ه/= رةJ * ا3 درا T واN آa*0 اد2 ت واSb اc` T واN آ3:# ا2 /رة وJا XU* اL و.d;= ضZ 3`8 b وا9= 9:L ن اه2/8 L ` *) 3 آ *ب و آن إ *ري/ا ا6 ه3= Vesim وMicrosoft Excel *)ام+ 9H6 ا " إ9=UAf Q ا2 اT وإ *)ا9 اTKد2H و/ره اP* إ2 هa اا56T .X*A " ا# Vesim aب ا: 9J 9) د2Hو اP* g% وا آة ا9H6 ا3= Excel N^ P`ت ا% aإن إ *)ام ا Spreadsheet P`ت ا% 9H6 X% وا،تJ آ^ ا3= $ L و3J ا* ا3= 98ورiا واH 9 T اQUا2 اModelling NL =ة2* /K رة2 T/! أN/ ات9J 9 @T اTKوذ] ;ر .9bر ا ت اP*إ Prototype Models 9و0*ب إ" إ *اض ا ذج ا/ا ا6 ه3= XL:K ; اN^ K 3*^ ا ذج ا ;ة ا/ ء ا3= 9و0ت ا2/ * اK 3*ا تS ا دN^ ء ا ذج3= 9 T ا9U8دوات ا0 اXU* آ إ،3;;ا Difference Equations 9L وا`وDifferential Equations 9 U`*ا 9ء اi= N^ K وMatrix Algebra and Cagculus =ت2` و ب ا إ *اضK *ب/ ة ا0ء اGJ ا3= و.State Space Representation .Case Studies 9 ت دراS 3= 9` *) 9 آ9 O0 ا ذجj ء 3
9*/ اء اYA و8/ ا$TH2 N ا ا6ز هJ إ3= 3;=28 ا ا ان2Hا وار6ه .*ب/ا ا6 هN^ " ا`;ة إ9 ا 3= X*A ا9/! " # Hا2* ن2/ و# RD ي0 3J *ب/ا ا6ن ه2/ http://www.abarry.ws/books/ModelBuildingBook.pdf QL2 ا . =2 وا ا lB ا ا ي# H ن# .د د2 ] ا9H هـ1423 RHر م2002 *
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ت اب ;...................................................................................... 9
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ا اول ..............................................................................
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أ ت ا :9H6اZض و ا2U2ح و ا 2ارد ..................................
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#ا;; 9و #ا 2ذج .............................................................
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ا*` /ا 3Oوآ 9اOم ........................................................
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Kر l8أ ........................................................................... 9
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l8Kا 2ذج و O 9THا*` /ا.................................... 3O
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N^ KاOم = 3ا*` /ا 3Oوآ 9اOم ...................................
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ا ا .............................................................................
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ا دSت ا*`.................................................................... 9 U
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ا دSت ا*` 9 Uار 9Hا0و" ا)..................................... 9:
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ا دSت ا*` 9 UارHت ا ..............................................
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ا دSت ا) 9:ت ..................................................... 9* Y
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ا 9ا *.......................................................................... 9 J
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ا \ 9ا *.................................................................... 9 J
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2اص N#ا*`............................................................. D NU
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98Oا* 2 Nا................................................................... N
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ا6Jور ا ` ......................................................................... 9
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ا6Jور ا /رة ..........................................................................
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ا6Jور ا آ 9ا *;ر............................................................... 9
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ا ا .............................................................................
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ا دSت ا`و....................................................................... 9L
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ا دSت ا`و 9Lا) 9:ار 9Hا0و" ......................................
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Nا دSت ا`و 9Lا) 9:ار 9Hا0و" )* +ام ........... Excel
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R8;Kد 9 U`K 9د= 9و............................................... 9L
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ا دSت ا`و 9Lا) 9:ت .......................................... 9* Y
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ا Nام د 9ا *.......................................................... 9 J
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ا Nا)ص د 9ا.......................................................... 9 /
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ا*ف ا2 3@Tل )* +ام .......................................... Excel
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ا اا .............................................................................
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Hو ب ا `=2ت ...............................................................
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ا ت ا " # 9 0ا `=2ت ..................................................
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jا)2اص ا `ة دات ......................................................
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ا; ا Gة .......................................................... Eigenvalues
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ا *TJت ا Gة .................................................. Eigenvectors
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ر=2; 9 9=2` Qة ...............................................................
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ر=2; e Qة `............................................................ 9 9=2
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]/`Kا; 9اPذة .............................................................. SVD
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ب ا `=2ت ......................................................................
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9;*Pا `..........................................................................9=2
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N/Kا `.......................................................................... 9=2
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9;*Pا 2/س ..........................................................................
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ا ا ...........................................................................
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i= N^ Kء ا....................................................................... 9
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دSت =iء ا 9و 3= Tا Jل ا.................................... 3G
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ا* N82إ" ! N/ا................................................................ 9
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N82Kا دSت ا*` 9 Uار........................................... n 9H
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O N82Kم ار 3 9Hإ" ! N/ا.......................................... 9
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9ا دSت ا*` 9 Uوا 98Jا )* ....................................... 9:
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N82Kا \ 9 O0ا) 9:إ" ............................................... 9:
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Nا 9 O0ا) 9:ا *;ة ..........................................................
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ا 9 O0ا) 9:ا *;ة ا *.................................................. 9 J
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ا 9 O0ا) 9:ا *;ة \ ا *........................................... 9 J
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i= N^ Kء ا 9دSت ا`و............................................... 9L
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ا)* + Nام .................................................................. Excel
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i= N^ Kء ا 9دSت ا`و2 9Lا ....................... Vensim 9:
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ا ا دس ...........................................................................
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jا ذج ا................................................................. 9 0
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2ذج ا ) 2أو اAل( ا)................................................... 3:
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2ذج ا ) 2أو اAل( ا.................................................... 3 0
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2ذج ا 2واAل ا)........................................................ 3:
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2ذج ا 2وا0ل ا......................................................... 3 0
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ا 2ذج ا* ......................................................................... 3
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2ذج اS2دة وا 2ت ..................................................................
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2ذج " ا 2ا ......................................................... 3* H2
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2ذج ا 2ا ود Pوط ) ........................................... (3* H2
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2ذج l%28دSت ا*`Z* 9 Uي ............................... 9
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2ذج l%28دSت ا*`^ 9 Uث *Zات ....................... 9
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ا ا .............................................................................
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Sت درا 9ء jا ذج ....................................................
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(1ء 2ذج آ2 9ق ...........................................................
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Kا 2ذج )* +ام ................................... Curve Expert
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Kا 2ذج )* +ام .................................... Excel Solver
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2 (2ذج )Gون ;2ة ......................................................... 9 #
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اLت ا 3= 9ا 2ذج ...........................................................
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دورات ا* 986Zا) `....................................... Feedback Loops 9
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دSت ا 2ذج ........................................................................
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2 (3ذج ** ........................................................... 3P*2= 9
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2ذج !i= N/ء ا)* + 3P*2= 9 ** 9ام ............... Vensim
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(4ا ذج اآ 9ا2Pا@......................................................... 9
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ا) 9%ا رآ..................................................................... 9=2
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N^ Kا ذج اآ 9ا2Pا@ 9ا رآ)* + 9=2ام .............. Vensim
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..................................................... bifurcation RP*ذج ا2 (5
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............................................ VenSim d:) RP*ذج ا2 N^ K
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..................................................... R L 3= 9 i 38;K ذج2 (6
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.................................................... Vensim a N^ ذج2 ا
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................. Lorenz Attractors Models c*8ر2 ذ تH ذج2 (7
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..................................................... Vensim 9: ا2 ذج2 اN^ K
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............... Prey and Predator Model 9iذج ا `*س وا2 (8
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.................................................... Vensim *)ام+ ذج2 اN^ K
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......................................................................... 9 ت اLأ
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.......................................................................... 9` )اورات ا
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....................................... 9; *رات اA ت اH وإ9 ( أ1)
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.................................................................................... QHا ا
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ا اول: أ&&%ت ا : !"#$ا+ض و ا ) ح و ا #ارد 8ف ا #$ذج " #ا# N 2 ^ = ، /%/ %#0 : $ة أ!)ص ه 2ا 2ذج RJ8 L اه :ه 2د 9ر ، 9U8و : & RJ8 Lه:L 2ر 9و& 2/Kن إ : $* Hه2 2ذج " 8اد إ $@Pو : & RJ8دKK 9ي 2Yب = 3أ ت ا).9D fK 2ا Nا *Pك = 3إ H2 TK Hا = ، "9;; N^ K " $دSت اN^ K 9U8 ) 9//آ XK Q *J 2 (9ارا ^ 9و :Lر ا :L N^ 8 9ر ;; 3و 2ذج ا " N^ 8ا N/Pا6ي و $ #ا " ا;; 3إ ، $@Pا N/! N^ K 9ا cوآl # TOرKا@.$ ا* l8ا 2ذج \ آ= ;K / 8 l/= 3و;ر 9ا ذج ا )* `9؟ = N/ا9 وا;:ر ا 9;; N^ K 9و /هك إ *ف آ Tو; / 8Sر*d:) ^ = .T !2ارع 98و)D d:ق ا ` = 3ا ة ا* T= Q;K 3ا 98آ9;; N^ K T N و8وان * T Pو / 8S /إ *)ام أه /ن ا ، bإذا هك ! 2Hهي *; ا 2ذج أ Sوه 2ا+ض Purposeو l8K g8 3Kا #$ذج " #ا %#0 : $ذا 1ض d:) = ، /%/ 2ا2Pارع ه 9;; N^ K 2ا*K 3و 2! Tارع ] Kا Z 98ض اS* Aل 2! " #ار T#وهS* g 8S D 2ل D " #ق ا ` = ] K 3ا ة وg 8 "*S * ;ط إزدم ا ور = ] K 3ا 98و /ه6ا NJ8Sا 2ذج .هك أ2اع )* `9 :Lرات ا0ب Ti ،ص Rا`D0ل و Q H 3 Tiذج ا;:رات )^N Q Hا2:ا (Qوه6/ا \ = ،اض ا )* ` a*K 9ذج )* `.9 هك ! 2Hهي & = 3ا 2ذج و 2ف *^ $Uل ا* 2 :3د#ك أ اL%0ء اJد إ" 0 $Gول ة واKS D Xف ا ];8 / = ، 8:أن 9# ] d)8 )2P d:ارع ا*BK 3دي إ" 2/8 L ، $Gن ) ;* \ d d:أو )8 Lج ] )d: ه 9;: O 3ا* T= / 8 3أو ) N P8 d:ا T / 98و، $G QL2 ] !B8 ه 56ا )::ت 2/8 Lن = TدرHت )* ` 9ا*` N%و ; c8ر )* ` ، 9وه6ا B8دي إ" " 8ر 9Hا2U2ح أو ا ) ح ^ = ، ResolutionاN%`K 8K R ا!0ء 2%رة ا 6ت 8 82K 9ل ه اBري ) ون =2آ (Focus cو =9 ة و2/* /ن ا*` N%واH 9Uا ودة إذا ا 6ت 2 82K 9ع Hو !) *) = 3ا* 82و = .Rذج ا T i8در 9Hو2Uح وه6ا *8
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;ار ا*` N%اiور 98وا 9Tا* 3= N K 3ء ا 2ذجT= .ك ذج 9 Lا2U2ح Low resolutionو ذج 9#ا2U2ح . High resolution ا Pا2Jهي ا 3= 0ء 2ذج ه 2ا #ارد Resourcesوا/Aت و N PKا 2اد ا :ة وا Gا *ح وا)ة ا *=2ة أو ا /إآ* U Tا Gا *ح وا #اPي وا L " = ، 3ر ا 2ارد وا/Aت * Q:ان 2 3ذج Zض ا 2 :ب و 2U2ح ا ^ = ، Rإذا ا X:#ورi 9Lء و Lر%ص و :ة =; dو ] R Dر )d: 2Pارع وا ا= 2ل T)* 8 3/ ]*Hا:ب ا * 8Jوا Gا *ح ] c د2 = ، @Lارد ا :ة 2ف 2:) 9:8 Kط *; 9أ 2د " #ا jوj liK ات ا*K 3ل " #ا ا= ا ، 9 Tاbن 2ا ] 3:#أLم * ]+= 92ا:ق ا@ 2 9ن وا`2 9#ن & وا ا= 2fان Kل ^ = T*#2 " #ا 2 Hن وا 2 #:ن وا 2 R#ن & وه6/ا 8GKا*` N%وا2U2ح ;ر اTف ا 2 :ب وا 2ارد ا * ^ = 9ر أر 9`%أو أ!Jر أو زل )KSم اTف ا 2 :ب و K L ا) N ;K N%`* 9:8و T2Uآ ا*K Tج ا" و XLا2Dل ا *ح. ا دئ ا 3= $ Tء ا ذج وا6ي * JT 3= $ه89 2أ ا 6 7%#ا& 3245 Bottom up designوه 2اء 2ذج N L dا2U2ح Yز8دة ا*` N%ا ; 9ر g Kا 2ارد و*" ;*8اZض ا 2ذج .ه6ا c/#أ ا* اN` " #0 Top down designوا6ي 8أ *` N%آ^ة و;ة T N8G8 Yا*` \ N%ا ، 9 T وا6ي * 5ه. = 3درا * 2ف *:ق =;#$2 dذج ا ) %وه 3ذج )* + "Kام ا:ق ا9U8 )L#ت L# ، 98Hت 2K ، 9;:ز8ت إ* 2 ، 9ارزت و ا 9 2 aا(. و 2ف `f Tن ا 9H6ه 9 # 3ء 2ذج ر.3U8
74ا /%/و 74ا #$ذج : Real World and Model World ا ا;; 3ا6ي =H ; $ا " # 3* 8 N/Pا Sن = Tا Pآ Nا;;N/P 9 آ Nاو *" 3@GHو6Tا fJ 8ا Sن ا" ا*; R8وا* P dح 9Dه 56ا Pآ3/ N 8ول N^ K / 8 ، Tا ا;; 9 U 9 N/P 3آ = 3ا N/Pا*:3
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Occam's razor
Real Worl
Model World
Interpreting and Testing Formulating model world problem
Model results
Model
Mathematical analysis
(مTس اوآ2 5 وود$=اDى أ2 K) d 3 هN/! " # ب6T8 أو3;;;ب ا ا8و و*;ل ا3;; اd K وR8;K 2ذج ه2 ا# و، ذج2 ا# N^ 8 ي6وا 3 # 3` = أ2 وه، Occam's Razor *)م س اوآ;م8 ذج2 ا# " إ3;;ا 7هD أ8 إF ا324 ور أوG ا%1 %H آ ا8I9&آ( إA#ء )ا%>? ا% 9 :ل2;8 KAة ا84F أ او$ # 3 ا ا أ6 ه، ( 2A# )اJ%Aد ه"ا ا8 0 ا%Hا ذج# 6 ك أآ$ إذا آن ه: ل2;8 ي6 وا3 اV ا3= Parsimony Principle
ه7I#ات وا%+# ا6 د84 F ذج ا"ي ي ا#$P QR$ ا9 /0 SI 8Tوا 2ي ه6ر وا2TP اE = mc 2 *P8ن ا2L ا ا أ6 " ه# أدلS و.GP ذج ا#$ا و\هmxɺɺ = F K2 9ن اآ2L ]6 وآ، قDA " ا# T وأه9U8ا ا2; اd ا .92 ا9 O اR ا*آ9: ا9U8ا ا2; ا ا2L و% QU و3= l /K ^ ا ء واj $ م2;8 ي انO 9TH و 52:* أl%2 (^ او اآ9`% Ti ) ر2: ة# و;ة92: 9U8ر م# وT P= ا2:\ وg *ط اA ا اN وا آL ، TU2 \*ه *;ه و ه ب6 f8 j وا9` /* ا وا ا2; ا56 هN^ 9;; ا9 /P T * إ و ط0 ا3= * وS2; ن2/8 $D* *; ان إ8 N 5\ N # آN 8( و9U2 )ا
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ا 9ون دSت ر; 9U8ة و2K 9 82Dي ا ^/ا *Zات وا \ اiور .98وآهن " #ذ] = 3اAار ا *د GK R2 9 Lداد آ NأN;* Z* `U إ" ا د2 "* 9آن ه6ا ا *f X 8S Zي 9 9`%أو 9;:إ" ا * Zا* N^ Q د U 9= Nط ا Jو أ2Dال ا`D0ل 5إ" c` 3= 9 15ا ) Q *JآT N G8داد وه6ا K 3:8ا 2L dي و * 3ز8دة = . ( R2 3وه6/ا ى ا ^/ا^ GKد د ^/ TKSا*` \ N%اiور 98وه 2 3Kf8س أوآTم :إ * آ Nا*`N% \ اiورH 98ا وا ; 3ا 2ذج = 3ا dا!.$/ N82K 9 #ا ا;; 3إ" #ا 2ذج ه 3= 3ا;; 9أه وأ2: R%ة = 3ء ا ذج وا*2; 3م 8T* Tأ 2س اوآTم =` 9 # 3ء ا 2ذج * آ Nا*`\ N% ا Z 8!* $ Tض وا2U2ح او ا*2 8!* T*J / 8S 3ارد ا *.9 = ; Q:ا ا;; 3ا" / 8 Jا* $ Nآ 2 Q:L 2س .أ 2س اوآTم أداة 9و :ة ،إذا ا X:*Lا+= ^/ن ا Nا ** aا 2ذج 2/8ن $اي 9 % 9 /Pا;; 9وإذا ا X:*Lا; += Nن ا 2ذج 2 9 Rارد ا *2 .9ف Pح ا N/Pا و2س اوآTم ^ل ا*:3 ; در 3= XاU8ت =2U2 3ع ا*` NUوا*2U2 N/ع ا;6ا@ Projectiles lوا*3 Q*Kر ، =/ Q:Lا;*!Aق 8أ ل g X Y NJKو!وط او 9= 9وا s 0 9#
و
v0و N/ا د9
" # N K a (t ) = gاAزا 9آاG 9
2U " # . s (t ) = 12 gt 2 + v0t + s 0ء ا N/Pا و2س اوآTم =+ن ا 3= 9 /Pا ا;;2;K 3م * # Bآ 9`86L 9أو ا!ء 9 Yآ J 3او ! 9\ L ، $T ا 9 /Pا;; 9و; Tإ" #ا 2ذج *) 2س اوآTم ا6ي ا=*ض ان ا ا;;3 g:و ل NJKاو Hذ X Y 9و= 3ا`اغ )#م و2Hد \ف 2Hي( # Qم و2Hد اي 2Lى YBة ا ى U ،ه 56ا`وض او ه6ا ا =+ن 2ذج ا ر=/ Q:L N/! " # ا 9 /PK /و Y ، $ا * = 3ا Nو ت ا آ6T ، ^ 9ة ا\0اض وا ا 2ذج Hا ،ا) :ه 3= TO8ا`f 9Uن ا 2ذج f*8ان Q Hا;6ا@ T lآ9 Q*Kر 3= =/ Q:L N/! " #ا ا;; 3ذا 2ر ر 9P8أو آة Dو(a2 a ) 9 = 3و2Hد ر8ح؟ .ا # T S i8ا;6ا@ lان = 9Uل ا* NJا^ Xوإام ;و 9ا2Tاء \ 9%و 3:Kأ :ء N`K Lا وا .9 8GTوZ /ض ا*;+= R8ن رJ 3رة أو ا!ء 28 3= 9 ;Yم = $ا8ح ``+= 9ن 2ذج ا ر ا;=/ Q:
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H Rا .ا H Tا اYء آ* * *; N a@* #8ا 2ذج ان Qiا`Uت ا*3 T # 3ا 2ذج =`^ 3ل ا;6ا@ lأ=* Uان ا ا;;\ H28S g: 3ف 2Hي )أي = 3ا`اغ( وHذ 9* Y 9و2L H2KSى ا ى .ه 56ا`Uت هGH 3ء Tو2ي ا 2ذج = )! 38ر 9P8و2;8ل ] :ا2 !Oذ* = ، f: ]Hد ! : $ # ا 9P8آ Q** Xر 2 =/ Q:L N/! " #ر 3= Xا`اغ 8 L .د ] #ان 2ذ]H \ وا= 3= S 0 3Lاغ و 2/ن رد ! : $ #ء ا^; 9وا*= "K 3ت Lة =+ن #YfKم و2Hد =اغ ه / 8 YfK 2إه $وه $*^K 2ا*Jرب ا .9 ا`Uت وا* Aرات 2ذج ه 3اه 2:ط ا=ع Uا //Pوا2 8Lذج. Pوع g8) :ا N R:ه6ا ا Pوع H T $0ا * Aب ا دة( `*ض ا 2ف ;2م Pوع 2 9ز N%2* 8ا 98 @ Gا" اLب :ر. ا=*ض ان ا = 9ا 98وا :ر 50آ 5JK *2ا2ا 8 .ان 9` /K R ا6T 8Gة ا) R 3/ 9ا ا U; Rة ا2 Gن .إذا ا=* XUان ا رة T * d 2إ *Tك 25آ 2H/ن*#A 6 .ر :3 8 ا ر ا 8GاZ*K L 9 اوLت ا*2H2 Yf*K N%2د زم وري أو $# 9ا ) 2Jا:ر\ ،ر ،ر8ح 982Lا( إ *ف ا =ت دا Nا ) 98إذا آن ا :ر ^ = ! 3ل ا += 98ن ا 3= @ Gا2JبZ* Kق ر * 9= Tا2Dل( و2Hد /ن =رغ GAال ا2 Gن #ا2%2ل :ر ام اO*Aر `اغ /ن * Q:إJ8د إ*#رات ا ى آ^ة و /ذا 2 #س اوآTم؟ ه Nآ Nا*` N%ا ;9 9 T؟ ذا T 6 fواي Q:*; T؟ Zض ا 2 :ب و Uا 2ارد ا *9؟
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مO ا9 وآ3O ا/`*ا Cause Yf* واR ا9L# " # م2;8 ا*; ي5JKA اR 3;; ا اNآP Q N*ا .j اTi # لG N/P 9 /P اء اGH أ6 f8 و9: اand Effect " إ9 /P اQّ:;8S System Approach/ Thinking 3O ا/`* أو ا3O ا5JKAا S 3* *دة واd رواH2K م إذO اT= 3* ا9 ا8* N/ آT O8 N 9G اءGHأ T JK دة# إY " ة# ءGH N إ" آO انS T=P/* *ر وإ#A ا3= ه6 ا م لO =تK وRاآK T= ا*ف و9 " و# Gآ8 3O ا/`*= .;S 9و0 اR ا*اآ56 " ه# مOف اK # اتB*" اK Y و9 أوRاآK إ *)ام 9 ا*آ$%ا2 9 مO ا" اO *K مO ا9 آ ان آ.T # ا *ف اءH+ g 8 ا6 وه9 اراXK مOف اP/* A ء ذجR :*K "* واG اQ ةZ* ا او/`* " ا*; ا# # K وره3*م واO ا9H6 9 ;* إ *رات R *فKS T*: 9 O0د إ" ان ا28 3O ا5JKA ا3K 3= R ا.3O ا5JKAا 9H6 ا. ه ا اg 8S ا*; ي5JKAن ا+= ا6T و$T8 = *فK L N 3T8 ا/`K /`* ا3= ت8ورi ا ا3 هModeling or System dynamics مO ا9أو آ .ل2 ت واU`*; اK T0 3Oا
: %&& اV رI0 : System مW$ا .j اTi " # *K وN#`*K !ء0 ا9#2 J : Entity 6Rآ . إه* م ص$= مO ا3= ! 2وه : Attribute H .@/ 9`% او9% 2وه : Activity طA .مO ا9 3= ZK R K 9 # أي : State of the System or State Variables ات ا%+ م اوW$ اT ر2:K رس8 و.9 9O # مO ا3= 9:P0 واTK`%@ت و/ اN آlK اتZ* 3وه .$* 3= اتZ* اQ** مOا
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15
:مW$[ ا% اتZ* ا56 هN^ $2 d ا3 # YB8 $ آ ا$Hث رK 3*ات اZ* مO اYf*8 ا6 وه$* م وO اود اG م انO ا9H6 # اH T = ا.مO ا9 3 # YBK .مOا ا6 ه9 هاف وراء درا0 ا9= *د8 ا6Tم وO اYfK * رجK تJ* ا3 # ت: /*K 3* اNا2 اQ ^ل ا3`= *ر إذ ان#0 ] ذ6 اRJ= R : ا3 # YfK إذا آن ض/م وO ا9 ءGH 3T= .مO ا9:Pط اP 9Lة ا6* هK ت و:ل ا2% ووQ ت اH) 9L# هك :Endogenous Activities %2\ا8 اSAا .مO اN دا9:P0 اlKو :Exogenous Activities %! ارSAا 9H ر9:Pf Yf*8S ي6م اO ا.مOا ا6 ه3 # YBK 3*م واO ا9 3= 9:P0 اlKو .ح2*` مO $f l%28 ي6 وا9H ا)ر9:P0 Yf*8 ي6م اO اc/ Z مO 3 8 :Deterministic Activities دة8 # اSAا .TK مK N/P TJ@* 8K / 8 3* ا3وه : Stochastic Activities %R اAI اSAا 3 * إQ8ز2* l%2K ت *دة/ إTJ@* ن2/K و3@ا2P# N/P هYfK Z*8 3* ا3وه 9& ل:# إG آ ان ا3 * إQ8ز2* l%28 Q J* 9& $LZ* K ي6 اXL2= ^ ا .3@ا2P# N/P Z*8 :Continuous Systems ]2# ا#Wا ي2H ر3= @ةD 9 آ5JK و * = ^ إN* N/P مO ا9 T= Z*K 3* ا3وه .5JKA اg* Smooth # N/P ثK 3bر ا: ا/K XK :Discrete Systems 2$# ا#Wا N/P ث8 Q ا3= 9#i = ^ إآ لG اQ Q:;* N/P مO ا9 T= Z*K 3*وا . اQ:;* N/P ث8 Q@i ا3 # R D ل2% و،Q:;* :System Modeling مW$"! ا# اءHض إZ مOا ا6 هl%2 Model ذج2 3 ن او2/ انRJ8 مO 9 را 3* ]م !ة وذO ا3 # T@اH إ/ 8S تU وإ=*ا9 أ3 # 9 H ربJ*ا $%ا2) $م و=;اO اZK 3دي اB8 $ # 3= كKث إر8 و3 %0م اOب ا:i8S 16
ا 9 %0آ ان درا 9ا 2ذج SاOم # R8JK / Kة 2ارات 2ل اOم وذ] #+دة ا 2ذج ا 3ا 9ا # 9 %0إHاء آ2 Nار c/اOم ا 3 %0ا6ي إذا ZK / 8Sإ#د 9Kة ا ى * 9ا ^ = 9 %0را O 9م إ*Lدي * Zت اض واB8 L R :دي ا .T /# / 8S a@* 3آ أن ا 2ذج / 8ان 8رس = 3أز 9إ=*ا9U = ^ / 8إHاء آة Oم )* +ام ا 2ذج و==K j 9ت اOم `*ات #ة ا! Tاو = 3د .9 L @Lوآ 8D # / 8 ]6ا 2ذج درا 9اOم NLإ $@Pوو2Hد5 ا 8 ^ = %ء Qو# 8ة رات ء = * 8اي ر ا=2/ Niن 2ذج N/ ر وآK 3ف ا XK Qه 56ا)رات. أ اع ا#$ذج: ) %R _%Pد ( :Physical Models وه 3ا*2 3K 3اد N^ 9ء 2ذج @:ة = 9 3ا* وذ] * Aر هT / XKوف .9 ر ) %2%2 0 ) %او :Mathematical Models ( 8 Z0 وا*)* 3م @L# Tت ر2K ) 9U8ز8ت إ* ،9دوالH ،اول ،ر 2ت ا( هك ا lK i8ذج اJة Static Modelsوا* Q T* Z*KS 3ا Gوا ذج اآ 9أو ا Dynamic Models 9/8وا* Q T* Z*K 3ا.G وآ ^ل 3 #ا ذج ا` 9@8GاJة 2ذج ء ا Jاام2 ،ذج ` 9او @Dة =3 =6ة 3 R*/ا وآ ^ل 2ذج = 3@8Gآ 3ذج & 9ا*Aاق اا 3= 3ور!9 آ 9ا 9 Tوآ^ ا ذج ا` 9@8Gاآ L N 3= 9ا`8Gء 9 /ا 2م. :8 Kأذآ أ^ j 9ا ذج ا 9U8اJة وآ ]6اآ ).9أ8ذ :ا2Oاه ا*3 l%2KدSت 9 U`Kأو =و2/K 9Lن آ(9 2ف رس 9;Kء ا ذج S 8D #ت درا 3= 9` *) 9 O0 Case Study 9 SJت ا #0ل وا*LAد وا 9وا`8Gء وا R:وا Q *Jا
17
N^ KاOم = 3ا*` /ا 3Oوآ 9اOم: Kر l8أ :9 ا#ع : Stock أي ! Q Z*8ا) Gد* ،3/8ك( G8داد و ;8و " 8ا2* i8ى Levelاو * ، State Variable 9 Zا *ع R#2* 8ا!0ء `*ة ز 9وه 56ا!0ء ا *9#2 TOKSأو O 3`*)Kإذ اiK Tف أو (N;K) R Kل =*ة ز.9 أ^ :9آ 9ا G 3= 5ان# ،د ا T 0ا* ، ^* T/ * 8 3ا 9L:ا ): 3= 9Gر# ،98د ا@/ت = ،Q *J 3آ 9ا;2د = 3ب ،3/ا)2ف ،ا ،RiZا/اه ،9ا* Aب ،أ ا %ب : Flow ه 2ل 9 Z8 Rateا *ع 8Gُ8 2T= ،أو ِ;ُ8ا *ع .ا Aب ا6ي 8G8ا *ع " 8أ Aب اا Nأو ا ر ،Sourceوا6ي ;8ا *ع " 8أ Aب ا)رج أو ا2Zر . Sink أ^ :9آ 9ا 5ا G) 986Zان )إ ب دا Nأو ر( او ا ) $ 9 2إ ب رج أو \2ر(# ،د ا T 0ا *Pاة )إ ب دا Nأو ر( أو ا ) 9#إ ب رج أو \2ر(، ا:ر8ت ا T 9=Jأ ب رج =; dوه 3ا 9L:ا 9@ T/ا *)# ،9د ا@/ت =3 S2 8GK Q *Jدة )إ ب دا Nأو ر( و N;Kأو 2 ;Kت )إ ب رج أو \2ر(، آ 9ا;2د = 3ب GK 3/داد Aد ر )إ ب دا Nأو ر( و ;Kف )إ ب رج أو \2ر( ،ا)2ف G8داد م ا2Pر 0ن )إ ب دا Nأو ر( و;8 ) 9 :إ ب رج أو \2ر( ،اG8 RiZداد *ش )إ ب دا Nأو ر( وN;8 * Nا) R:إ ب رج أو \2ر( ،ا/اه) 9إ ب دا Nأو ر( GKداد *=c و* N;Kون )إ ب رج أو \2ر( ،ا* Aب DA 8G8ع )إ ب دا Nأو ر( وA N;8ه ل )إ ب رج أو \2ر( ،ا / 8 .ان Oا" ا Aب " #ا "N=" $إذ ا$ 2;8م ` Nاو 3= N #ا 2ذج " " #أ ء" )ا *ع( ا` #ت Convertersأو ا%+#ات ا84 #ة : Auxiliary Variables 2Kى أرLم2;K ،م ت 9أو 98Hأو 9;:و *Kا *3= Controllers 9 / ا 2ذج.
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،9H ا# @اG اN ا، بA ا،TJ ا،T ا،5 ان اG 3= 9=ا2: ا:9 ^أ O8 ا و،عDAت ا# د# ،c=* ا9#2 ، ا*ش9#2 ،9G ا*ض ر ا adverb آل او ف$ا : Information Link ت2I# اS أو راConnectors تaH #ا أن انT ا،ذج2 ا3= & "ء إGH ت2 اd K أوN K 3* ا3وه 9 ;= d;= ت2 N;K 3 هN (د2; او ا9L: او ا5 اN^) 98 آ ت دN;KS ت%2 ا . & ءGJ N;*K ذج2 ء اGH د# 3 # YB8 T ا، ا" ا م5 `ع اK ى إرN;8 انG) ا3= 9=ا2: ذراع ا:9 ^أ 9 آ،بJA ا3 # ث ا;دراتAد ا# ،زTH NZP* TJ ا9 آ،T@ !ا/ 8 3* اT 0ا . ا،ل =*ة ا*ش2D ،9G ] ر اUK ;ار،9H ا# @اG اN ا :م آW$ ا%آT وW$ ا% مW$ ا# و
Stock Source/Sink Flow
Information Link
AuxiliaryVariable
19
:ا ا Differential Equations %2)د`ت اI#ا Z* 9*;ت اP ن وده2/*K 98H تS د# رة# 3 ه9 U`*ت اSا د x = f ( t ) Z* 9 U`K تS د3*ن ا+= x = f ( t ) آن2 ^ = NU`* 9 L يH dx + tx + 3 = 0 dt d 2x dx + 5 − 14 x − 10 = 0 2 dt dt 3 2 d x d x dx − − +x+2=0 dt 3 dt 2 dt
"ق إ:* ف2 وLinearity T*: وDegree T*H ر9 U`*ت اS ا دG *Kو قD ة9 U`*ت اS ا دNK و.9:) وا9^و" وا0ت اHت ارSا د .ا ا ;ر6ى ه2* R 8 T *ض
: %S ا3ر! او8 ا6 %2)د`ت اI# ا:`او N/P " ا# 3وه f (t, x)
dx + h (t, x ) = 0 dt
. x وt 3= 9: دوالh ( t , x ) وf ( t , x ) V :N` 9 L اتZ* ي2K تS دN N/P اTU و/ 8 h ( t , x ) وf ( t , x ) N آXإذا آ f (t, x ) = p (t ) q ( x )
h (t, x ) = r (t ) s ( x )
2 ه9 U`* ا9 ا دN/! ن2/8 أي p (t ) q ( x )
dx + r (t ) s ( x ) = 0 dt
لJ ا3= t L Q J p ( t ) ≠ 0 " و: ل اJ ا3= x L Q J s ( x ) ≠ 0 " !ط ان#و 3*ات آZ* اN= Q:* s ( x ) p ( t ) ≠ 0 " # 9 ا د3=D 9 ; ،": ا q ( x) r (t ) dx + dt = 0 s ( x) p (t )
! اN/* N " ا# Nو 20
q ( x)
r (t )
∫ s ( x ) dx + ∫ p ( t ) dt = C .9و0وط اP " ا# *8 إ *ريX Y C V : ل 9* ا9 U`* ا9 ا دN dx 1 − x − =0 dt 1 + t
: ا " # N= x ≠ 1 " !ط أن# 1 − x " # 9 ا د3=D ; 1 dx 1 − = 0, x ≠ 1 1 − x dt 1 + t
J اودRKK وdt NU`* ا3= =:ب اi dx dt = , x ≠1 1− x 1+ t
فD N/ N/* ا6 أ/ 8 وj اTi # 92` X%ن اbات اZ* ا dx
dt
∫ 1 − x + ℓn C = ∫ 1 + t ,
x ≠1
أو − ℓn 1 − x + ℓ n C = ℓn 1 + t , x ≠ 1
أي
(1 − x )(1 + t ) = C , x ≠ 1 t N;* اZ* ا9S x Q *اZ* Nو x = 1±
C , x ≠1 1+ t
:\b ل 9* ا9 U`* ا9 ا دN t
dy − ( x − t ) = 0, y ≠ 0 dt
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9;ل أن اا8 ) "و0 ا9H ار9 J* دوال3 هf ( t , x ) = t , h ( t , x ) = x − t اوال 9;; ا56 ( وهg ( kx, ky ) = k n g ( x, y ) 9L اX;; إذاn 9H ار9 J* g ( x, y ) :3* آN ا3= # K J تU`K N/! " # 9 ا دQU ووx = st j82* t ( s − 1) dt − t ( s dt + t ds ) = 0, t ≠ 0
J N/* واt Z* 9 اودRK*
∫ ds = − ∫
dt + C, t ≠ 0 t
أي s = C − ℓn t , t ≠ 0
x Z* ا9S أو x = t ( C − ℓn t ) , t ≠ 0
:"و0 ا9H ار9:) ا9ا د امN/P اTو dx + P (t ) x = Q (t ) dt
9L ":8 Nا − P ( t ) dt − P ( t ) dt ∫ P(t ) dt +e ∫ x = Ce ∫ ∫ Q ( t )e dt
واxCF $ G8 وComplementary Function 9 / ا9 " اا8 ول0 ا اV N/P " ا# ن2/8 N أي اxPS $ G8 وParticular Solution ا)صN ا3^ا . x = xCF + xPS : ل 9 U`* ا9 ا دN dx + tx = t dt
22
N واxCF = Ce − ن2/8 وxPS = e −
1 t2 2
1 t2 2
− P ( t ) dt T أJ= P ( t ) = t 6 f ] وذxCF = Ce ∫ 9 / ا9 ااH2
∫ te
1t2 2
− P ( t ) dt ∫ P( t )dt dt = 1 J= Q ( t ) = t V xPS = e ∫ ∫ Q ( t )e dt ا)ص
General Solution امNا x = Ce
− 12 t 2
+1
*ري *ج إ" !وطA اX ^د اJ8A . C إ *ري واX Y ي28 امN ان اS 0 = C + 1 ⇒ C = −1 J j82*= t = 0 # x = 0 /* Initial Values (IV) 9أو
. x = 1 − e−
1 t2 2
9و0وط اP اXK امNن ا2/8و : ل 9 U`* ا9 ام دN اHأو
d 2 x 1 dx − = te 2t dt 2 t dt
J y =
dx QU2 / و9^ ا9Hو ارK 9ا د dt
dy 1 − y = te 2t dt t
.R: 8 * آT *ك8 . y 3= "و0 ا9H ار3وه
Higher Order Differential :%2Iر!ت ا8 ا6 %2)د`ت اI#ا Equations Linear Equations with Constant c تa I# %Sد`ت اI# ا:`أو Homogeneous Case Z# ا اCoefficients N/P " ا# 3وه dnx d n −1 x + a + ⋯ + an x = f ( t ) 1 dt n dt n −1
م# N/P وɺɺx =
d 2x dx وxɺ = 2 ه9 ا دN/! d * صGK ف *)م2 2 dt dt
N/P " ا# 9; ا9 ا دg*= x( n ) = x( n ) + a1 x ( n −1) + ⋯ + an x = f ( t )
23
dnx dt n
ن2/K \) X ا2Y a1 , a2 ,..., an V ،9* Y 9: ت9 U`K تS " دKو أيf ( t ) = 0 T= ن2/8 3*ت واS ا د56T cJ* اN/P اSف رس او2 .(9;; N/P " ا# تSا د x( n) + a1 x ( n −1) + ⋯ + an x = 0
3= j82* ، C إ *ريX ^ وeλt ≠ 0 V x = Ceλt Nب اJ تS ا د56 هN a*8 9; ا9ا د
(λ
n
+ a1λ n −1 + ⋯ + an ) eλt = 0
.Characteristic Polynomial ةG " آ^ة اود اK P ( λ ) ≡ λ n + a1λ n −1 + ⋯ + an 9ا د
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Auxiliary Equation ة# ا9 " ا دK 3* واP ( λ ) ≡ λ n + a1λ n−1 + ⋯ + an = 0 ور6J اn T P ( λ ) = 0 ن+= n 9H آ^ة ود ار3 هP ( λ ) أنVو او ازواج9;; ن2/K ور إ ان6J ا56ن ه+= 9;; a1 , a2 ,..., an X وإذا آλ1 , λ2 ,..., λn xi = Ci eλit , i = 1, 2,..., n * اN آ. Complex Conjugates 9 ا *;ر9اد ا آ#0ا
ن2/8 وCi إ *ريX ^ 9 U`* ا9 دN 2ه x = C1e λ1t + C2 eλ2t + ⋯ + Cn e λnt
تS ا د9 3= وComplementary Function
9 / ا9 اا3 8 وi8أ
.9 / ا9 اا2 ام هNن ا2/8 9 J* ا : ل ɺɺ x + 3 xɺ + 2 x = 0 9 U`* ا9 ا دN
: ا ن2/8 وλ1 = −1, λ2 = −2 3 هP ( λ ) = 0 ور6H وP ( λ ) ≡ λ 2 + 3λ + 2 ة# ا9ا د امNا x = C1e − t + C2 e −2t
.9و0وط اPدان ا8 8 ^ * إ *ر : Repeated Roots ر6Jار ا/K 9 3ر ه6Jا ا6T 9:K ل ا2 ن ا+= ا اتr د# λ = λ1 ^ ر6H ر/K إذا eλ1t , teλ1t , t 2 eλ1t ,..., t r −1eλ1t
24
: ل ɺɺɺ x + 4 ɺɺ x + 5 xɺ + 2 x = 0 9 U`* ا9 ا دN
: ا 3 هP ( λ ) = 0 ور6H وP ( λ ) = λ 3 + 4λ 2 + 5λ + 2 = ( λ + 1) ( λ + 2 ) ة# ا9ا د 2
امNن ا2/= K ر/ −1 ر6J اλ1 = −1, λ2 = −1, λ3 = −2 x = C1e − t + C2te− t + C3e −2t t = 0 # x = 1, xɺ = 0, ɺɺ x = 1 9و0وط اP 98 *رA اX ا2^ اL Hأو
NU`* ا8D # H2 امN ا xɺ = −C1e− t + C2 (1 − t ) e − t − 2C3e−2t
ɺɺ x = C1e − t − C2 ( 2 − t ) e −t + 4C3e −2 t
J 9و0 ; اj82* و C1 + C3 = 1 −C1 + C2 − 2C3 = 0 C1 − 2C2 + 4C3 = 1
ا)صN " ا# N j82* وC1 = −1, C2 = 3, C3 = 2 ل2 اT 3*وا x = ( 3t − 1) e− t + 2e −2 t
Linear Equations with Constant c تa I# %Sد`ت اI# ا:%c : Inhomogeneous Case Z# ا%1 اCoefficients 9 J* ت \ اS ام دNا x( n ) + a1 x ( n −1) + ⋯ + an x = f ( t )
Complementary Function 9 / ا9 " اا8 xCF V x = xCF + xPS N/P " ا# 2ه .Particular Solution ا)صN " ا8 xPS و . ا)صNد اJ8ن إbف *ض ا2 و،9 / ا9د ااJ8 إ9` آ9; ا`;ة ا3= 8; رأ : ا)صNد اJ8ق إD : D N ا9;8D . Dn ≡
dn d2 d 2 و D ≡ ^ = D ≡ 9L ف8 وD NU`* اN# :l8K n 2 dt dt dt
9L P ( D ) ≡ D n + a1 D n −1 + ⋯ + an Polynomial Operator آ^ة اودN# ف
25
P ( D ) f ( t ) ≡ ( D n + a1 D n −1 + ⋯ + an ) f ( t ) = f ( n ) ( t ) + a1 f ( n −1) ( t ) + ⋯ + an f ( t )
أيD 0 x ( t ) = x ( t ) وan D0 2 هan أن اS) ةn NU`* 9 L 9 داf ( t ) V .(ة2 اNU`K N# : ل D NU`* اN# N/! " # 9* ا9 U`* ا9 ا دQU d 2x dx − 3 + 2 x = te − t 2 dt dt ɺɺ x − 3 xɺ + 2 x = te− t
: ا D 2 x − 3Dx + 2 x = te − t
(D
2
− 3D + 2 ) x = te − t
P ( D ) ≡ D 2 − 3D + 2 ه
: D ) ا4 \ اص ل آ^ات# P ( D ) , Q ( D ) , R ( D ) وآنNU`* 9 L دوالf ( x ) , g ( x ) Nإذا آن آ 9*اص ا2) اT ل آ^ات اود# ن+= ود 1) P ( D ) + Q ( D ) f ( x ) = P ( D ) f ( x ) + Q ( D ) f ( x ) 2 ) P ( D ) Q ( D ) f ( x ) = P ( D ) Q ( D ) f ( x )
3) D ( aD r ) f ( x ) ≡ aD r +1 f ( x )
4 ) P ( D ) f ( x ) + g ( x ) ≡ P ( D ) f ( x ) + g ( x ) f ( x ) 5) P ( D ) + Q ( D ) ≡ Q ( D ) + P ( D ) 6 ) P ( D ) Q ( D ) R ( D ) ≡ P ( D ) Q ( D ) R ( D ) 7 ) P ( D ) Q ( D ) + R ( D ) ≡ P ( D ) Q ( D ) + P ( D ) R ( D ) 8 ) ( D − λ ) Q ( D ) ≡ DQ ( D ) − λ Q ( D )
The Factorization Theorem اI %2 اW
( λ − λ1 ) , ( λ − λ2 ) ,..., ( λ − λn ) Nا2# T λ n + a1λ n −1 + ⋯ + an `*ض أن آ^ة اود 6@# D n + a1 D n −1 + ⋯ + an ≡ ( D − λ1 )( D − λ2 )⋯ ( D − λn )
.$ Permutaion N8دK يf *ل8 ان/ 8 80ف ا: ان اS 26
: ل ɺɺ x + 5 xɺ + 6 x = cos t
(D
2
+ 5 D + 6 ) x = cos t
( D + 2 )( D + 3) x = cos t ( D + 3)( D + 2 ) x = cos t Distinct Roots 2$#"ور اZ ا: ل T ا2# N/! 3= 9^ ا9H ار9 ا9 ا" ا دO
( D − λ1 )( D − λ2 ) x = 0, λ1 ≠ λ2 g*= 9; ا9L ا3= ( D − λ2 ) x = u ض2#
( D − λ1 ) u = 0 أو du − λ1u = 0 dt
N اT وu 3= "و0 ا9H ار9 J* 9 U`K 9 د3وه u = C1eλ1t
J ( D − λ2 ) x = u # j82* و
( D − λ2 ) x = C1eλ t 1
dx − λ2 x = C1eλ1t dt
N/P " ا# وx 3= "و0 ا9H ار9 J*\ 9 U`K 9 د3وه dx + P (t ) x = Q (t ) dt
2 هT و − P ( t ) dt − P ( t ) dt ∫ P (t ) dt x = Ce ∫ +e ∫ ∫ Q ( t )e dt
(ب: 8 * آN%`**ك اK) Nن ا2/8 وQ ( t ) = C1eλ t وP ( t ) = −λ2 V 1
x=
C1 eλ1t + C2 eλ2t λ − λ ( 1 2)
x = C1′eλ1t + C2 eλ2t
27
Repeated Roots رة#"ور اZ ا: ل 9^^ ا9H ار9 J* ا9* ا د
( D − λ1 ) ( D − λ2 ) x = 0 2
g8 وu = C1eλ t T 3* ( واD − λ2 ) u = 0 a*8 ( D − λ1 ) x = u QU ووNا2 اN8* 2
2
( D − λ1 )
2
x = C1eλ2t
9; ا9 ا دgK ( D − λ1 ) x = v 9 */ و
( D − λ1 ) v = C1eλ t 2
N اT 3*وا v = C1′eλ2t + C2 e λ1t
وا ا
( D − λ1 ) x = C1′eλ t + C2eλ t 2
1
" # N Tو x = C1′′eλ2t + ( C2 x + C3 ) eλ1t
.N N%`* إآ ل اR: " ا# :9O Complex Conjugate Roots ر/# ا9آ#"ور اZ ا: ل N/P " ا# 9;; تT 3* اP ( λ ) = 0 9 دRر آ6H د2H و9 3= ا)صN اN/! ن2/8 وλ = µ − iν N/P " ا# & ر6H ف ان هك+= λ = µ + iν e µ t ( C1 cosν t + C2 sin ν t )
N/P ا$ ن2/8 ا)صNن ا+= ةm ور6J ا98دK 9 3=و e µ t Pm−1 ( t ) cosν t + Qm−1 ( t ) sin ν t
فiK و98 إ *رX ا2Y T وm − 1 9H ارt 3= آ^ات ودQm−1 ( t ) وPm−1 ( t ) V 9 اا3:K 3/ P ( λ ) = 0 9 د9;; ور6H a*K 3* ي ا0 اود إ" اود ا56ه . xCF 9 / ا : ل 9 إ" ا دO
(D
5
− 5 D 4 + 12 D 3 − 16 D 2 + 12 D − 4 ) x = 0
28
Nا2# N/! 3=و
( D − 1)( D − 1 − i ) ( D − 1 + i ) 2
2
x=0
3 ه9 / ا9 اا. 2 98دK T ν = 1 وµ = 1 9 ا *;ر9 ا آNا2ا xCF = et C1 + ( C2 + C3t ) cos t + ( C4 + C5t ) sin t
/ و9 / ا9د ااJ8A gU واN/P تS ا دN d 8 D N ا:K آ أن . ا)صNد اJ8 إ3= ^ اآgi*K ف2 D Nة ا2L *) اN/P 9 J* \ 9 د9 */ P ( D ) x = f (t )
9L ":8 ا)صNوا xPS = P −1 ( D ) f ( t )
:3*` آ8س و2/ اN اP −1 ( D ) V / xɺ =
dx =t dt
أو Dx = t
J x Z* N 1 x = D −1t = t 2 + C 2
]6 وآ. ة واةN/* اN^ K D −1 أي D2 x = t
J س2/ د اJ8+ x = D −2t = D −1 ( D −1t )
J ر/* اN/* و 1 1 x = D −2t = D −1 t 2 + C1 = t 3 + C1t + C2 2 6
. ةr ر/* اN/* ا3K D − r أي أن
29
س2/ ن2/K وD −1 3= ن آ^ة ود2/K P −1 ( D ) ن+= D 3= آ^ة ودP ( D ) أن أيP ( D ) P −1 ( D ) P ( D ) ≡ 1 xPS = P −1 ( D ) f ( t ) أيf ( t ) " # P ( D ) س2/ :K 9D 2 ا)ص هNد اJ8إذا إ
.N/* 9 L f ( t ) ن2/K " أن# Inverse Operator سI# اI ا:V I0 9L 3:8 f ( t ) 9 " دا# N 8 ي6 ( واD − λ ) س2/ اN اYfK −1
(D − λ)
−1
f ( t ) = eλt ∫ f ( t )e − λt dt
: ل 9 ا)ص دN اHأو
( D − 1)( D − 2 ) x = et ن2/8 ( وD − 2 ) ( D − 1) 2س ه2/ اNا −1
x = ( D − 2)
−1
( D − 1)
−1
−1
et
a*8 ( D − 1) et 3 # اl8* ا:* −1
( D − 1)
−1
et = tet
a*8 ( D − 2 ) tet " # l8* ا:K Y −1
xPS = − ( t + 1) et
.ب2 : ا2وه :\b ل 9 ا)ص دN اHأو
(D
2
− 2 D + 2 ) x = et
:Nا Nا2# N/! " # 9 ا دQU2
( D − 1 − i )( D − 1 + i ) x = et :3* آN اQ *و
30
( D −1+ i ) x = ( D −1− i ) = e(
1+ i )t
−1
et
t −(1+ i )t
∫e e
dt = iet
ن2/8 3* و x = ( D − 1 + i ) iet −1
= e(
1−i )t
t −(1−i )t
∫ ie e
dt = et
2 ا)ص هNأي ان ا xPS = et
.ب2 : ا2وه
31
:ا ا Difference Equations %Fد`ت اوI#ا يH Z* تLن وده =و2/*K 98H تS د# رة# 3 ه9Lت ا`وSا د Z* 9Lت =وS د3*ن ا+= xk = x ( tk ) , k = 0,1, 2,..., n آن2 ^ = 9 ` L 6 f8 xk , k = 0,1, 2,..., n xk − xk −1 − xk − 2 = 0 xk + 2 xk −1 = 5 xk − 3 = 0 xk − 3k − 7 = 0
9*اص ا2) ا$ وB 3` ) ا9زاA اN# :V I0 1) Bxk = xk −1 2 ) B 2 xk = B ( Bxk ) = B ( xk −1 ) = xk − 2 3) B m xk = xk − m 4 ) Bc = c
5 ) B ( cxk ) = cxk −1
9*اص ا2) ا$ ≡ ∇ و1 − B 8`* اN# :V I0 1) ∆xk = xk − xk −1 2 ) ∆ 2 xk = ∆ ( ∆xk ) = ∆ ( xk − xk −1 ) = xk − 2 xk −1 + xk − 2 3) ∆ m xk = ∆ m −1 ( ∆xk ) 4 ) ∆c = 0
5 ) ∆cxk = c∆xk
6 ) ∆ ( xk + yk ) = ∆xk + ∆yk
:ة84F m m m− j ∆ m xk = ∑ ( −1) xk − j j =0 j
S ≡ ∆ −1 3@Tb اQ J* اN# وF ≡ B −1 30 ا9زاA اN# ه8 & # هك
.9H اXL وT)* ف2 3*وا :3ر! او8 ا6 %S ا%Fد اوI#ا N/P " ا# 3وه a0,k xk + a1,k xk −1 = ck , k = 1, 2,..., n
32
N/P " ا# i8 اR*/Kو xk = −
a1,k a0, k
xk −1 +
ck , k = 1, 2,..., n a0,k
xk = Axk −1 + C , k = 1, 2,..., n
.9* Y وA ≠ 0 V J k = 1 QU2 ،ة: x0 " =ض ان# xk = Axk −1 + C , k = 1, 2,..., n 9 ا دN x1 = Ax0 + C k = 2 9 ;و x2 = Ax1 + C
= A ( Ax0 + C ) + C = A2 x0 + (1 + A ) C k = 3 9 ;و
x3 = Ax2 + C
= A ( A2 x0 + ( A + 1) C ) + C = A3 x0 + (1 + A + A2 ) C
م# N/P و xk = Ak x0 + C (1 + A + A2 + ⋯ + Ak −1 ) , k = 1, 2,...
انS 1+ A + A +⋯ + A 2
k −1
1 − Ak , if = 1− A k, if
A ≠1 A =1
N اg8و k 1 − Ak , if A ≠ 1 A x0 + C , xk = 1− A x0 + Ck , if A = 1
k = 0,1, 2,...
:1 ل Xإذا آ xk = 2 xk −1 + 1, k = 1, 2,...
J x0 = 5 9 او9 ;و
33
, k = 0,1, 2,...
1 − 2k xk = 5 ( 2k ) + 1 1− 2
= 6 ( 2 k ) − 1, k = 0,1, 2,...
و :#+ء kا; J 0,1, 2,...ان ا2/8 Nن ا ** . 5,11, 23, 47,... 9 TاI#د`ت او %Fا 6 %Sا8ر! او8&f 3ام : Excel Nا دSت ا`و/K 9Lار / 8 8ا;م 2 9ا Excel 9:آ*:3 أد Nا*) 3ا : 0ا0ول و*" ا^ Y Vا ا :ا^ "* Vا ى ا :(R
ا*:9J
و Tا N/Pا*:3
34
x(k) 7000 6000 5000 4000 3000 2000 1000 0 0
2
4
6
8
10
12
:2 ل Xإذا آ 2 xk − xk −1 = 4, k = 1, 2,...
J x0 = 3 9 او9 ;و xk =
1 xk −1 + 2, k = 1, 2,... 2
=(
)
1 k 2
3+ 2
1 − ( 12 )
k
1 − ( 12 )
, k = 0,1, 2,...
= 4 − ( 12 ) , k = 0,1, 2,... k
. 3,3 12 ,3 43 , 3 87 ,3 1615 ,... 9 ** ن ا2/8 N ان اJ 0,1, 2,... ; اk ء:#+ و : Excel ام8&f ا :(R *" ا ى اV^ ا: ا اY V^ول و*" ا0 ا: 0 )ا3* اN أد
9J*ا
35
:3* اN/P اTو 4.5 4 3.5
x(k)
3 2.5 2 1.5 1 0.5 0 0
1
2
3
4
5
k
36
6
7
8
:3 ل Xإذا آ xk + xk −1 = 1, k = 1, 2,...
J x0 = 1 9 او9 ;و xk = − xk −1 + 1, k = 1, 2,... = ( −1) + k
=
1 − ( −1)
k
1 − ( −1)
, k = 0,1, 2,...
1 k 1 + ( −1) , k = 0,1, 2,... 2
. 1, 0,1, 0,1, 0,... 9 ** ن ا2/8 N ان اJ 0,1, 2,... ; اk ء:#+ و : Excel ام8&f ا :(R *" ا ى اV^ ا: ا اY V^ول و*" ا0 ا: 0 )ا3* اN أد
9J*ا
:N/P اTو
37
1.2 1 0.8 0.6 0.4 0.2 0 30
25
20
15
10
5
0
k
: AF$ رأ 3= 8ا 9 ^0ا ; 9YY 9ا دSت ا`و 9Lآن Nا0و" ; 9ا0و 9ا :ة ** 9ا0رLم *#ة ،و Nا^ 9 ; 9ا0و 9ا :ة ** 9ا#0اد ا*;*K 3رب إ 3اد ، 4و Nا^^ 9 ; 9ا0و 9ا :ة ** 9ر 0 Lو 1ا */رة ) أي ا; 6 6*Kب 0و .(1 :8 Kأو L Hاو 9دSت ا`و 9Lا ;K 3:K V 9ف \ ا6ي OSة ;. " ه 3ا; 9ا0و 9ا* 9 ** 3:K 3ا0رLم أو ا#0اد ا *;ر 3= 9ا ^ل 1وه6/ا. YfKا; ا0و " # 9ا ** 9ا:9JK 2ف *ض YfKا; 9ا0وK " # 9ف ا ** 9ا 9JKا^ Nل ا*:3 ل:4 Nا د 9ا`و 9Lا*9 1 1 xk −1 + , k = 1, 2,... 2 2
; ا0و. x0 = 0,1, 2 9 ا ا&: Excel S
38
= xk
ان هك 3= ;* Nآ NاSت إذ ;*Kرب ** 9ا Nإ" .1 ل:5 Nا د 9ا`و 9Lا*9 xk = 3 xk −1 − 1, k = 1, 2,...
; ا0و. x0 = 0,1, 2 9 39
: Excel S&ا ا
.N ا9 ** #*K إذ9 اي3= ;* N هكc ان
: %FوP دI# %2)0 دI k /0 *; ف2 .T 9 K 9L =و9 دN 98T آ9 U`K 9 دN دJ8 إ/ ا ; 9= 9 داx /* .9* Y و" ت0 ا9H ار9:)ت اS " ا د# 9PL ا 9 U`* ا9; ا دK 3* واa ≤ t ≤ b =*ة3= t 9;; Dx ( t ) =
d x ( t ) = Ax ( t ) + C , a ≤ t ≤ b dt
40
.ة: x0 = x ( a ) 9و0 ا9 ; و`ض أن ا98 إ *رX ا2Y C وA ≠ 0 V 9 ` ; a ≤ t ≤ b إ *ال ا`*ةS اوR :*K 9L =و9 د9 U`* ا9 ا د56 هR8;* " إa و; ا`*ةn RH2 اg اL ا6 f .T # 9L ا`و9 ا دl8K / 8 ;* ;ط اh = ( b − a ) n ل2: ذات ا98اء ا * وGH0 اn " إb t0 = a, t1 = a + h, t2 = a + 2h,..., tn = a + nh = b
. xk = x ( tk ) = x ( t0 + kh ) 9 */ ز2 اd #د انNU`* ا وف ;ر ا ∆xk h →0 h
Dx ( tk ) = lim
Dx ( t ) =
d x ( t ) = Ax ( t ) + C , a ≤ t ≤ b 9 U`* ا9وا إ *ال ا د8 ا6Tو dt
9L ا`و9 د ∆xk = Axk + C , k = 0,1, 2,..., n − 1 h
أو xk +1 = (1 + Ah ) xk + Ch, k = 0,1, 2,..., n − 1
و" ت0 ا9H ار9L =و9: 3 ة ه0 ا9 ا د.": x0 "و0ط اP اQ آT و9* Y C C k xk = (1 + Ah ) x0 + − , k = 0,1, 2,..., n − 1 A A C = 0 # :9%) ا9ا
9 U`* ا9 ا د.9 T اTK;:* ] وذ9 اهT 9 3وه Dx ( t ) =
d x ( t ) = Ax ( t ) , a ≤ t ≤ b dt
. x 9 ااQ R *8 t ـ9 x 9 ا3O اZ*أو ل ا 3 هT 9 * ا98;* ا9L ا`و9ا د xk +1 = (1 + Ah ) xk , k = 0,1, 2,..., n − 1
T 3*وا xk = x0 (1 + Ah )
k
41
2 ه9 U`* ا9 ا دN انS x ( t ) = x0 e
A( t − t0 )
ث8 A < 0 # وx 9 اExponential Growth 3 ا2 a*8 A > 0 9 3= . Exponential Decay 3 ش ا/إ
42
Linear Difference Equations c تa I# %S ا%Fد`ت اوI#ا : with Constant Coefficients N/P " ا# X إذا آ9: T اk = 0,1, 2,... ; " ا# 9= 9L =و9;ل ان د8 f 0 ( k ) xk + n + f1 ( k ) xk + n −1 + ⋯ + f n −1 ( k ) xk +1 + f n ( k ) xk = g ( k )
ا تX إذا آ. k = 0,1, 2,... " # 9= k دوال ـf 0 , f1 ,..., f n−1 , f n , g V 9* Y ن2/K م انG 8S g 9 )ا9* Y ت9: ن2/K 9ن ا د+= X ا2Y T آf 0 , f1 ,..., f n . n 9Hن ار2/K 9ن ا د+= f n ≠ 0 وf 0 ≠ 0 N آX إذا آ.(9: او :9 ^أ 9* Y ت9: 9Lت =وS د3*ا 2 xk +1 − xk = 6 3 xk + 2 + 2 xk +1 + xk = 3k
xk +3 − xk = k
9 ا د3=D ;K Ni=0 = اf 0 ≠ 1 ن2/K # .RK* " ا# 3 و2 و1 تH درT 9 ا دN/! g8ا و2 8 وxk +n N NJ 3/ T # xk + n + a1 xk + n −1 + ⋯ + an −1 xk +1 + an xk = r ( k )
. ai = fi f 0 , i = 1,..., n; r ( k ) = g ( k ) f 0 V ف `*ض ان2 .9* Y ت9:) ا9Lت ا`وS د0 اN/Pف *)م ا2 ; " ا# 9= 98 إ *ر9 داr ( k ) , k = 0,1, 2,... 9 وااan ≠ 0 i8 واX ا2Y a1 , a2 ,..., an 3 و2 و1 تH ار9* Y ت9: 9Lت =وS د3 ه3*= ا6 ه3 # و.ة: ا 3ا2* " ا# xk +1 + a1 xk = r ( k )
xk + 2 + a1 xk +1 + a2 xk = r ( k )
xk +3 + a1 xk + 2 + a2 xk +1 + a3 xk = r ( k )
9ا د xk + n + a1 xk + n −1 + ⋯ + an −1 xk +1 + an xk = 0
9:) ا9L ا`و9 د9 * وا9* Y ت9 J* ا9:) ا9L ا`و9 " ا دK . xk + n + a1 xk + n −1 + ⋯ + an −1 xk +1 + an xk = r ( k ) 9* Y ت9 J* \ا
43
: W ن+=
xk + n + a1 xk + n −1 + ⋯ + an −1 xk +1 + an xk = 0
9 د
x( 2)
و
x(1)
إذا آن
. C2 وC1 98 إ *رX ا2^ T N i8 ا2 هC1 x(1) + C2 x( 2) :98O N xPS وآنxk + n + a1 xk + n−1 + ⋯ + an −1 xk +1 + an xk = 0 9 J* ا9 دN xCF إذا آن N xCF + xPS ن+= xk + n + a1 xk + n −1 + ⋯ + an −1 xk +1 + an xk = r ( k ) 9 J* \ ا9 د . xk + n + a1 xk + n −1 + ⋯ + an −1 xk +1 + an xk = r ( k ) 9 J* \ ا9 د General Solution of the Homogeneous Z#د اI#2 مIا ا : Equation xk + n + a1 xk + n −1 + ⋯ + an −1 xk +1 + an xk = 0 9 J* ا9 ام دNد اJ8ف *ض إ2
:3* آ9 ^f 9 J* ا9 ام دN اH( او1 xk − xk − 2 = 0
Auxiliary Equation ة# ا9 " ا د# N= xk = λ k , λ ≠ 0 Nب اJ λ k − λ k −1 = 0 λ k −1 ( λ − 1) = 0
امNن ا2/8 وλ1 = λ2 = 1 8 * و8رGH T ة# ا9ن ا د+= λ ≠ 0 و ان 2 ه9 J* ا9 د xk = ( C1 + C2 k )1k = C1 + C2 k
.9و0وط اPه ا8K *8 C2 وC1 8 * إ *رY Tو 9 J* ا9 ام دN اH( او2 xk − 3 xk −1 + 2 xk − 2 = 0
Auxiliary Equation ة# ا9 " ا د# N= xk = λ k , λ ≠ 0 Nب اJ λ k − 3λ k −1 + 2λ k − 2 = 0
λ k − 2 ( λ 2 − 3λ + 2 ) = 0
44
λ1 = 1, λ2 = 2 ورGJ اT ( وλ 2 − 3λ + 2 ) = 0 gK ة# ا9ن ا د+= λ ≠ 0 و ان
2 ه9 J* ا9 ام دNن ا2/8و xk = C1λ1k + C2λ2k = C1 + C2 2k
.9و0وط اPه ا8K *8 C2 وC1 8 * إ *رY Tو 9 J* ا9 ام دN اH( او3 xk + 3 xk −1 + xk − 2 = 0
Auxiliary Equation ة# ا9 " ا د# N= xk = λ k , λ ≠ 0 Nب اJ λ k + 3λ k −1 + λ k − 2 = 0
λ k − 2 ( λ 2 + 3λ + 1) = 0
ورGJ اTو
(λ
2
+ 3λ + 1) = 0
gK ة# ا9ن ا د+=
2 ه9 J* ا9 ام دNن ا2/8 وλ1 =
λ≠0
و ان
−3 + 5 −3 − 5 , λ2 = 2 2
xk = C1λ1k + C2λ2k k
−3 + 5 −3 − 5 = C1 + C2 2 2
k
.9و0وط اPه ا8K *8 C2 وC1 8 * إ *رY Tو 9 J* ا9 ام دN اH( او4 xk + xk −1 + 14 xk − 2 = 0
Auxiliary Equation ة# ا9 " ا د# N= xk = λ k , λ ≠ 0 Nب اJ λ k + λ k −1 + 14 λ k − 2 = 0
λ k − 2 ( λ 2 + λ + 14 ) = 0
λ1 = λ2 = − 12 ورGJ اT ( وλ 2 + λ + 14 ) = 0 gK ة# ا9ن ا د+= λ ≠ 0 و ان
2 ه9 J* ا9 ام دNن ا2/8و xk = C1λ1k + C2λ2k = ( C1 + C2 k ) ( − 12 )
k
.9و0وط اPه ا8K *8 C2 وC1 8 * إ *رY Tو 9 J* ا9 ام دN اH( او5 45
xk + xk − 2 = 0
Auxiliary Equation ة# ا9 " ا د# N= xk = λ k , λ ≠ 0 Nب اJ λ k + λ k −2 = 0
λ k − 2 ( λ 2 + 1) = 0
ورGJا
Tو
(λ
2
+ 1) = 0
gK
ة# ا
9ا د
ن+=
λ≠0
ان
و
2 ه9 J* ا9 ام دNن ا2/8 وλ1 = i = −1, λ2 = −i = − −1 xk = C1λ1k + C2λ2k
πk = C1 cos + C2 2
.9و0وط اPه ا8K *8 C2 وC1 8 * إ *رY Tو Polar Form 3:L N/! " إ9اد ا آ#0 اN82K :9O 3* اj82* r ( cos θ + i sin θ ) ":; اN/P إ" اa + bi Rل اد ا آ28 r = a 2 + b2 a b cosθ = , sin θ = , −π < θ ≤ π 2 2 2 a +b a + b2
R*/K λ1 = i = 0 + 1i
= r ( cos θ + i sin θ )
r = 02 + 12 = 1 cosθ =
0 1 π = 0, sin θ = = 1 ⇒ θ = 1 1 2
λ2 N^ و
2 ه9 J* ا9 ام دNن ا2/8و xk = Ar k cos ( kθ + C )
N/P " ا# T J xk = Ar k cos ( kθ + C ) 9 ا د3= j82* ا8 * آR: *ك8 . xk = C1 cos
πk
+ C2 2
. اN xk = sin
kπ ه أنx0 = 0, x1 = 1 9و0 ; ا:8 K 2
46
إذا آن:9J* xk + a1 xk −1 + a2 xk − 2 = 0
ة# ا9ري ا د6H λ2 وλ1 وآنa2 ≠ 0 وX ا2Y a2 وa1 V λ 2 + a1λ + a2 = 0
تL ":8 xk + a1 xk −1 + a2 xk −2 = 0 9L ا`و9 ام دNن ا+= xk = C1λ1k + C2 λ2k
;; و)* ` وλ2 وλ1 وذ] إذا آن xk = ( C1 + C2 ) λ1k
و8 ;; و* وλ2 وλ1 وذ] إذا آن xk = Ar k cos ( kθ + C )
N/P " ا# آ *;ر أيλ2 وλ1 وذ] إذا آن λ1 = a + bi = r ( cosθ + i sin θ )
λ2 = a − bi = r ( cos θ − i sin θ )
: Particular Solution of the Complete Equation 2 د اI#2 ا اص 9 / ا9 ا)ص دNد اJ8ف *ض إ2 xk + n + a1 xk + n −1 + ⋯ + an −1 xk +1 + an xk = r ( k )
:3* آ9 ^f 9( * ا د1 xk + 2 − 3 xk +1 + 2 xk = 3k
Method of Undetermined Coefficients ا ت \ ا دة9;8: T ف2 :3*آ 9 L دJ8 ول إ. د8 A X ^ اN اV xk* = A3k N/P " ا# N دJ8ول ا ض2 ، xk* T= ن2/8 (تH )إن وA ـ
47
x*k +2 − 3 xk* +1 + 2 xk* = A3k + 2 − 3 A3k +1 + 2 A3k = A3k ( 9 − 9 + 2 ) = 2 A3k ∴A=
1 2 1 2
9 / ا9 اا2 وه9 J* ا9 ام دN اH ان و. ا)صN اxk* = 3k ن2/8و 2 ه9 / ا9 ام دNن ا2/8 و، 9 / ا9 د xk = C1 + C2 2k + 12 3k
.9و0وط اPه ا8K *8 C2 وC1 8 * إ *رY Tو K ^آ0 ا9( * ا د2 xk + 2 − 3xk +1 + 2 xk = a k
ا ^ل ا3= آ.( a = 3 6 f ]ا وذ6 ه9% 9 ) ا ^ل اX Y a V ض2 وxk* = Aa k 38J* اN ا6 f x*k +2 − 3 xk* +1 + 2 xk* = Aa k + 2 − 3 Aa k +1 + 2 Aa k = Aa k ( a 2 − 3a + 2 )
= A3k ( a − 1)( a − 2 ) ∴A =
1
( a − 1)( a − 2 )
, a ≠ 1, 2
3ة ه# ا9ور ا د6H انS . ا)صN اxk* =
1
( a − 1)( a − 2 )
a k , a ≠ 1, 2 ن2/8و
/* ^ أيa = 2 أوa = 1 Xآ2 ذا. 2 و1 xk + 2 − 3xk +1 + 2 xk = 1
ض2 وxk* = Ak ن2/8 a = 1 QU2 وxk* = Aka k Nب اJ x*k +2 − 3xk* +1 + 2 xk* = A ( k + 2 ) − 3 A ( k + 1) + 2 Ak = −A ∴ A = −1
. xk* = −k ا)صNن ا2/8و صxk* = Ak 2k ه انxk + 2 − 3xk +1 + 2 xk = a k 9 ا د3= a = 2 9 3= :8 K امN اT xk + 2 − 3xk +1 + 2 xk = 2k 9L ا`و9 وان ا دA = 1 2 #
48
xk = C1 + C2 2 k + k 2 k −1
9 ا)ص دN اH( او3 8 xk + 2 − 6 xk +1 + xk = 5sin
kπ 2
Nب اJ xk* = A sin
kπ kπ + C cos 2 2
38J* اN ض2 ،ه8K * 9* Y تC وA V ( k + 2 ) π + C cos ( k + 2 ) π − 6 A sin ( k + 1) π + C cos ( k + 1) π 8 xk*+ 2 − 6 xk*+1 + xk* = 8 A sin 2 2 2 2 kπ kπ + A sin + C cos 2 2
9^ ^ ت اL *)ام ا+ و sin
( k + 2)π
cos
( k + 2)π
sin
( k + 1) π
cos
( k + 1) π
2 2 2 2
kπ kπ = sin + π = − sin 2 2 kπ kπ = cos + π = − cos 2 2 kπ kπ π = sin + = cos 2 2 2 kπ kπ π = cos + = − sin 2 2 2
a*8 8 xk*+ 2 − 6 xk*+1 + xk* = ( −7 A + 6C ) sin
kπ kπ + ( −6 A − 7C ) cos 2 2
انJ 8 xk + 2 − 6 xk +1 + xk = 5sin
kπ 9 %0 ا9 ا دQ 56 ه9 ا د9; : 2
−7 A + 6C = 5 −6 A − 7C = 0 7 6 A=− , C = 17 17
ا)صNن ا2/8و xk* =
1 kπ kπ + 6 cos −7 sin 17 2 2
49
: AF$ 9;8:= ، r ( k ) 9 ااN/! $D* إ/ 8 ا)صN اN/! انa** 9; ا9 ^0 ا بU N% ع او2 J r ( k ) 9 ااXح إذا آJ T إ *)ا/ 8 ا ت \ ا دة د# n وX ا2Y ايb وa V a k , sin bk , cos bk , k n N/P وال ا9:) اRاآK .RH2 g% ا ;*ح38J* اN واr ( k ) 9 اN/! 3:8 3*ول اJا r (k )
xk*
ak sin b k / co s b k
Aa k A sin b k + C co s b k A 0 + A1 k + A 2 k 2 + ⋯ + A n k n
kn k nak
a k ( A0 + A1 k + A 2 k 2 + ⋯ + A n k n )
a k sin b k / a k co s b k
a k ( A sin b k + C co s b k )
: ل 9 ام دN اHأو xk + 2 − 4 xk +1 + 4 xk = 3k + 2k
9ن اا2/K 9 # وλ1 = λ2 = 2 رة/ ا8ر6J اT λ 2 − 4λ + 4 = 0 ة# ا9ا د 9 ا د80ف ا: ا" اO اa** 8 ا)صN ا. xCF = ( C1 + C2 k ) 2k 9%)ا A0 , A1 , A L ودxk* = A0 + A1k + Ak 2 2 k N/P " ا# $ اJ= ولJ 9* Aوا
j82* xk* + 2 − 4 xk* +1 + 4 xk* = ( A0 − 2 A1 ) + A1k + 8 A2k
J xk + 2 − 4 xk +1 + 4 xk = 3k + 2k Q 9 ;ر A0 − 2 A1 = 0,
A1 = 3, 8 A = 1
امNن ا2/8 وA0 = 6, A1 = 3, A = 18 أي xk = ( C1 + C2 k ) 2k + 6 + 3k + 18 k 2 2k
50
اف ا 2 2 R;$ل 8&fام : Excel 2ف رس Kف ا 2ل ا N 9JKد= 9و ; 9Lأو: 9ة ورس ا*ف ا ** 3@Tت اد " # 98أ^ Excel )* 9آ*:3 (1أدرس Kف ا د 9ا`و 9Lا*9 xk + 2 − 3 xk +1 + 2 xk = 0 ; أو x0 = 0, x1 = 1 9و x0 = −1, x1 = −2و x0 = x1 = 2 ا ; :Nا0وx0 = 0, x1 = 1 9
ان ا #*8 Nإ" ∞ . و ; ا0وx0 = −1, x1 = −2 9
ان ا #*8 Nإ" ∞ . − و ; ا0وx0 = x1 = 2 9
51
ان ا 9* Y 9 L Nوه6ا Q H " # :8ا; ا0و. x0 = x1 9 (2أدرس Kف ا د 9ا`و 9Lا*9 2 xk + 2 + 3 xk +1 − 2 xk = 0
3 1 ; أو x0 = 1, x1 = 9و x0 = 1, x1 = −2و 2 2 1 ا ; :Nا0و9 2
= x0 = 1, x1
ان ا;*8 Nرب إ" .0 و ; ا0وx0 = 1, x1 = −2 9
52
. x0 = 2, x1 = −
ان ا* #*8 Nددا 9H2 Lو .9 3 و ; ا0و9 2
x0 = 2, x1 = −
` cا*ف ا 3@Tا . (3أدرس Kف ا د 9ا`و 9Lا*9 xk + 2 + xk = 0
; أو. x0 = 1, x1 = 0 9
53
.*دد ودا8 Nا أنS 1, if k = 0, 4,8,12,... kπ cos = −1, if k = 2, 6,10,14,... 2 0, if k = 1,3,5, 7,...
9* ا9L ا`و9ف ا دK ( أدرس4 4 xk + 2 + xk = 0
. x0 = 1, x1 = 0 9 ; أو
.0 " إH )*8 Nا
54
1 kπ 9 و; أوxk = A cos + C ; وآن9 ا د56T امN اH; و 2 2 k
kπ 1 kπ انS . xk = cos امN اg8 x0 = 1, x1 = 0 2 2 2 k
ا*دد وR 8 cos
.)* اR 8 (1 2 )
k
: AF$ أ إذا.H )*K ء اوd L*K او8اG*K او9* Y NOK L 9 *;رN ا9 ** X( إذا آ1 أو− ∞ " * إN/P L*K أو98TS ∞ " * إN/P 8اG*K L T+= ة#* Xآ .3@TS اJ` او3@T * دداK يK .9و0 وا; ا9L ا`و9 ا دN " آ# N ا9 ** فK *8 (2 .N 3@T ا*ف ا8K 3= 9 T اNا2* اK ة# ا9ور ا د6H (3 : هWTa N واxCF 9 / ا9ن اا2/*8 9L وا`و9 U`*ت اS دxGS امN; ذآ أن ا وxCF Nف آK " # *8 N 3@T ا*ف ا. xGS = xCF + xPS أيxPS ا)ص N i8ة وL `*ةTransient 3;* او إ3 xCF فK ن2/8 \ وxPS . xPS Yf* ا3= * 8 و9 82D *!" =*ة8و
55
:ا اا :Matrix Algebra and Calculus تP # ب اT و9! ف2 ا6T و، و ء ا ذج3U8 اN * ا3= اH 9 T دوات ا0=ت ا2` * اK .ا ا ;ر6)م ه8 =ت2` و ب اH N ;ق إ" ا:* ص2) " ا# و. ة#ر وأ2: N/! 3= !ء0 اRآK T " ا# 9=2` ف اK :3* آA 9=2` ف اK aij ∈ ℝ 9;;اد ا# a11 ⋯ a1n A = aij = ⋮ ⋮ ⋮ am1 ⋯ amn
:تP # ا324 %&&ت ا%2#Iا =ت2` Q H -1 1- C = A + B ⇒ cij = aij + bij ب دi ا-2 2- C = α A ⇒ cij = α aij 9=2` بi ا-3 n
3- C = AB ⇒ cij = ∑ aik bkj k =1
9=2` (ل2;) س2/ -4 4- C = AT ⇒ cij = a ji 9 ا9=2` ا-5 5- A = aij , i = 1,..., n; j = 1,..., n ة2 ا9=2` -6 6- I n = ek( n ) , ek( n) = ( 0,..., 0,1( k th position ) , 0,..., 0 )
T
9=2` ب2 ; -7 7- AB = I ⇒ B = A−1 Singular !ذةA ن2/KS " !ط ا# ب2 ; س ا2/ -8
56
8- ( A−1 ) = ( AT ) = A−T −1
T
9=2` دة ا-9 det ( A ) = ∑ ( −1) a1 j det ( A1 j ) n
j +1
j =1
j د2 ول وا0 ا: ف ا6 T # N ( n − 1) × ( n − 1) 9=2` A1 j V . A 9=2` ا :دات8 #2 ة8%# ا اص اYI
( i ) det ( AB ) = det ( A ) det ( B ) ( ii ) det ( AT ) = det ( A ) ( iii ) det ( cA ) = c n det ( A) ( iv ) det ( A ) ≠ 0 ⇔ A is nonsingular . y = [ yi ] , i = 1, 2,..., m أيm × 1 9=2` # رة# 2د ه2 ا$J* :l8K 38 y = Ax ن+= $J* x و9=2` A إذا آن n
yi = ∑ aij x j , i = 1, 2,..., m j =1
":8 y = y j , j = 1,..., n وx = [ xi ] , i = 1,..., m TJ* 3Hب ا)رi ا:l8K 9L x1 y1 ⋯ x1 yn xy = ⋮ ⋮ ⋮ xm y1 ⋯ xm yn T
":8 y = y j , j = 1,..., n وx = x j , j = 1,..., n TJ* 3 ب ااi ا:l8K 9L n
xT y = ∑ xi yi = y T x i =1
. A 3= k د2 اck V A = [ c1 ,..., cn ] 2 هA 9=2` دي2 اN^ * ا:l8K r1T . A 3= k : اrkT V A = ⋮ 2 هA 9=2` ي: اN^ * ا:l8K rmT
. aij = a ji , ∀i, j = 1,..., n , i ≠ j *ة إذا آنA 9 ا9=2` ;ل ان ا8 :l8K 57
;K *ة إذاA ا *ة9 ا9=2` ;ل ان ا8 :l8K AT A = AAT = I
أو A−1 = AT
أي ckT c j = 0, , k = 1,..., n ; j = 1,..., n, k ≠ j
أو rk rjT = 0, k = 1,..., n ; j = 1,..., n, k ≠ j
9=2` ا
ن+=
1 ≤ j1 < ⋯ < js ≤ n
9@GH 9=2` 3 هbpq = ai
p jq
و
1 ≤ i1 < ⋯ < ir ≤ m
Xآ
إذا
:l8K
9L 9= اB = bpq , p = 1,..., r ; q = 1,..., s
9@GH 9=2` B ن+= p = 1,..., r ; i p = j p وr = s X وإذا آ، A Submatrix ن+= p = 1,..., r ; i p = j p = p ] " ذ# 9=U وإذا آن إPrincipal Submatrix 9 @ر .Leading Principal Submatrix 9;* 9 @ ر9@GH 9=2` B T= ن2/8 9 9=2` 3 وهDiagonal Matrix 98:; ا9=2` ا:l8K أيaij = 0, ∀i ≠ j a11 0 ⋯ ⋮ ⋮ ⋮ A = 0 ⋮ aii ⋮ ⋮ ⋮ 0 0 ⋯
0 0 ⋮ ⋮ ⋮ 0 ⋮ ⋮ 0 ann
. A = diag ( aii ) , i = 1,..., n T G8و
{λ1 ,..., λn } ; ا9#2 J 3ه
A 9 9=2` Eigenvalues _ة%## ا7%/ ا:V I0
9; ا دK 3*وا det ( A − λi I ) = 0, ∀i = 1,..., n
تTJ* ا9#2 J 3 هA 9 9=2` Eigenvectors _ة%##;ت اZ# ا:V I0 ;K 3*{ واλ1 ,..., λn } ةG ; ا9 *ا Avi = λi vi , ∀i = 1,..., n
أو 58
( A − λi I ) vi = 0,
∀i = 1,..., n
.G وج اG ن ا8 T Q K G 9J*ة وG ا9 ; ا:l8K 3* ا9=2` ا3 هA 9 ا9=2` 9 * اModal Matrix ا;س9=2` :l8K ;K 3* واA 9=2` ةG تTJ* TK #ن ا2/*K A = M ΛM −1
. A 9=2` ةG ه ا; ا%# 92/ ا98:; ا9=2` ا3 هΛ = diag ( λi ) V . AM = M Λ أنS 9=2` سL 9=2`ة وG تTJ*ة وG ا; اH أو:^ل −2 −3 4 A = 0 1 0 −2 −2 4
:Nا −2 − λ −3 det ( A − λ I ) = 0 1− λ −2 −2
(1 − λ )
−2 − λ
4
−2
4−λ
4 0 =0 4−λ
= (1 − λ ) ( λ 2 − 2λ ) = 0
∴ λ = [1 2 0] T
1 0 0 Λ = 0 2 0 0 0 0
3* ا;س آ9=2`ة وG ت اTJ* اH2 −3 4 vi1 −2 − λi ( A − λi I ) vi = 0 1 − λi 0 vi 2 = 0 −2 −2 4 − λi vi 3 −3 −3 4 v11 1 λ1 = 1 ⇒ 0 0 0 v12 = 0 ⇒ v1 = −1 −2 −2 3 v13 0
59
−4 λ2 = 2 ⇒ 0 −2 −2 λ3 = 0 ⇒ 0 −2
−3 4 v21 1 −1 0 v22 = 0 ⇒ v1 = 0 1 −2 2 v23 −3 4 v31 2 1 0 v32 = 0 ⇒ v1 = 0 1 −2 4 v33
1 1 2 M = −1 0 0 0 1 1
. A = M ΛM −1 وAM = M Λ ; أنK :8 K :3* آR 8 R \ g% د# k V Ak ة/ I P P ر:V I0 A k = M Λ k M −1
. Λ k = diag ( λik ) و 2 هA 9 ا9=2` اTrace Y أ:l8K n
tr ( A ) = ∑ aii i =1
: أن9 ه/ 8 n
tr ( A ) = ∑ λi i =1
و n
det ( A ) = ∏ λi i =1
:تO N/ ( A − λi I ) vi = 0 ;8 vi و$J* H28S ا6T !ذة و9=2` ( A − λ I ) أنS (1 انgU28 ا6 وهα ≠ 0 V α vi ]6/= λi ةG ا9 ; Q K G $J* vi ذا آن+= λi .5 ;ارc و$هJK إ3= ن2/K G ا$J* ا9 أه Right and Left Eigenvectors 98 وا ر9 ة اG ت اTJ* ا#2 ( هك2 9L 3 اG ا$J* = ا# أن، {λ1 ,..., λn } ةG ; ا9 *ا ف8 ا ريG ا$J* ا، AM = M Λ T # ي6 واAvi = λi vi , ∀i = 1,..., n N ي6 واuiT ( A − λi I ) = 0, ∀i = 1,..., n أوuiT Avi = uiT λi , ∀i = 1,..., n 9L ا 60
ن2/*K M −1 ر2: ن+= vi ن2/*K M ة# أXذا آ+= M −1 A = ΛM −1 9L ا$ # .ة وةG أن ا; اS . uiT I P ة/ e P ر:V I0 e
At
( At ) = I + At +
2
2!
( At ) + 3!
( )
3
+⋯
= M diag eλi t M −1
: (SVD) Singular Value Decomposition ذةA ا#%/ اm%0 K* *=2` H28 $+= A ∈ ℝ m×n Xإذا آ U = [u1 ,..., um ] ∈ ℝ m×m
و V = [ v1 ,..., vn ] ∈ ℝ n×n
V U T AV = diag (σ 1 ,..., σ p ) ,
p = min ( m, n )
vi وui تTJ* واA 9=2` ذةP ا; اσ i 3 K . σ 1 ≥ σ 2 ≥ ⋯ ≥ σ p ≥ 0 V و
.3ا2* " ا# ذP ا80 اi $J* ذ واP ا80 اi $J* ا : Matrix Calculus تP # ب اT ه%# V t Z* " ا# *K =ت2` B ( t ) = bij ( t ) وA ( t ) = aij ( t ) /* . !*;ق9 L ن دوال2/K bij ( t ) وaij ( t ) P # ا/A :V I0 d d A ( t ) = aij ( t ) dt dt
P # ا0 :V I0
∫ A ( t ) dt = ∫ a ( t ) dt ij
بiة ا#L :l8K d d d A (t ) B (t ) = A (t ) B (t ) + A (t ) B (t ) dt dt dt d
∫ A ( t ) dt B ( t ) dt = A ( t ) B ( t )
limits
d − ∫ A ( t ) B ( t ) dt dt
61
سI# ا/A :V I0 d −1 d A ( t ) = − A−1 ( t ) A ( t ) A−1 ( t ) dt dt
ن+= 9* Y =ت2` C = cij وA = aij Xإذا آ d At e = Ae At dt
∫e
At
dt = A−1e At + C
62
: ا ا : State Space Representation ء اGP %#0 The State Space Equations and $ _ل اZ# اP ;2Tء ا وGP د`تI : their Time Domain Solution :3 ;ط ه9YY " # هGف آ2 .3 ; ا9ء اi= N/! " 8 و9 اآ9 O0 ا3= 9 اZ* امN^ * ا-1 .3 ; اN/P " ا# 9 ا9 O0 اQUق وD -2 .3Gل اJ ا3= 9ت اS دN دJ8ق اD -3 State
9ات اZ* 9: ا2 9 اآ9 O0 ا9J 9ء اi= N^ K *)م8
. Standard State Space Form 3 ; ا9ء اi= N/! 3= TU ووVariables =ت2` واx, y, u, f , g $J* وال اR*/8 * 3م آO 9 اZ* امN/Pا N/P " ا# A, C , D 9* ^ا d x ( t ) = Ax ( t ) + f ( x, t ) + Bu ( t ) dt y ( t ) = Cx ( t ) + Du ( t )
أنRJ8 A مO ا9=2` .9:)ة واود \ اZ* ا ت اNي آ2K f ( x, t ) V .م# N/! ايT ن2/8 ان/ 8 D وC وB =ت2` وا9 ن2/K :1^ل سL N/! 3= 9* ا9^ ا9Hم ارO ا9 دQU d x1 ( t ) = 7 x1 ( t ) + 3 x2 ( t ) + 4tx1 ( t ) + x1 ( t ) x2 ( t ) − u1 ( t ) + 2u3 ( t ) dt d x2 ( t ) = 9 x1 ( t ) − 5 x2 ( t ) − 3 x22 ( t ) + 4u2 ( t ) dt y ( t ) = x1 ( t ) + x2 ( t ) − 2u2 ( t )
:Nا d x ( t ) = Ax ( t ) + f ( x, t ) + Bu ( t ) dt y ( t ) = cT x ( t ) + d T u ( t )
V
63
u1 ( t ) x1 ( t ) x (t ) = , u ( t ) = u2 ( t ) x2 ( t ) u3 ( t ) 4tx ( t ) + x1 ( t ) x2 ( t ) −1 0 2 f ( x, t ) = 1 , B = 2 −3 x2 ( t ) 0 4 0 cT = [1 1] , d T = [ 0 −2 0] 7 3 A= , 9 −5
: > ا3ا ا ءi= N^ K ص2) " ا# و9 اN/! " إ9 O0 اj N82* 9;8D ء:# إ2ه= ه ه \ 9 O0 اN82K و9: *) ا98J وا9 U`*ت اS وا دn 9H ار9 O0 9ا .9: " إ9: : n !ر8 ا6 %S ا%2)د`ت اI# ا0 -1 n 9H ار3*م اO" ا: dn d n −1 d y t + a y ( t ) + ⋯ + an −1 y ( t ) + an y ( t ) = ( ) 1 n n −1 dt dt dt n n −1 d d d b0 n u ( t ) + b1 n −1 u ( t ) + ⋯ + bn −1 u ( t ) + bnu ( t ) dt dt dt
Qi x1 ( t ) = y ( t ) − β 0u ( t ) d d d y ( t ) − β 0 u ( t ) − β1u ( t ) = x1 ( t ) − β1u ( t ) dt dt dt 2 2 d d d d x3 ( t ) = 2 y ( t ) − β 0 2 u ( t ) − β1 u ( t ) − β 2u ( t ) = x2 ( t ) − β 2u ( t ) dt dt dt dt ⋮ x2 ( t ) =
x j (t ) =
d j −1 d j −1 d j −2 d β β y t − u t − u ( t ) − ⋯ − β j −1u ( t ) = x j −1 ( t ) − β j −1u ( t ) ( ) ( ) 0 1 j −1 j −1 j −2 dt dt dt dt
V β 0 = b0 β1 = b1 − a1β 0 β 2 = b2 − a1β1 − a2 β 0 ⋮
β j = b j − a1β j −1 − ⋯ − a j −1β1 − a j β 0
" # N ا6/وه
64
d x ( t ) = Ax ( t ) + bu ( t ) dt y ( t ) = cT x ( t ) + du ( t )
V x1 ( t ) β1 β x2 ( t ) x (t ) = , b = 2 , cT = [1 0 0 ⋯] , d = β 0 = b0 ⋮ ⋮ β n xn ( t ) 0 0 A= ⋮ 0 −an
1
0
⋯
0
1
⋯
⋮
⋮
⋯
0
0
⋯
−an −1
− an − 2 ⋯
0 0 ⋮ 1 −a1
:2^ل 9ء اi= N^ K " إ9* ا9ل ا د2 d3 d2 d d y ( t ) + 6 2 y ( t ) − 8 y ( t ) + 4 y ( t ) = 2 u ( t ) + 7u ( t ) 3 dt dt dt dt
j82* 9; ا9;8: *) ا x j (t ) =
d x j −1 ( t ) − β j −1u ( t ) dt
J x1 ( t ) = y ( t ) − β 0u ( t ) = y ( t ) d d d y ( t ) − β 0 u ( t ) − β1u ( t ) = y ( t ) dt dt dt 2 2 d d d d2 x3 ( t ) = 2 y ( t ) − β 0 2 u ( t ) − β1 u ( t ) − β 2u ( t ) = 2 y ( t ) − 2u ( t ) dt dt dt dt x2 ( t ) =
]6وآ β j = b j − a1β j −1 − ⋯ − a1β 0 β 0 = b0 = 0 β1 = b1 − a1β 0 = 0 β 2 = b2 − a1β1 − a2 β 0 = 2
β 3 = b3 − a1β 2 − a2 β1 − a3 β 0 = 7 − 6 ( 2 ) − 5
J 9 U`* ا9 ا د3= j82* و 65
x1 ( t ) 0 1 0 x1 ( t ) 0 d x2 ( t ) = 0 0 1 x2 ( t ) + 2 u ( t ) dt x3 ( t ) −4 8 −6 x3 ( t ) −5 x1 ( t ) y ( t ) = [1 0 0] x2 ( t ) + [ 0] u ( t ) x3 ( t )
: > ا3 إ3 !ر8 ا6 مW 0 ": d3 d2 d y t + a y ( t ) + a2 y ( t ) + a3 y ( t ) = u ( t ) ( ) 1 3 2 dt dt dt
انi8 ا. u ( t ) *;تP ى28S 80ف ا: ان اO ،9 N/! "م اOا ا6ل ه2 N 9JKة ا# ا9ور ا د6H Q 9; :* 9JKم اO ا9=2` ةG ا; ا .ة: ا9 د9 * ا9 J* ا9ا د ت/ض ا2 9 ا56T :Nا x1 ( t ) = y ( t ) x2 ( t ) = y′ ( t ) =
d x1 ( t ) = x1′ ( t ) dt
م# N/P و x j (t ) =
d x j −1 ( t ) dt
J y′′′ ( ' t ) =
d x3 ( t ) = x3′ ( t ) QU2 9= ا9ن ا دb اx1 ( t ) = y ( t ) Q dt
d x3 ( t ) = −a1 x3 ( t ) − a2 x2 ( t ) − a3 x1 ( t ) + u ( t ) dt
3* آ3=2` N/! " # QU2K x3′ ( t ) وx2′ ( t ) وx1′ ( t ) 3= تSن ا د+= ا6/وه x1 ( t ) 0 d x2 ( t ) = 0 dt x3 ( t ) − a3
1 0 − a2
0 x1 ( t ) 0 1 x2 ( t ) + 0 u ( t ) −a1 x3 ( t ) 1
و x1 ( t ) y ( t ) = [1 0 0] x2 ( t ) + [ 0] u ( t ) x3 ( t )
66
3 ه9 J* ا9ة د# ا9ا د λ 3 + a1λ 2 + a2λ + a3 = 0
N ":K 9 ا9=2` ةG وا; ا det ( A − λ I ) = 0 −λ
1
0
0
−λ
1
− a3
− a2
−a1 − λ
=0
J ول0 ا: و `] ا ة −λ
−λ − a2
0 1 −1 − a3 −a1 − λ
1 −a1 − λ
= −λ ( λ 2 + a1λ + a2 ) − 1( a3 ) = λ 3 + a1λ 2 + a2λ + a3 = 0
3: مO 9 ا9=2` ةG ; أي ان ا; اT # 3* ا9J* اc` 3وه .9 J* ا9ة د# ا9ور ا د6H Q :*K ;* : S2# ا9Z وا%2)د`ت اI# اT -2 3 ; ا9ء اi= N/! " # 3*م اO اQU d d x1 ( t ) + x2 ( t ) = −4 x1 ( t ) + x4 ( t ) + x5 ( t ) + u ( t ) dt dt d x2 ( t ) = x2 ( t ) − 5 x3 ( t ) + 3u ( t ) dt 0 = x3 ( t ) + x4 ( t ) = x1 ( t ) − 3x2 ( t ) + x3 ( t )
0
+ 7u ( t )
= x1 ( t ) − x2 ( t )
0
+ x5 ( t )
3=2` اN/P " ا# تS ا د56 هR*/
1 0 0 0 0
1 0 0 1 0 0 0 0 0 0 0 0 0 0 0
d dt x1 ( t ) 0 d x ( t ) −4 0 0 2 0 dt 0 1 −5 d 0 x3 ( t ) = 0 0 1 dt 0 1 −3 1 d x4 ( t ) 1 −1 0 0 dt d x5 ( t ) dt
67
1 1 x1 ( t ) 1 0 0 x2 ( t ) 3 1 0 x3 ( t ) + 0 u ( t ) 0 0 x4 ( t ) 7 0 1 x5 ( t ) 0
3*=ت آ2` ; ا
1 0 0 0 0
1 1 0 0 0
0 0 0 0 0
0 0 0 0 0
d dt x1 ( t ) 0 d x ( t ) −4 2 0 dt 0 d 0 x3 ( t ) = 0 dt 1 0 d 0 x4 ( t ) 1 dt d x5 ( t ) dt
0 1 0 −3 −1
0 −5 1 1 0
1 1 x1 ( t ) 1 0 0 x2 ( t ) 3 1 0 x3 ( t ) + 0 u ( t ) 0 0 x4 ( t ) 7 0 1 x5 ( t ) 0
N/P " ا# X%; ا E11 0
d xd ( t ) 0 dt C11 C12 xd ( t ) bd u (t ) = + C21 C22 xa ( t ) ba 0 d x t dt a ( )
N اT 3*وا d xd ( t ) = Axd ( t ) + wu ( t ) dt
V A = E11−1 ( C11 − C12C22−1C21 )
w = E11−1 ( bd − C12C22−1ba )
: %S\ 3 إ%S\ %1 #W ا0 -3 مO 3: R8;K Hأو −1 2 A= , −1 −3
x2 (t ) 1 f = 2 , bu ( t ) = u ( t ) 0 0
Equilibrium Point ازن2* ا9:; Reference State QH ا9 6 + ف2 . u0 ( t ) Q dx0 ( t ) dt = 0 :Nا 3 وهQH ا9 ود اu0 ( t ) وx0 ( t ) ل2 مOف `] ا2 d x0 ( t ) = 0 dt Ax0 ( t ) + f ( x0 ( t ) , u0 ( t ) ) + 0 = 0
68
أو 2 0 = − x10 ( t ) + 2 x20 ( t ) + x20 (t )
0 = − x10 ( t ) − 3 x20 ( t )
J RK*دة ا#+ و x10 ( t ) = −3 x20 ( t )
2 3 x20 ( t ) + 2 x20 ( t ) + x20 ( t ) = ( 5 + x20 ( t ) ) x20 ( t ) = 0
3 *;ار هA ا3* ا6/وه 0 15 x10 ( t ) = , x20 ( t ) = 0 −5
نf # $# * ا/ 8 9 ا$J* 9 * اJacobian Matrix 9 2آJ ا9=2` ا J u (t ) = 0
J x(t )
∂ ∂x ( t ) f1 ( x ( t ) , u ( t ) ) 1 = ∂ f2 ( x (t ) , u (t )) ∂x1 ( t )
∂
f1 ( x ( t ) , u ( t ) ) 0 2 x2 ( t ) = ∂ 0 0 f 2 ( x ( t ) , u ( t ) ) ∂x2 ( t ) x ( t ), u (t ) 0 0 ∂x2 ( t )
x0 ( t ),u0 ( t )
2 ( ه: NH ي6 ) ا3:)م اO اx0T ( t ) = [ 0 0] Q QH ا9 −1 2 1 d δ x (t ) = δ x (t ) + δ u (t ) dt −1 3 0
2 ه3:)م اO اx0T ( t ) = [15 −5] QH ا9 و −1 −8 1 d δ x (t ) = δ x (t ) + δ u (t ) dt −1 −3 0
.9` *) 9ت اوS ل2 3 %0م اO 3: R8;K N^ K ا *;ة9:) ا9 O0 ا56ه :ة/ # ا%S ا#W اT : Z#ة ا/ # ا%S ا#Wا (9J*) 3=2` Z* 9 دScalar دي# Z* 3= 9 دNف ;رن ا2 :98 اد9ا d x ( t ) = ax ( t ) , x0 ( 0 ) = x0 dt
N اJ 9 U`*ت اS ا دN 9; ق ا:
69
x ( t ) = x0e at
:9=2` ا9ا d x ( t ) = Ax ( t ) , x0 ( 0 ) = x0 dt
2 هN ا9;8: اc` x ( t ) = e At x0
State Transition Matrix 9 إ*;ل ا9=2` " K e At 9=2` ا :9 إ*;ل ا9=2` اص2 1.
d At e = Ae At = e At A dt
2.
∫e
3. e
At
dt = A−1e At = e At A−1
A ( t +τ )
= e At e Aτ
4. let τ = −t , e 5. e(
A+ B)t
A ( t +τ )
= e At e − At = I
= e At e Bt , if
or
−1
e At = e − At
AB = BA
: Z#ا%1 ة/ # ا%S ا#Wا :98 اد9ا d x ( t ) − ax ( t ) = bu ( t ) dt
N اJ 9 U`*ت اS ا دN 9; ق ا: x ( t ) = x0e at + ∫ e a ( t −τ )bu (τ ) dτ t
0
:9=2` ا9ا d x ( t ) − Ax ( t ) = Bu ( t ) dt
2 هN ا9;8: اc` x ( t ) = e At x0 + ∫ e A( t −τ ) Bu (τ ) dτ t
0
70
: %Fد`ت اوI#2 ء اGP %#0 N آ ان9 U`*ت اS د5O ^/ NT أ9Lت ا`وS د9ء اi= N^ K آNد اJ8A Excel N^ a *)م8م وO اري/K N 8D # *8 aKم اOا .;S ى :3* آ9ء اi= N^ K N/P 9H أي در9L =و9 أي دQU2K xk +1 = Gxk + Huk
\ ود9=2` H ( و9 ) 9 ا9=2` G وk ار/* ا# 9 ا$J* xk V .cJ*ا :1^ل "و0 ا9H إ" د* ار9 Z* و9^ ا9H ار9* ا9L ا`و9ل ا د2 .9 يZ*و xk + 2 + 5 xk +1 − 7 xk = 2k
:Nا :N/P " ا# 9 ا دS أوQi xk + 2 = −5 xk +1 + 7 xk + 2k
a*= xk +1 = yk QU xk +1 = yk yk +1 = −5 yk + 7 xk + 2k
أي xk +1 1 0 xk 0 y = −5 7 y + 2 k k k +1
V xk +1 = Gxk + Huk N/P " ا# 3وه 1 0 G= −5 7
و 0 H = 2
م اO N H أوx0 = 1, y0 = 1 9; أو
71
:Excel ام8&f ا :3* اN أد
J
N^ 9 اآ9 O0 وآة اN K a ا3= T*J 92T 2 ه9ء اi= N^ K 9 أه . و\هSTELLA وVensim a :Vensim 9: ا2 9Lت ا`وS د9ء اi= N^ K : Vensim 9: ا2 م اO اN^ K x dx
y dy
k
dx = y dy = -5*y+7*x+2*k k = LOOKUP( [(0,0)-(10,10)],(0,0),(1,1),(10,10)) x = INTEG ( dx, 1) y = INTEG ( dy, 1)
72
:ا ا دس : %&&ذج ا#$ اYI ;اK ^ ء ذج أآ9آت اوK *K 3* ا9 0 ا ذج اj ورس3 ف2 3;;ب إ" ا اLوا
: Sل( اa ` )ا#$ ذج ا# -1 # ى او ا *ع2* ان اU= 2 = ،X Y لL*8 داد أوG8 مO RآK d ا ا6ه 9L ":8 (a < 0) ;( أو اa > 0) دة8Gن ل ا+= x (t ) ;ارN^ 8 t Gا dx (t ) = xɺ (t ) = a dt
:N/P N^ 8و x(t) dx(t)/dt
a
N/P ":K (a > 0) و ـ
Graph for x(t) 40
30
20
10
0 0
10
20
30
40
50 60 Time (Month)
"x(t)" : Current
73
70
80
90
100
xɺ (t ) G
dx (t ) دا@ ـG ف2 :9O dt
:&`ل( اa ` )ا#$ ذج ا# -2 2 = ،دة2H2 ا9 / اQ R *8 ; ل8 أو8اG*8 ى أو *ع2* *8 ذج2 ا ا6ه ( اوa > 0) دة8Gن ل ا+= x (t ) ;ارN^ 8 t G ا# ى او ا *ع2* ان اU= 9L ":8 (a < 0) ;ا xɺ (t ) = ax (t )
:N/P N^ 8و x(t) dx(t)/dt
a
N/P ":K (a > 0) و ـ
Graph for x(t) 200
150
100
50
0 0
10
20
30
40
50 60 Time (Month)
70
80
90
100
"x(t)" : Current
(a < 0) 9 ; ; ا/P أر ا:8 K
74
: Sل اa ` وا#$ ذج ا# -3 & X Y لL*8 XL2 اc` 3=( وa > 0) X Y ل8اG*8 ى2* *ع اوN^ 8و :3* آN^ 8( وb < 0) xɺ (t ) = a − b
x(t) dx(t)/dt
-dx(t)/dt
a
b
:&`ل اa ` وا#$ ذج ا# -4 2 = ،دة2H2 ا9 / اQ R *8 ; ل8 و8اG*8 ى أو *ع2* *8 ذج2 ا ا6ه ( وa > 0) دة8Gن ل ا+= x (t ) ;ارN^ 8 t G ا# ى او ا *ع2* ان اU= 9L ":8 (b < 0) ;ا xɺ (t ) = (a − b ) x (t )
:N/P N^ 8و x(t) dx(t)/dt
-dx(t)/dt
a
b
75
:I% ذج ا#$ ا-5 QL2 N^ 8 x (t ) آن2 ^ = ، 9 آ9J* 3:8 3/ *: * # Q J8 ذج2 ا ا6ه أنS) v (t ) G 9# G ر2 و9* ^ ا9;=S اT*# N^ 8 xɺ (t ) ن+= 9آ N/P 9ت اآS دR*/Kو
dv (t ) = vɺ (t ) = xɺɺ (t ) ن2/8 NJ*ن ا+= ( v (t ) = xɺ (t ) dt
9ا د xɺɺ (t ) = a
; NJK a < 0 دة و8 زNJK a > 0 V
x(t) dx(t)/dt
v(t) dv(t)/dt a
"و0 ا9H ار9 U`K K داN/! " # 9^ ا9H ار9 U`* ا9 ا دR*/Kو :3* ( آState Equations 9ت اS)د xɺ (t ) = v (t ) vɺ (t ) = a
9و; او x ( 0) = 0 k v (0) = 0 k/sec xɺ (t ) = 5 k/sec vɺ (t ) = 3 k/sec 2
76
Graph for x(t) 80,000
60,000
40,000
20,000
0 0
10
20
30
40 50 60 Time (Second)
70
80
90
"x(t)" : Current
100 k
Graph for v(t) 400
300
200
100
0 0
10
20
30
40 50 60 Time (Second)
"v(t)" : Current
70
80
90
100
k/Second
77
Birth-Death Model ت2 دة واS2ذج ا2 (3) 9 U`* ا9 دl%28و xɺ = ( b − d ) x
9Lأو ا`و xt = xt −1 + ( b − d ) xt −1
x
dx/dt
b
d
dx/dt=(b-d)x
Logistic Growth Model 3* H2 ا2 ذج " ا2 (1) 9 U`* ا9 دl%28و xɺ = k ( H − x )
9Lأو ا`و xt = xt −1 + k ( H − xt −1 )
78
Inventory x dx/dt
k
Sales rate
H
Market limit
dx/dt=k(H-x)
Product Limit Growth (3* H2) وطP ا ود2 ذج ا2 (5) 9 U`* ا9 دl%28و xɺ = kx ( h − x )
9Lاو ا`و xt = xt −1 + kxt −1 ( h − xt −1 )
79
Product sold x dx/dt Sales rate
k z
Residual market z=H-x
H Market limit dx/dt=kx(h-x)
:9 يZ* 9 U`*ت اS دl%28 ذج2 (6) yɺ = k1 ( h − y )
xɺ = k2 ( y − x )
9Lأو ا`و yt = yt −1 + k1 ( h − yt −1 )
xt = xt −1 + k2 ( yt − xt −1 )
80
Houses sold y= Number of houses sold Houses supply h= Number of housholds
y dy/dt
x= Number of airconditions
k1 h dy/dt=k1(h-y)
Airconditions sold
Airconditions supply
x dx/dt
k2 dx/dt=k2(y-x)
:9 اتZ* ^ث9 U`*ت اS دl%28 ذج2 (7) hɺ = k4 h
yɺ = k1 ( h − y )
xɺ = k2 ( y − x ) − k3 x
9Lأو ا`و ht = ht −1 + k4 ht −1
yt = yt −1 + k1 ( ht − yt −1 )
xt = xt −1 + k 2 ( yt − xt −1 ) − k3 xt −1
81
Hous holds h
dy/dt=k1(h-y)
dh/dt dx/dt=k2(y-x)-k3x
k4
dh/dt=k4h Houses sold y dy/dt k1 Airconditions sold Broken aircond.
x
dx/dt
k3
k2
82
: ا ا :ذج#$ اYI ء$9 %&`ت دراT :ق2 ا9ذج آ2 " ء# ^ل-1 ق2 ا2 gU28 ذج2 ا.@ داa* 82 K 9ذج آ2 ف *ض ء2 ا ا ^ل6 ه3= 9= ر اP*ف * ان ا2 .ن2 زN/ ة واة$@* !ا8 V ر2D 8H a* 3 /ا *8 a* ن ل !اء ا+= ا6T وa* ا ا6 ا!*وا ه86 @ اG " ا# *K a* ا ا6 ه# :" # a* ا!*وا ا86 @ اGد ا# (1) a* *وا اP8 انQL2*8 86 اL @ اGد ا# (2) Tآ/* إ# a* إ!*وا ا86 @ اGء اU L2* @ اG*ع اL إ9H( در3) j اTi : ف2 ،(3@TS) اH آL2* @ اGد ا# T= ن2/8 3* ا9 اL ف2 :Sأو انQL2*8 86 " ا#( و8*P) ActualCustomers = a* ا!*وا ا86 @ اG " ا# اآL2* د ا# T= 3* ا9ف * ا2 ( وL2*) PotentialCustomers *واP8 .8*P دة ا8 ز# O اjZ X Y 8;K NO8 و8*P ^ ا/ 3 إL2* اc∆t ل28 8*P وا اN∆ آt ةL 9 `*ة ز$`*ض ا 3:8 ي6( واconversion coefficient N82* اN) RH2 X Y c V ،8*P .8*P ا28 3/ L2* ع اLA 8*P ا9= ; ى L2* اc∆t ل2 T واN آ8*P اn ( t ) H28 t G ا# انU*=إذا إ g8 8*P د ا# t + ∆t G∆ = اt 9G ا`*ة ا3= 8*P "إ n ( t + ∆ t ) = n ( t ) + n ( t ) c∆ t
أو n ( t + ∆t ) − n ( t ) dn ( t ) = = cn ( t ) ∆ t →0 ∆t dt lim
(1) 9 /K 9 دN/! " # أو
t
n ( t ) = n0 + ∫ cn (τ ) dτ
(2)
0
2 ه9; ا9 ا دN ، t = 0 G ا# 8*P د ا# 2 هn0 V 83
n ( t ) = n0e ct , t ≥ 0
( 3)
:9L =و9 دN/! 3 # (3)-(1) 9; ت اLف ا2 :9O n ( t + ∆ t ) − n ( t ) = n ( t ) c∆ t for ∆t = 1 (one time unit) nt +1 = nt + nt c nt +1 = (1 + c ) nt
when t = 0, nt = n0
n1 = (1 + c ) n0
n2 = (1 + c ) n1 = (1 + c )(1 + c ) n0 = (1 + c ) n0 2
n3 = (1 + c ) n2 = n0 (1 + c )
3
or nt = n0 (1 + c ) , t ≥ 0 t
Vensim 9Z ذج2 اN^ 8 3* اN/Pا
n(t)
n(0)
dn(t)/dt
c
(1)
c = 1 Units: **undefined**
(2)
"dn(t)/dt" = c*"n(t)" Units: **undefined**
(3)
FINAL TIME = 100 Units: Month The final time for the simulation.
(4)
INITIAL TIME = 0 Units: Month The initial time for the simulation.
84
(5)
"n(0)" = 1
Units: **undefined**
(6)
"n(t)"= INTEG ( "dn(t)/dt", "n(0)") Units: **undefined**
(7)
SAVEPER = TIME STEP Units: Month The frequency with which output is stored.
(8)
TIME STEP = 1 Units: Month
The time step for the simulation.
:أو
Actual Customers
InitialActualCustomers
ConversionFlow
ConversionConstant
(1)
ActualCustomers = INTEG ( ConversionFlow, InitialActualCustomers) Units: **undefined**
(2)
ConversionConstant = 1 Units: **undefined**
(3)
ConversionFlow = ConversionConstant*ActualCustomers Units: **undefined**
(4)
FINAL TIME = 10
Units: Month The final time for the simulation.
(5)
INITIAL TIME = 0 Units: Month The initial time for the simulation.
(6)
InitialActualCustomers = 1 Units: **undefined**
(7)
SAVEPER = TIME STEP Units: Month The frequency with which output is stored.
(8)
TIME STEP = 1 Units: Month
The time step for the simulation.
85
1 Current 1 ActualCustomers 2,000
1
1
1
1
1
1
1
1,500 1
1,000 1
500 1 1 0 ConversionFlow 2,000
1
1
1
1
1
1,500 1
1,000 1
500 1
0
1
0
1
1
2.5
1
1
1
5 7.5 Time (Month)
10
.8*P " إL2* و ل إ ب ا8*P د ا# N/ 3 S ا2 اd S 2 ا9;; ا3= ن0 ،(3@T) ودL2* د ا# T= ن2/8 3* ا9 اL نb ا:Y ده# إذا آن8*P " إT ا آ22* L2* ن ا0 ] * وذ8 ان/ 8S 3 Sا .ود 2 هt G ا# ;* @ اGد ا# ن2/= M 2 هL2* 3 /`*ض ان اد ا N/ N82* ل ا2 هc و `*ض انt G ا# 8*P د ا# 2 هn ( t ) V M − n ( t ) 3 / اد اQ : R *8 N82*ف `*ض ان ل ا2 ،L2* M H28 # *يP N82*ن ل ا+= L2* اl ";K آن2 ^ = ،9 ز9O اي# ;* اL2* .ا6/ وهc / 4 2*ي هP N/ N82*ن ل ا+= L2* اQ ر3;K وإذاc / 2 2*ي هP N/ 2∆ هt 9G ا`*ة ا3= *ى واP 2 اL2* د ا# ن+= تU` ا56 هXK
{ M − n ( t ) M } × c∆t 2*ى هP n ( t ) N/ 2 اL2* د ا# ن2/8و
86
{
}
n ( t ) × M − n ( t ) M × c∆t
g8 8*P د ا# t + ∆t G ا#
{
}
n ( t + ∆t ) = n ( t ) + n ( t ) × M − n ( t ) M × c∆t n ( t + ∆t ) − n ( t ) = c × M − n ( t ) M × n ( t ) ∆t
{
}
أو dn ( t ) M − n (t ) = c× × n (t ) dt M
(4) 9 /K 9 دN/P أو
t
n ( t ) = n0 + ∫ c × 0
M − n (τ ) × n (τ ) dτ M
( 5)
2 ه9; ا9 ا دN ، t = 0 G ا# 8*P د ا# 2 هn0 V n (t ) =
M , t≥0 1 + ( M − n0 ) n0 e − ct
(6)
logistic curve 3* H2 " ا " ا8 " (6) 9ا د :9L =و9 دN/! 3 # (6)-(4) 9; ت اLف ا2 :9O M − nt × nt M n nt +1 = nt + c 1 − t nt M nt +1 = nt + c (1 − α nt ) nt nt +1 − nt = c ×
where α = 1 M
د# 2 هn0 = 1 أنU= ذا+= ار/* ا9;8: T ف2 9: \ 9L =و9 د56وه M = 100 2 هL2* 3 / واد اc = 1 N82* ول اt = 0 G ا# 8*P ا
ن+= α = 1 M = 0.01 nt +1 = nt + c (1 − α nt ) nt n1 = n0 + c × (1 − α n0 ) n0
= 1 + 1 × (1 − 0.01 × 1) × 1 = 1.99
87
n2 = n1 + c × (1 − α n1 ) n1
= 1.99 + 1 × (1 − 0.01 × 1.99 ) × 1.99 = 3.94
n3 = n2 + c × (1 − α n2 ) n2
= 3.94 + 1 × (1 − 0.01 × 3.94 ) × 3.94 = 7.73
n4 = 14.85 n5 = 27.50 n6 = 47.44 n7 = 72.37 n8 = 92.37 n9 = 99.42 n10 = 99.996
:3* اN/P اTو
Graph for ActualCustomers 100
1
1
9
10
1
75
1
50
1
1
25 1
0 0
1
1
1
2
1
3
ActualCustomers : Current
4 1
5 6 Time (Month)
1
1
1
1
7 1
8 1
1
1
1
1
:3* آVensim *)ام+ ذج ا2 ا6`
88
TotalMarket Potential Customers
Actual Customers
InitialActualCustomers
ConversionFlow
ConversionConstant
(01)
ActualCustomers = INTEG ( ConversionFlow, InitialActualCustomers) Units: **undefined**
(02)
ConversionConstant = 1 Units: **undefined**
(03)
ConversionFlow = ConversionConstant*(PotentialCustomers/TotalMarket)*ActualCustomers Units: **undefined**
(04)
FINAL TIME = 10
Units: Month The final time for the simulation.
(05)
INITIAL TIME = 0 Units: Month The initial time for the simulation.
(06)
InitialActualCustomers = 1 Units: **undefined**
(07)
PotentialCustomers = INTEG ( -ConversionFlow, TotalMarket-InitialActualCustomers) Units: **undefined**
(08)
SAVEPER = TIME STEP Units: Month The frequency with which output is stored.
(09)
TIME STEP = 1 Units: Month The time step for the simulation.
(10)
TotalMarket = 100 Units: **undefined**
89
n(0)
M
M-n(t)
n(t) dn(t)/dt
c
(01)
c = 1 Units: **undefined**
(02)
"dn(t)/dt" = c*("M-n(t)"/M)*"n(t)"
(03)
FINAL TIME = 10
(04)
INITIAL TIME = 0 Units: Month The initial time for the simulation.
(05)
M = 100
(06)
"M-n(t)" = INTEG ( -"dn(t)/dt", M-"n(0)") Units: **undefined**
(07)
"n(0)" = 1 Units: **undefined**
(08)
"n(t)" = INTEG ( "dn(t)/dt", "n(0)")
(09)
SAVEPER = TIME STEP Units: Month The frequency with which output
Units: **undefined**
Units: Month The final time for the simulation.
Units: **undefined**
Units: **undefined**
is stored. (10)
TIME STEP = 1 Units: Month The time step for the simulation.
90
1 Current 1 ActualCustomers 100
1
1
1
1
1
1
1
1
1 1
75
1
50
1 1
25 1 1
1
1
0 ConversionFlow 40 30
1 1
20 1
10 0
1
1 1
0
1
1 1
2.5
5 7.5 Time (Month)
10
9* H2 ا2 ت اdK اn ( t ) 8*P ا*; يS فd 8 " #0 اN/Pا i8 ا2ي ه6ب( وا2 ; ة ا2T;ن اJ= س )اوJ اN/! $P8 ي6 اN82* ل ا3 N/!و .3* H2 ا2 ا3= N82*ت اS @ ا : Determining Model Parameters ذج2 اK وريi ا3* H2 = " ا،ذج2 *)ام ا3/ اL K اH وريi ا :3* ا ^ل ا56 ه8* 9;8D ف *ض2 . M وn0 وc 8K 9 "* و1984 9 N* و إX=2 و/ تJ* ق2 ا9 L 9 3 ه9*ات ا 1994 Year
1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994
Market 3.0
2.5
4.0
7.5
7.0
13.0
17.0
29.0
46.5
50.0
49.5
Value
3= ق2 ا9 L وx XK د2 ا3= 9 ت اN K Curve Expert a *)ام+ Sاو :3* آUser-Defined Model *ر+ ذج2 ا8 وy XK د2 ا
91
User-Defined Model: y=a/(1+b*exp(-c*x))
9 اوL د Coefficient Data: a=
57.760559
b=
286.77098
c=
0.72892383 S = 3.46006530 r = 0.98725204 75 54.
Y Axis (units)
67 45. 58 36. 50 27. 42 18. 3 9.3 5 0.2
0.0
2.0
4.0
6.0
X Axis (units) User-Defined Model: y=a/(1+b*exp(-c*x)) Coefficient Data: a=
57.760567
b=
138.3462
c=
0.72892364
92
8.0
10.0
12.0
S = 3.46006530 r = 0.98725204 75 54.
Y Axis (units)
67 45. 58 36. 50 27. 42 18. 3 9.3 5 0.2
0.0
1.8
3.7
5.5
7.3
9.2
11.0
X Axis (units)
Excel ام8&f ذج#$ ا7I 6%%I0 ة8H 9`% 3= 3* اN أد
3*دوات أ *ر ا0 ا:
93
Solver ة6= TOK
Equal To 3= و5ZK ع ا ت ا اد2 J 9 L ي2K Set Target Cell أنS 3* اBy Changing Cells ; اZ* ع ا ت2 J ZK *8 وZ* Min * أ " # N Solve " # dZU و$D$1:$D$3 9و0 ا; ا8 3=
94
Sان ا) D1 8و D2و 2K D3ى اbن ا; ا 9@Tا ;رة و 2K E16ى اNL 2 J 9 Lع ت ا: 0ء.
95
The Workforce Inventory Example : 2 4 ة/ ذج _ون# -2 8 2T= 3/8ذج د2 ء9// # ةH ة/= ء:#A اH T d ذج ا2 ا ا6ه م# "دي إBK L 9 ة ا2; اl2K ب2 ون واG) ت إدارة ا3= N#`* ا9`آ l2K 3= =عA ان ا3 وه9;: \ 9J* 8 i8 ا9 آ ا.*جA ا3= *;ارAا 9/8 دgU28 3* اN/P ا. اآ^ إ *;ارا9 # ة2L 3دي اB8 ان/ 8 ا لg8 Kو :ذج2 ا
Inventory Production
Sales
InventoryCoverage
Productivity
TargetInventory InventoryCorrection TargetProduction
Workforce NetHireRate
TimeToCorrectInventory
TargeWorkforce TimeToAdjustWorkforce
" # ي2*8 ذج2 ا ( Level ى2* ) أو اStocks ا *ع#2 (ا Inventory ونG) ( ا1 Workforce 9 ة ا2;( ا2 Flows بAاع ا2 أ9 ب( أر ( Inflow N ون ) إ ب داG) *ع ا8GK وProduction *جA( ا1 ( Outflow 3H) أ ب ر
ونG) *ع اN ;K وSales ( ا ت2
9 ة ا2; *ع ا8G8 أوN ;8 2 وهNetHireRate *)امA ل ا3=% (4 ( و3 Auxiliary Variables ة# اتZ* (ج TargetInventory فT* ون اG) ( ا1 InventoryCorrection ونG) اgK (2 TargetProduction فT* *ج اA( ا3 TargetWorkforce 9=T* ا9 ة ا2;( ا4 Constants ذج2 اX ا2Y (د Productivity 9H*A( ا1 96
InventoryCoverage ونG) ا9:ZK (2 TimeToCorrectInventory ونG) اg* ب2 : اG( ا3 TimeToAdjustWorkforce 9 ة ا2; اN8* ب2 : اG( ا4
Causal Relationships between Variables :ذج2 ا3= 9 ت اLا :ونG) *ع ا:Sأو Workforce Production Productivity
Inventory Sales
Inventory
InventoryCorrection
TargetProduction
:9 ة ا2; *ع ا:Y (Workforce) (TargeWorkforce)
NetHireRate
TimeToAdjustWorkforce
Workforce
Productivity TargeWorkforce TargetProduction
NetHireRate
(Workforce)
Workforce Production
Inventory فT* ون اG) ا# اZ* ا:^Y
InventoryCoverage TargetInventory Sales
97
TargetInventory
InventoryCorrection
TargetProduction
ونG) اgK # اZ* ا:f را Production (Sales)
Inventory
InventoryCoverage Sales
InventoryCorrection TargetInventory
TimeToCorrectInventory
InventoryCorrection
TargetProduction
TargeWorkforce
فT* *ج اA ا# اZ* ا: Inventory TargetInventory
InventoryCorrection TargetProduction
TimeToCorrectInventory Sales
Workforce TargetProduction
TargeWorkforce NetHireRate
9=T* ا9 ة ا2; ا: د Productivity InventoryCorrection Sales
TargeWorkforce TargetProduction
98
(NetHireRate) Workforce TargeWorkforce
Production NetHireRate
(Workforce)
Feedback Loops %2" ا+دورات ا :ونG) *ع ا9` ) ا986Z*دورة ا Loop Number 1 of length 6 Inventory InventoryCorrection TargetProduction TargeWorkforce NetHireRate Workforce Production
:9 ة ا2; *ع ا9` ) ا986Z*دورة ا Loop Number 1 of length 1 Workforce NetHireRate Loop Number 2 of length 6 Workforce Production Inventory InventoryCorrection TargetProduction TargeWorkforce NetHireRate
:فT* *ج اA ا# اZ* 9` ) ا986Z*دورة ا Loop Number 1 of length 6 TargetProduction TargeWorkforce NetHireRate Workforce
99
Production Inventory InventoryCorrection
:ونG) اgK # اZ* 9` ) ا986Z*دورة ا Loop Number 1 of length 6 TargetProduction TargeWorkforce NetHireRate Workforce Production Inventory InventoryCorrection
:9=T* ا9 ة ا2; ا# اZ* 9` ) ا986Z*دورة ا Loop Number 1 of length 6 TargetProduction TargeWorkforce NetHireRate Workforce Production Inventory InventoryCorrection
:ذج2 ت اSد (01)
FINAL TIME = 100
Units: Month
The final time for the simulation. (02)
INITIAL TIME = 0
Units: Month
The initial time for the simulation. (03)
Inventory = INTEG(Production-Sales ,300) Units: Widget
(04)
InventoryCorrection = (TargetInventory - Inventory)/ TimeToCorrectInventory Units: Widget/Month
(05)
InventoryCoverage = 3
Units: Month
(06)
NetHireRate = (TargeWorkforce Workforce)/TimeToAdjustWorkforce Units: Person/Month
100
(07)
Production = Workforce*Productivity Units: Widget/Month
(08)
Productivity = 1 Units: Widget/Month/Person
(09)
Sales = 100 + STEP(50,20)
Units: Widget/Month
(10)
SAVEPER = TIME STEP
Units: Month
The frequency with which output is stored. (11)
TargetInventory = Sales * InventoryCoverage Units: Widget
(12)
TargetProduction = Sales + InventoryCorrection Units: Widget/Month
(13)
TargeWorkforce = TargetProduction/Productivity Units: Person
(14)
TIME STEP = 1 Units: Month The time step for the simulation.
(15)
TimeToAdjustWorkforce = 3
Units: Month
(16)
TimeToCorrectInventory = 2
Units: Month
(17)
Workforce = INTEG(NetHireRate, TargeWorkforce) Units: Person
:9; ا`;ة ا3= ة: ا9و0ذج ; ا2 آت ا9J* 9*ل ا/!0ا runinv01 Inventory 600 450 300 1 150 0 Production 400 300 200 100 1 0 Sales 200 170 140 110 80
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
50 Time (Month)
100
101
1
runinv01 Workforce 400 300 200 100
1
1
1
1
1
1
1
1
1
1
1
1
0 NetHireRate 60 30 1 0 -30 -60
1
0
50 100 Time (Month)
Graph for Inventory 600 1
1
1
450
1
1
1
1
1
1
1
1
1
300
1
1
1
1
150
0 0
10
20
Inventory : runinv01
30 1
40 50 60 Time (Month) 1
1
1
102
1
1
70 1
80 1
1
90 1
100 Widget
Graph for Workforce 400
300
1
200
1
100
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0 0
10
20
Workforce : runinv01
Time (Month)
30 1
40 50 60 Time (Month) 1
1
1
1
1
70 1
80 1
1
90
100 Person
1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
"Inventory"
Runs:
Inventory
runinv01
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
250
241.667 269.444 322.685 388.272 453.215 506.799 541.985 555.977 549.973
528.308 497.203 463.414 433.022 410.524 398.355 396.822 404.408 418.328 435.206 451.737 465.224 473.925 477.189 475.377 469.638 461.582 452.939 445.246 439.628 436.675 436.435 438.496 442.13 446.471 450.676 454.068 456.216 456.971 456.438 454.92 452.836 450.626 448.68 447.279 446.565 446.542 447.099 448.047 449.163 450.232 451.084 451.614 451.786 451.632 451.232 450.693 450.128 449.636 449.287 449.115 449.119 449.269 449.516 449.802 450.074 450.288 450.418 450.457 450.414 450.308 450.169 450.025 449.9
449.813 449.772 449.776 449.816 449.88 449.954
103
Time (Month)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
"Workforce" Workforce 100
Runs:
runinv01
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
141.667 177.778 203.241 215.586 214.943 203.584 185.187 163.991 143.997 128.335 118.894 116.212 119.607 127.502 137.831 148.467 157.585 163.92 166.879 166.531 163.486 158.701 153.264 148.188 144.261 141.944 141.357 142.307 144.382 147.047 149.76 152.061 153.635 154.341 154.205 153.392 152.148 150.754 149.467 148.483 147.916 147.79 148.054 148.598 149.286 149.977 150.557 150.948 151.115 151.069 150.852 150.53 150.172 149.846 149.6
149.461 149.435 149.508 149.651 149.828
150.004 150.15 150.247 150.286 150.272 150.214 150.13 150.039 149.956 149.895 149.861 149.856 149.876 149.913 149.959 150.004 150.04 150.064 150.074 150.069
104
Fibonacci Sequence Model :A 9%P I ذج# -3 وا9 Z* و9^ ا9H ار y ( k + 2 ) = y ( k + 1) + y ( k ) ,
y (1) = 1, y ( 2 ) = 1, k = 1, 2,3,⋯
k = 1 ⇒ y ( 3) = y ( 2 ) + y (1) = 1 + 1 = 2
k = 2 ⇒ y ( 4 ) = y ( 3) + y ( 2 ) = 2 + 1 = 3 k = 3 ⇒ y ( 5 ) = y ( 4 ) + y ( 3) = 3 + 2 = 5 ⋮
:ض2 9 8Z* و" و0 ا9H إ" ارT2 x ( k + 1) = y ( k ) , k = 1,2,3,⋯ ,
x ( 2 ) = y (1) = 1
:9ء اi= N/! 3= j82* ا3P*2= 9 ** 9= ا9 ا دg*= y ( k + 2 ) = y ( k + 1) + x ( k + 1)
x ( k + 2 ) = y ( k + 1) k = 1, 2,3,...
y (1) = 1, y ( 2 ) = 1, x ( 2 ) = y (1) = 1
k = 1 ⇒ y ( 3) = y ( 2 ) + x ( 2 ) = 1 + 1 = 2 x ( 3) = y ( 2 ) = 1
k = 2 ⇒ y ( 4 ) = y ( 3) + x ( 3) = 2 + 1 = 3 x ( 4 ) = y ( 3) = 2
k = 3 ⇒ y ( 5) = y ( 4 ) + x ( 4 ) = 3 + 2 = 5 x ( 5) = y ( 4 ) = 3
⋮
Vensim ام8&f A 9%P I# ء اGP > ذج#
y dy
x dx
105
(1)
dx = y
(2)
dy = y+x
(3)
FINAL TIME = 10
(4)
INITIAL TIME = 1
(5)
SAVEPER = TIME STEP
(6)
TIME STEP = 1
(7)
x= INTEG ( dx,1)
(8)
y= INTEG ( dy,1)
k
1
2
3
4
5
6
7
8
9
y
1
3
8
21
55
144
377
987
2584 6765
x
1
2
5
13
34
89
233
610
1597 4181
10
ي اود2K) x ( و9HوGى اود ا2K) y 9 ; 3P*2= 9 ** انS (98ا`د k
1
2
3
4
5
6
y
1
3
8
21
55
144
x
↑ ց ↑ ց ↑ ց
↑
1
13
2
5
ց
↑
ց
34
106
↑ 89
…
Dynamic Stochastic Models %R اAI ا%% $ 8ذج ا#$ ا-4 Markovian Property of order k 9H ار%P رآ# ا%H ا:l8K
9ز0 ا# T L " # *K t G ا# 9@ا2Pهة اO ا9 L " ان# ل2;K ن+= { yt , −∞ < t < ∞} X = ^ إذا آ.d;= t − 1, t − 2,..., t − k
(
)
P yt < s | yt −1 , yt −2 ,..., yt −k , yt −( k +1) ,... = P ( yt < s | yt −1 , yt −2 ,..., yt −k )
.9=2 ا رآ9%) اQ*K 3* ا98*دLAاه ا2O^ ا/هك ا : Vensim ام8&f %P رآ# ا%R اAI ا%ذج ا آ#$ ا%#0 م28 98T 3= اد2 ا0 `لLAهة وا;س ان اP 8*دLA أ اH و:1 ^ل Q8ز2K $ 3@ا2P# f: Q ; ا2 ا98T 3= `ل دةLA " ا# *8 اولK .هةO ا56 هl8 ذج2 ن2 آ.م2] ا6 σ 2 8K`ي و% d 2* 3D R ن2/= ε t 2 ه3@ا2P اf:) واyt 2م ه2 ا98T 3= `لLA `*ض أن ا:Nا 9=2 ا آ9%)ا yt = φ1 yt −1 + φ2 yt −2 + ε t , ε t ~ N ( 0,σ 2 ) , ∀t
9 Z* 9 J* \ 9^ ا9H ارDifference Equation 9L =و9 د56وه ن2/= xt −1 = yt −2 `*ض،* 3 او9H در3 إT2 ف2 .وا yt = φ1 yt −1 + φ2 xt −1 + ε t , ε t ~ N ( 0, σ 2 ) , ∀t xt = yt −1
9; ت اS ا دN= φ1 = 1.7, φ2 = −0.72, and σ 2 = 1 X ا ^ل ا إذا آ3= :2 ^ل y0 = 0, x0 = 0
9; أو
Vensim *)ام+ :Nا
107
x dx
y dy
phi2
eps phi1
(01) dx = y (02) dy = phi1*y+phi2*x+eps (03) eps= RANDOM NORMAL(-3.99,3.99 ,0 ,1 ,19 ) (04) FINAL TIME = 2
The final time for the simulation.
(05) INITIAL TIME = 0
The initial time for the simulation.
(06) phi1 =
1.7
(07) phi2 =
-0.72
(08) SAVEPER = TIME STEP
The frequency with which output
is stored. (09) TIME STEP = 0.01
The time step for the simulation.
(10) x= INTEG ( dx, 0) (11) y= INTEG ( dy, 0)
108
dy v y 4
2
0
-2
-4 -0.100
-0.050
0 y
0.050
0.100
dy : Current
xvy 0.008
-0.009
-0.026
-0.043
-0.06 -0.100
-0.050
0 y
x : Current
109
0.050
0.100
y 0.2
0.1
0
-0.1
-0.2 0
0.50
1 Time (Day)
1.50
2
1.50
2
y : Current
dy 4
2
0
-2
-4 0
0.50
1 Time (Day)
dy : Current
110
x&y 0.2 0.008
0 -0.026
-0.2 -0.06 0
0.50
1 Time (Day)
y : Current x : Current
111
1.50
2
: bifurcation kIA ذج ا# -5 ^ آa*K T2/ 3U8 رN/P l^/ N/P نbرس اK 3* أه ا ذج اRP* ذج ا . ا ذج56 أ هN^ 8 3*ذج ا2 وا،92/ وا9:اه ا2O ا 9:) ا9 U`*ت اS ا د9#2 J فK ف *ض2 ا ارس6 ه3= xɺ = −0.5 x + ay ,
a < 0.4
yɺ = x − 0.5 y
9ء اi= N/P T*/و a x xɺ −0.5 yɺ = 1 −0.5 y a −0.5 A= −0.5 1 −0.5 − λ det ( A − Iλ ) = det 1
( −0.5 − λ )
2
=0 −0.5 − λ a
−a =0
λ1 = − a − 0.5 λ2 = a − 0.5
:VenSim ?S# kIA ذج ا# %#0 x dx/dt b
a
y dy/dt
c
(01) a = -0.3
Units: **undefined**
(02) b = -0.5
Units: **undefined**
(03) c = -0.5
Units: **undefined**
112
(04) "dx/dt" = b*x+a*y
Units: **undefined**
(05) "dy/dt" = x+c*y Units: **undefined** (06) FINAL TIME = 10
Units: Month
The final time for the
Units: Month
The initial time for the
simulation. (07) INITIAL TIME = 0 simulation. (08) SAVEPER = TIME STEP Units: Month
The frequency with
which output is stored. (09) TIME STEP = 0.0625
Units: Month
The time step for the
simulation. (10) x = INTEG ( "dx/dt",
1)
Units: **undefined**
(11) y= INTEG ( "dy/dt",
1)
Units: **undefined** a = −0.3
1 2 0.06 0.6
3
3
1
3
3
3
3
3
3
3
2 3
2 41
-0.2 -0.08 -1 -0.6
2 2
1
2
4
4
4
2 1
0
4
4
3 4 1
4
4
4
4
1 2
1
3
4
2
5 6 Time (Month)
2
2
7
1
1
1
1
1
2
8
1 2
2
9
10
1 1 1 1 1 1 1 1 1 1 x : Current 2 2 2 2 2 2 2 2 2 2 y : Current 3 3 3 3 3 3 3 3 3 "dx/dt" : Current 3 4 4 4 4 4 4 4 4 4 4 "dy/dt" : Current
a = 0.3
113
2 4 0.06 0.6
3
3
3
3
3
3
3
3
3 3 3
4 3
0.8 0 -0.2 0
2
2
2
2
2
2 4
4
1 1
0
1
1
1
4
2
1
3
41
4 1
4
4
5 6 Time (Month)
4
4
4
4
1
1
1
1
1
2
2
2
2
2
2
7
8
9
10
1 1 1 1 1 1 1 1 1 1 x : Current 2 2 2 2 2 2 2 2 2 2 y : Current 3 3 3 3 3 3 3 3 3 "dx/dt" : Current 3 4 4 4 4 4 4 4 4 4 4 "dy/dt" : Current
a = −0.3, b = −0.5, c = 0.5
20 40 4 8
1
2 2 1
3 4
-20 -40 -4 -8
2 3
12
2 1
34 12
2
3
4
4
1
23
4
1
3
4
34
1
4
1
3
4
1
2
3 3 2
2
1 4
2
3
1
4 3
0
23
46
69
92 115 138 Time (Month)
161
184
207
230
1 1 1 1 1 1 1 1 1 1 x : Current 2 2 2 2 2 2 2 2 2 2 y : Current 3 3 3 3 3 3 3 3 3 "dx/dt" : Current 3 4 4 4 4 4 4 4 4 4 4 "dy/dt" : Current
a = −0.3, b = −0.3, c = 0.5
114
800 B 2e+012 200 B 400 B
4 4 123412341234123412341234123412341234123
3 12
-0.8 Tr -2e+012 -0.2 Tr -0.4 Tr 0
23
46
69
92 115 138 Time (Month)
161
184
207
230
1 1 1 1 1 1 1 1 1 1 x : Current 2 2 2 2 2 2 2 2 2 2 y : Current 3 3 3 3 3 3 3 3 3 "dx/dt" : Current 3 4 4 4 4 4 4 4 4 4 4 "dy/dt" : Current
a,b,c N/ 9` *) ; 9 U`*ت اS ا د9#2 J فK 3 # ربJK H أ:8 K
x dx/dt a
y r
z dz/dt b
115
dy/dt
xɺ = − a ( x − y ) yɺ = − xz + bx − y zɺ = xy − cz
a = −0.1, b = 0.02, c = −0.03
4 1 2 0.4 2 2
2
3 2
5
5
6
6
4 3
0 -1 -0.4 0 -2 -2
6 1
4
1
5
6
5
1
4
12
1
1
4 5
2
6
45 6
2
1 2
4
1
4
5
2
56
4 3
3
2
3
3
3 3
0 x : Current y : Current 2 z : Current 3 "dx/dt" : Current "dy/dt" : Current "dz/dt" : Current
1
2
3
1
1 2 5
5 6
5 6
5 6
116
3 4
5 6
2 3
4
4 5
6
10
1 2
3 4
9
1 2
3 4
8
1 2
3 4
7
1 2
3 4
6
1 2
3 5
4 5 6 Time (Second)
5 6
6
The Pumping Heart Model :k2/ اP 2GI 9 /0 ذج# -6 . Jاء اGH اQ H " إ9@ اJ آS 3Z ام اN;K 9)i ن آA اR L *ر# إ/ 8 *دد8 مO آR ; ا. Oscillator ا( آ *ددH 38;K N/P ) نA اR L 9H6 / 8 آ . ;K 9 أيSystole )ء و إ;ضK إر9 أيDiastole إ ط:م# N/P * Electro-Chemical
3@ وآT آ6` R ;ت اi# إ;ض وإ ط3= R *8
9 آv وR ; ا3= Muscle Fiber 9 i# 9` ل2D 2 هx أنU*=ذا إ+= . Stimulus R ; ا9 i# 9` ل2D 3= ;ضA ط واA أن ل ا9 رب اJ* اH =; و،6` ا 9` اJ V Y ا`قQ ( µ > 0 R K X ^ ) * ;8 و6` ا9 آQ دادG8 .9` ل ا2D Q L*K 6` ا9 أن ل آH آ و.T2Dو ت: " ا# * اR ; ا9 i# 9` 9ذج آ2 9 U`*ت اSن ا د2 آ:Sأو .9; ا :a@* اL و9* ; اVensim ذج2 ن2 آ:Y µ = 2 cm / sec, x ( 0 ) = 2 cm
v ( 0 ) = 1 microgrm, t = 0 ( 0.1) 100 sec
:Nا ;8 و6` ا9 آQ دادG8 R ; ا9 i# 9` ل2D 3= ;ضA ط واA أن ل ا:Sأو إذاT2D و9` اJ V Y ا`قQ ( µ > 0 R K X ^ ) * dx ( t ) = v ( t ) − µ x 3 ( t ) 3 − x ( t ) dt
إذا9` ل ا2D Q L*K 6` ا9 و أن ل آ dv ( t ) = − x (t ) dt
d N/P T*/و dx = v − µ ( x3 3 − x ) dt dv = −x dt
Vensim Q 9 # ذج#$ ا:%c
117
x dx
mu
v dv
(01) dv = -x
Units: microgrm/sec
(02) dx = v-mu*(((x^3)/3)-x) Units: cm/sec (03) FINAL TIME = 100
Units: Second
The final time for the
simulation. (04) INITIAL TIME = 0
Units: Second The initial time for the
simulation. (05) mu = 2
Units: cm/sec
(06) SAVEPER = TIME STEP
Units: Second
The frequency with which output is stored. (07) TIME STEP = 0.1
Units: Second
The time step for the
simulation. (08) v = INTEG (
dv, 1)
Units: microgrm
(09) x = INTEG (
dx, 2)
Units: cm
118
Current 1 x 4 2 1 0 -2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1 1
1
1
1
1
-4 dx 1
4 2 0 -2 -4
1
1 1
1
1
1
1
1
1
1
1
1 1
1
1
0
25
50 Time (Second)
75
100
(1) N/! 1
Current v 4 2 0 -2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
-4 dv 4 2 0 -2 -4
1
1
0
1
1
1
25
1
1
1
1
50 Time (Second)
1
1
1
1
75
100
(2) N/!
119
Heart Rate v Fiber Length 4
2
0
-2
-4 -3 dx : Current
-2
-1
1
1
1
0 x 1
1
1
1 1
1
2
1
1
1
3 cm/sec
1
(3) N/!
Heart Rate v Stimulus 4
2 1 1
1
1
1
0
-2
-4 -3 dx : Current
-2 1
-1 1
1
0 v 1
1
1
1 1
1
1
2 1
1
1
3 cm/sec
(4) N/!
120
Stimulus v Fiber Length 4
2
0
-2
-4 -3
-2
v : Current
1
-1 1
1
0 x 1
1
1
1 1
1
2
1
1
1
3 microgrm
1
(5) N/!
Stimulus v Heart Rate 4 1
1
1
1
1
1
1
2 1
0
-2
1
-4 -4 v : Current
-3
-2 1
1
-1 1
1
0 dx 1
1
1 1
1
2 1
1
3 1
1
4
microgrm
(6) N/! :a@* ا9PL Systole ;ضA( إ" ا9 82D )أفDiastole طA*;ل اA( أن ا1) N/! 8
ي6 وا3` إ ب3= R *8S 3* ) ول0 ا3= ءd ;ضA اT= ثK (ةL )أف 121
B8ذي ا; (Rو # /آ 9# 9ا `+= 6ن ا0ف Q= Q8 N/P j;Kام إ" ا)رج 8 (2) N/! .ا*دد 2* 3= Oscilationي ا ` 6ول .5ZKا/!0ل )(3 إ" ) 9D gU2K (6ا*ددات اور "= 98ا *82ت: -1ل iت 9 i#ا; 2D U Rل ا(3) N/! 9 i -2ل iت 9 i#ا; U Rآ 9ا `(4) N/! 6 -3آ 9ا `2D U 6ل ا(5) N/! 9 i -4آ 9ا ` U 6ل iت 9 i#ا; (6) N/! R :8 K H+ L -1اء JKرب " #ا 2ذج ) ون 2ف وث 2D0 (9 L 9*/ال )* `9 i 9 ا; Rوآ ت )* ` 6` 9وLرن ا*@a 8 3* -2ث إTHد 9 iا; Rو #اي *82ت `6؟ -3ه X Y YfK 2ا* µ R؟ -4د #أي *2ى ا ` 6أو #أي K µ X ^ 9 Lث 9 L 9*/؟ )أي lL2*K 9 i#ا; # Rا*دد 2 Rف او =ح =(3@J
122
:Lorenz Attractors Models $ ذج !ذت ر# -7 ا إدوارد9: ا2 رت2D ت ا دةS ا د9: 9#2 J 3 هc*8ر2 ذ تH ZK أن أي2J ا# B* 9 0 ا9 /P ا.ر/*KS 3* ا2J ط ا0 $* ء دراY أc*8ر2 * د#0 ا# 9# 8 ي6( وا9!ح =اH ` " 8 YfK ) 9و0 ط ا0 ا3= d 2و .9 /# أو9 رi* a@* "دي إB8 9و0وط اP ا3 # ا س \ 9 U`K تSث دY ن2/*8 3م آO رN/! # رة# 2 هc*ر2 9 ذH Z* ا ا6 ه6 إذا ا، واZ* 9TJ* 9 دا2ة ه6ت هS ا دN ،"و0 ا9H ار9: 3= 98 زاوT وG ن2/ ا ار. ر ارQ**8 Nن ا+= G ا$ " أ س ا# :3* آ3ت هS ا د. أ د9YY d x ( t ) = −ax ( t ) + ay ( t ) dt d y ( t ) = bx ( t ) − y ( t ) − z ( t ) x ( t ) dt d z ( t ) = −cz ( t ) + x ( t ) y ( t ) dt
Vensim S& ذج ا#$ ا%#0 x dx/dt a
y
b
dy/dt
z dz/dt
c
dx/dt=a(y-x)
a=10
x(0)=0
dy/dy=bx-y-xz
b=28
y(0)=0. 1
start time=0
c=2.67
z(0)=25
end time=100
dz/dt=xy-cz
dt=0.02
123
(01) a = 10
Units: **undefined**
(02) b = 28
Units: **undefined**
(03) c = 2.67
Units: **undefined**
(04) "dx/dt" = a*(y-x) Units: **undefined** (05) "dy/dt" = b*x-y-x*z
Units: **undefined**
(06) "dz/dt" = x*y-c*z
Units: **undefined**
(07) FINAL TIME = 100
Units: Second
The final time for the
Units: Second
The initial time for the
simulation. (08) INITIAL TIME = 0 simulation. (09) SAVEPER = TIME STEP Units: Second The frequency with which output is stored. (10) TIME STEP = 0.02
Units: Second
The time step for the
simulation. (11) x = INTEG ( "dx/dt",
0)
Units: **undefined**
(12) y = INTEG ( "dy/dt",
0.1) Units: **undefined**
(13) z = INTEG ( "dz/dt",
25)
Units: **undefined**
124
40 40 60 400 600 600
2
6
1 56
-40 -40 0 -400 -600 -600
1
1
1
4
4
12
2
4
5
3
56
2
5
5 2
6
3
4
456
345 1
3
4
6
5
12
2
2
2 1
1 6
4
56
6
3
5
12
3
3 4 3
3
4
3
0
1
2
3
4
5 6 Time (Second)
7
8
9
10
1 1 1 1 1 1 1 1 1 x : Current 2 2 2 2 2 2 2 2 2 y : Current 3 3 3 3 3 3 3 3 3 z : Current 4 4 4 4 4 4 4 4 "dx/dt" : Current 4 5 5 5 5 5 5 5 5 5 "dy/dt" : Current 6 6 6 6 6 6 6 6 "dz/dt" : Current
1
Current x 40 20
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
0
1
1
1
1
1
1 1
1
1
1
1
1
1
-20 -40 "dx/dt" 400 200 1
0
1
1
1
1
1
1
1
1
1
1
-200 -400
0
2.5
5 Time (Second)
125
7.5
10
1
Current y 40 20 0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1 1
1
-20 -40 "dy/dt" 600 300 1
0
1
1
1
1
1
1
1
1
1
1 1
-300 -600
0
2.5
5 Time (Second)
126
7.5
10
1
Current z 60
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
45 1
1
30 1
15
1 1
1
1
1
1
1
1
1
1
1
1
1
0 "dz/dt" 600 300 1
0
1
1
1
1
1
1
1
1
1
1
1
1
-300 -600
0
2.5
5 Time (Second)
7.5
10
xy-plane 40 1
1
1
1
20
1
1 1 1
0
-20
-40 -19 y : Current
-9 1
1
2 x 1
1
1
1
12 1
1
1
127
1
22 1
1
1
1
xz-plane 60
45 1 1
30 1
15
1 1 1
1
1
0 -19
-9
z : Current
1
1
2 x 1
1
1
12
1
1
1
1
1
22 1
1
1
1
yz-plane 60
45
30 1
15
1 1
1
1
1
1
1
0 -24
-20
-16
-12
-8
-4
0
4
8
12
16
20
24
28
y z : Current
1
1
1
1
1
1
1
1
128
1
1
1
1
1
1
# -8ذج ا#س وا: Prey and Predator Model % G ه6ا 2ذج Oم 2/*8ن 2رد و * ] Tوا6ي " # d;= 8ه6ا ا 2رد .ا 2ذج ه6ا * 8ض إ *Gاف ا 2ارد وYfKة " #ا Q *Jا 3 `:ا6ي 2/8 L ^ = .T # 8ن ا 2رد Q:Lا0را Rوا * Q:L ] Tا^ Y/*K ،Rا0را Rدورة ز 9ى ل # Q R *8ده = 3اورة ا ; 9O Q 9أن ا 2ارد ا *=2ة راg KS R fن 8أآ^ 500أر*;8 .Rت " #ه 56ا0را Q *J Rا^ Rوا6ي 2 8وG8داد آ آن هك #د آف آ ا0را .Rرس Oم ا دSت ا`و 9Lا* lK 3ه6ا اOم ا د 9اآ 9راR r rk +1 = rk + g 1 − k rk − 0.001 rk f k 500
ا د 9اآR^ 9 f k +1 = f k + 0.001 rk f k − 0.02 f k
ا د 9اآ 9را K Rأن ا0راGK Rداد ل # Q R *8ده " #ا# 8G8Sده 500 #أر Rو; L*Kار # 0.001د ا^ .Rا 9اآ K R^ 9ان ا^G8 Rداد #ده * # 0.001 Qد ا0را Rو; L*Kار # 0.02ده آN دورة. %#0ا #$ذج 8&fام :Vensim ا aا*)* + 3ام l8 Vensimآ 9اOم
Rabbits change changeRate Foxes foxesChange
)R(t)=R(t-1)+G[1-R(t-1)/500]R(t-1)-0.0001R(t-1)F(t-1 )F(t)=F(t-1)+0.0001R(t-1)F(t-1)-0.02F(t-1
129
0 200 400
3
1
3
3
1
3
3
3
3
1
3
3
1
3
1
3
3
1
3
1
-1 100 300
3 3
1 1 1
-2 0 200
2
0
2
10
change : Current Foxes : Current Rabbits : Current
2
2
2
2
2
2
2
2
1
2
2
2 1
1 1
20
30
1
1 2
40 1
2 3
50 60 Time (Day)
1 2
3
1 2
3
1 2
3
1 2
3
70 1
2 3
1 2
3
change
3
changeRate
(Rabbits)
Rabbits foxesChange
Loop Number 1 of length 1 Rabbits change Loop Number 2 of length 3 Rabbits foxesChange Foxes Change
130
1 2
3
Foxes
1
90
1 2
Rabbits
change
1
80
Foxes (Rabbits)
2
2
2
1 2
3
100 1 2
3
3
(01)
change = changeRate*(1-Rabbits/500)*Rabbits-0.0001*Rabbits*Foxes
(02)
changeRate = 0.01
(03)
FINAL TIME = 100 Units: Day
(04)
Foxes = INTEG ( foxesChange, 20)
(05)
foxesChange = 0.001*Rabbits*Foxes-0.02*Foxes
(06)
INITIAL TIME = 0 Units: Day
(07)
Rabbits = INTEG ( change, 400)
(08)
SAVEPER = TIME STEP Units: Day
The final time for the simulation.
The initial time for the simulation.
The frequency with which output is stored.
(09)
TIME STEP = 1 Units: Day
Current 1 Rabbits 400 1
1
1
1
1
1
1
The time step for the simulation.
1
1
1
1 1
350
1 1
300 250 200 change 0
1 1
-0.5
1 1
-1
1
-1.5
1 1
-2
0
25
50 75 Time (Day)
100
131
Graph for Rabbits 1
400
1
1
1
1
1
1
1
1
1 1
1
350
1 1 1 1
300
250
200 0
10
Rabbits : Current
20 1
30 1
40 1
50 60 Time (Day)
1
1
1
1
70 1
80 1
1
90 1
1
100 1
: %99 ت اFaIأ
(Foxes) foxesChange
Foxes
Rabbits
change
Rabbits
Foxes foxesChange
(Foxes) : %2ورات ا8ا
Loop Number 1 of length 1 Foxes foxesChange Loop Number 2 of length 3 Foxes change Rabbits FoxesChange
132
1
Current Foxes 200
1
1
1
1
1
1
1
150 1
100
1 1 1
50 1
1
1
1
0 foxesChange 2 1.5 1
1
1
1
1
1 1
0.5 0
1
1
0
25
50 75 Time (Day)
100
Graph for Foxes 200
150
100
50 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0 0
10
Foxes : Current
20 1
30 1
40 1
1
50 60 Time (Day) 1
133
1
1
70 1
80 1
1
90 1
1
100 1
: 1
ا 9 0وا HAت *رات ا ;9
134
ا ا ا ث ا ت2 ء وA اL V 203 ء ا ذج: ا دة 9 ` ل ا# ولS *ر اSا هـ1422/1421 3^ اN`ا *# : Gا :9* ا9 0 اQ H 3 # RHأ :ال اولr ا :T N/ S^ 3:# وأ9* ت ا: ف ا# ب ا)رجA )د( أInflow N ب ااA )ج( أFlow بA )ب( أStock )أ( ا *ع 9` 986ZK )ز( دورةSink (9#2ر )ا2Z )و( اSource )هـ( ا رOutflow System مO ا9H6 ( )طAuxiliary Variable # اZ* )ح( اFeedback Loop Dynamic Mathematical Model (3/8 *ك )د3U8ذج ر2 ( )يModeling
( =;ةN/ 9# l _ ت# 9 ) :ال اr ا د ا زل# 3 # * !ة8 تH^ ا$ Q8 ي6 أن ا ل ا9G تHY زع2 S إذا.تHY ] K 3*د ا زل ا# د8زد+ L*8 ا ا ل6ت وأن هHY T H28S 3*ا وT " إ*آ# ي2K / و9HY ] K S 3* زل ا3 %0 اد اN^ 8 H أنU*=إ .9HY ] K 3*د ا زل ا# x G وة ا6 ) م اO ا9H6 Difference Equation 9L ا`و9)أ( أن ا د :3م( ه2ا xt = xt −1 + k ( H − xt −1 )
(l* و#) وH = 20000 زل3 %0ذج د ا2 ا ا6 هN ، t = 0 # x = 0 )ب( إ=*ض أن (l* و#) t = 0,1, 2,⋯,10 ; ، k = 0.05 R K X Y
135
)ج( #ا *ع و ا Aب وا *Zات ا #ة وا ر ودورة ا* 986Zا) `9#) 9 واة( )د( أر !* 3::)K N/ع وأ Aب ، Stock and Flow Diagramو`)* + 56ام Vensimو ) .]J@* NJأر # 9ت(
136
ا HAت ا 2ذ:9H )– 1أ( ا *ع Stockأي ! Q Z*8ا) Gد* ،3/8ك( G8داد و ;8و " 8اi8 *2ى Levelاو * State Variable 9 Zو N^ 8آ*:3
Stock Source/Sink Flow
Information Link
AuxiliaryVariable
أ^ :9آ 9ا G 3= 5ان# ،د ا T 0ا* ^* T/ * 8 3ا )-1ب( ا 0ب Flowه 2ل 8G8 Rateأو * ;8ع .أ^ :9آ 9ا 5ا G) 986Zان او ا # ،$ 9 2د ا T 0ا *Pاة أو ا 9# )-1ج( أ Aب اا Nه 2ل 8G8ا *ع .أ^ :9آ 9ا 5ا G) 986Zان# ،د اT 0 ا *Pاة )-1د( أ Aب ا)رج ه 2ل ;8ا *ع .أ^ :9آ 9ا 5ا 9 2ا)Gان# ،د ا T 0ا 9# )-1هـ( ا ر ه 2ا Qاو ا N%0ا6ي 6Z8ي ا *ع .أ^ G :9ان 9ا ة ،اT 0 ا :و* 9اول )-1و( ا2Zر )ا (9#2ه 2ا;ع او ا = او ا Jري ا6ي ;8ا *ع .أ^ :9ا Jري او اف ا 3ة ا)ر 9Hا)Gان ،ا T 0ا :و* 9دل )2/8 L :9Oن ا ر وا2Zر ` cا Pآ = 3ا ^ل ا( 0 )-1ز( دورة 9` 986ZKه 3ا* T= YB8 3ا *ع " #ا 0ب ا 6Zي .$أ^ :9ا)Gان ا 6Zى 5ل ا2ب ،9=D $ /*K 5آ Q`KKا ة = 3ا)Gان $ Q`KK ا 9=:و N;Kآ 9ا 5اا 9 )-1ح( ا * Zا #ه 3= N 8 Z* 2و l%و N #اOم و 3= YB8ا Aب وYf*8 L *ع .أ^ :9ا 9ا;2ى )Gان ،رأس ا ل ا *ح ^* Aر 137
lK 9U8ت رL# N^ .مO اN # l8 3U8ذج ر2 QU و،مO ا9H6 (ط-1) T 0 " ا# R : اض وا9 lK 3 L *دLت إS د،5 انG)ل إ*ء ا ن2/*8 وG اQ Z*K تS دl%28 ي6 ا2( ه3/8 *ك )د3U8ذج ر2 (ي-1) 9 !*;قA " ا# لK Gق ا2= 9:; )اxɺ = kx N^ .9L او =و9 U`K تS د 3 ا2 9 3= مO N^ Kو" وS ا9H ار9 U`K 9 د3( وهG (أ-2) x ( t ) − x ( t − dt ) ∝ H − x ( t − dt ) dt =k H − x ( t − dt ) x ( t ) − x ( t − dt ) = k H − x ( t − dt ) dt x ( t ) = x ( t − dt ) + k H − x ( t − dt ) dt = x ( t − 1) + k H − x ( t − 1)
،9L =و9 إ" دT2 Y (92T ) 9 U`K 9 أ ء د2 Ni=0!*;ق اA ا3= xt = xt −1 + k ( H − xt −1 )
(ب-2) xt = xt −1 + 0.05 ( 20000 − xt −1 ) , x0 = 0, t = 0,1, 2,⋯ ,10 x1 = x0 + 0.05 ( 20000 − 0 ) = 1000
x2 = x1 + 0.05 ( 20000 − x1 ) = 1000 + 0.05 ( 20000 − 1000 ) = 1950 x3 = 1950 + 0.05 ( 20000 − 1950 ) = 2852.5 x4 = 3709.875
⋮ x 9HY ] K 3*د ا زل ا# 2ج( ا *ع ه-2) xt − xt −1 تH^ ا$ عK ي6 ا ل ا2 ب هAا H Tى إ*آ2K / و9HY ] KS 3* زل ا3 %0 اد ا# اZ* ا
N ب ااA واxt − xt −1 ;8 3* وH − x ;8 x دادG8 آ9` ) ا986Z*دورة ا .ا6/ وهx 8G8 ا6T و9 9 ن آ2/8 ان/ 8S (د-2)
138
x dx/dt
k
H
(x) H
dx/dt
x
k
x
dx/dt
(x)
Loop Number 1 of length 1 x dx/dt
139
1
Current x 20,000 15,000
1
1
1
1
1
1
1
1
1
1
1
10,000 5,000 1
0 "dx/dt" 1,000 750
1
500 250 0
1 1
0
50 Time (Day)
100
Graph for x 20,000
1
1
1
1
1
1
1
1
1
1
1 1
15,000 1 1
10,000 1
5,000
0
1
0
10
x : Current
x
1
0
20 1
30 1
1
1000
40 1
50 60 Time (Day) 1
1
1950
140
1
70 1
1
80 1
2852.5
90 1
1
100 1
3709.88
4524.38
5298.
ا ا ا
9Hا ] 2د آ 9ا 2م LاAء و 2ث ا ت ا* Aر ا^ #0 3ل ا` Nا^ 1422/1421 3هـ دة ) V 203ء ا ذج ( ا_ 4& 2 :6 أ Q H " # RHا 9 0ا*:9 ا Bال ا0ول: Nا دSت ا`و 9Lا* ; 9ا0و 9ا :ة /Kارt = 1, 2,...,15 ; (Iterate) 8 x0 = 5, x1 = −1 x0 = 3, x1 = −15
xt + xt −1 − 2 xt −2 = 0, xt + 6 xt −1 + 9 xt −2 = 0,
و= 3آ 9 Nأر . t = 1, 2,...,15 Q xt ا Bال ا^:3 هك ا ^/ا2Oاه ا N^ 9:ا2Oاه ا 982وا 9H2وا 9# *HAوا*LAد 98ا ) 2ت اآ^ إLأ 2/K ( Fibonacci Numbers #ن = Tآ O 9م د # 3/8اG ! *K tة 3 #آ 9اOم #ا0ز t − 1 9و += . t − 2ذا آN^ K T V xt , t ∈ T X =*ة ز ;K ،9 9أن ا د 9ا`و 9Lار 9Hا^ 9وا* lK 3د 9/8اOم #أي 9Oز 9ه:3 xt − xt −1 − xt −2 = 0, ∀t ∈ T
)أ( xt = λ t , λ ≠ 0 QU2أن ا د 9ا #ة " # R*/Kا λ 2 − λ − 1 = 0 N/PوT ا Nا*:3
141
1+ 5 ≈ 1.6180339 2 1− 5 λ2 = ≈ 0.6180339 2
λ1 =
N/P ا3 # R*/K xt )ب( أن t
t
1+ 5 1− 5 xt = A + B , ∀t ∈ T 2 2
:N/P ا3 # R*/K xt ن+= 3* وB = xt =
(
) (
−1 1 وA= ن+= x1 = 1 وx0 = 0 9و; أو 5 5
)
t t 1 1 + 5 − 1 − 5 , t = 0,1, 2,... 2 5
(T = {0,1, 2,...})
:QU2 ] وذt = 1, 2,..., 25 ; xt H أوExcel *)ام+ ()ج A1=1 A2=1 A3=A1+A2 . . . A(n)=A(n-1)+A(n-2) A4-A25 8) إ *)م ا وا ا:9O
; انK B1=A2/A1 QU2 B(n) → λ1 ≈ 1.6180339, As n → ∞
; انK C1=A1/A2 QU2 و C(n) → λ2 ≈ 0.6180339, As n → ∞
n=30 6 :9O
:V^ال اB ا
142
Markovian 9=2 ا آ9%) اT 9 98*دL*د أن هة إLAل أ اء ا2;8 ()أ
؟38 = ذا9^ ا9H ارProperty 9 ا دR 9^ ا9H ار9=2 ا آ9%) اQ*8 9 !آT `لLA)ب( ا 9Lا`و xt − axt −1 − bxt −2 = ε t , ∀t ∈ T , ε t ~ N ( 0, 4 )
(Second Order one State
وا9 Z* 9Y 9H در9 ا د56ل ه2
]( وذFirst Order two State Variables) 9 ىZ* " أو9H إ" درVariable) . yt −1 = xt −2 QU2 ; (Steady State) 5م وأ * إ *;ارOا ا6T ذج2 ن2 آVensim *)ام+ ()ج 1) a = 1.2, b = −0.7 2) a = −1.2, b = −0.7
9* إ *)م ا; ا:9O RANDOM NORMAL(-3.99,3.99 , mean ,standard deviation ,seed=Prime Number ) INITIAL TIME = 0 FINAL TIME = 2 SAVEPER = TIME STEP TIME STEP = 0.01
143
ا ا ا V 203 *رA 9/ ل ا2 أ ا :ول0ال اB $ Hإ xt = − xt −1 + 2 xt − 2 , x0 = 5, x1 = −1 5,-1,11,-13,35,-61,131,-253,515,-1021,2051,-4093,8195,-16381,32771
30000
C2
20000 10000 0 -10000 -20000 Index
5
10
15
xt = −6 xt −1 − 9 xt −2 , x0 = 3, x1 = −15 3,-15,-39,-57,-21,-93,51,-237,339,-813,1491,-3117,6099,-12333,24531
20000
C3
10000
0
-10000 Index
5
10
15
:3^ال اB 9 Hإ
144
أي9f ق ا2: R xt −2 وxt −1 3 # *K xt 8 xt = xt −1 + xt −2 xt − xt −1 − xt −2 = 0, ∀t ∈ T
J xt = λ t , λ ≠ 0 QU2 ()أ λ t − λ t −1 − λ t −2 = 0
λ t −2 ( λ 2 − λ − 1) = 0
∵λ ≠ 0 ∴λ 2 − λ − 1 = 0
λ1,2 = λ1 =
if
1± 1+ 4 2
ax 2 + bx + c = 0, then x1,2 =
−b ± b 2 − 4ac 2a
1 5 1 5 + ≈ 1.6180339, λ2 = − ≈ 0.6180339 2 2 2 2
2 ه9L ا`و9 ام دNن ا+= 9` *) و9;; ة# ا9ور ا دGH )ب( ان xt = Aλ1t + Bλ2t , ∀t ∈ T t
t
1+ 5 1− 5 xt = A + B , ∀t ∈ T 2 2 ∵ x0 = 0, x1 = 1 ∴ x0 = A + B = 0 ⇒ A = − B 1+ 5 1− 5 and x1 = A + B =1 2 2 1+ 5 1− 5 ∴− B + B =1 2 2 −1 1 ∴B = ⇒ A= 5 5 t t 1 ∴ xt = 1 + 5 + 1 − 5 , t = 0,1, 2,... 2 5
(
) (
)
()ج A
B
C
1
1
1
1
1
1
2
2
0.5
3
1.5
0.666666667
5
1.666666667
0.6
145
8
1.6
0.625
13
1.625
0.615384615
21
1.615384615
0.619047619
34
1.619047619
0.617647059
55
1.617647059
0.618181818
89
1.618181818
0.617977528
144
1.617977528
0.618055556
233
1.618055556
0.618025751
377
1.618025751
0.618037135
610
1.618037135
0.618032787
987
1.618032787
0.618034448
1597
1.618034448
0.618033813
2584
1.618033813
0.618034056
4181
1.618034056
0.618033963
6765
1.618033963
0.618033999
10946
1.618033999
0.618033985
17711
1.618033985
0.61803399
28657
1.61803399
0.618033988
46368
1.618033988
0.618033989
75025
1.618033989
0.618033989
121393
1.618033989
0.618033989
196418
1.618033989
0.618033989
317811
1.618033989
0.618033989
514229
1.618033989
0.618033989
832040
1.618033989
0.618033989
أن B ( n ) → λ1 ≈ 1.6180339, As n → ∞
C ( n ) → λ2 ≈ 0.6180339, As n → ∞
:V^ال اB 9 Hإ Markovian Property of order k 9H ار9=2 ا رآ9%))أ( ا
9ز0 ا# T L " # *K t G ا# 9@ا2Pهة اO ا9 L " ان# ل2;K ن+= { yt , −∞ < t < ∞} X = ^ إذا آ.d;= t − 1, t − 2,..., t − k
146
(
)
P yt < s | yt −1 , yt −2 ,..., yt −k , yt −( k +1) ,... = P ( yt < s | yt −1 , yt −2 ,..., yt −k )
()ب xt − axt −1 − bxt −2 = ε t , ∀t ∈ T , ε t ~ N ( 0, 4 )
∵ yt −1 = xt −2 ⇒ yt = xt −1 ∴ xt = axt −1 + byt −1 + ε t yt = xt −1
( )ج
x dx
a
eps b y dy
(01) a = 1.2
(02) b = -0.7
(03) dx = a*x+b*y+eps
(04) dy = x
(05) eps = RANDOM NORMAL(-3.99,3.99 , 0 ,2 ,19 )
(06) FINAL TIME = 2
147
(07) INITIAL TIME = 0
(08) SAVEPER = TIME STEP
(09) TIME STEP = 0.01
(10) x = INTEG ( dx, 0)
(11) y= INTEG ( dy, 0)
Current x 0.2 0.1 0 -0.1 -0.2 dx 4 2 0 -2 -4
0
0.50
1 Time (Day)
1.50
2
;* \ مOا
(01) a = -1.2 148
(02) b = -0.7
(03) dx = a*x+b*y+eps
(04) dy = x
(05) eps = RANDOM NORMAL(-3.99,3.99 , 0 ,2 ,19 )
(06) FINAL TIME = 2
(07) INITIAL TIME = 0
(08) SAVEPER = TIME STEP
(09) TIME STEP = 0.01
(10) x = INTEG ( dx, 0)
(11) y = INTEG ( dy, 0)
149
Current x 0.2 0.1 0 -0.1 -0.2 dx 4 2 0 -2 -4
0
0.50
1 Time (Day)
1.50
2
;* مOا
150
ا ا ا
ﻗﺴﻢ ﺍﻹﺣﺼﺎﺀ ﻭﺑﺤﻮﺙ ﺍﻟﻌﻤﻠﻴﺎﺕ ﻛﻠﻴﺔ ﺍﻟﻌﻠﻮﻡ ﺟﺎﻣﻌﺔ ﺍﻟﻤﻠﻚ ﺳﻌﻮﺩ
هـ1422/1421 3^ اN` 3@T *ر اAا ( ) ء ا ذجV 203 دة ت4& 3 6 _ا :9* ا9 0 اQ H " # RHأ :ول0ال اB ا :9* اLinear Differential Equations 9:) ا9 U`*ت اS ا د9#2 J xɺ = −0.5 x + ay ,
a < 0.4
yɺ = x − 0.5 y
ةG ور اGJ اH وأوxɺ = Ax State Space 9ء اi= N/! " # T*)أ( أآ .9 U`*ت اS *;ا د3:K ة: اa 9 L و أنA 9=2` Eigenvalues 9#2 J 9=/ اLinear Difference Equations 9:) ا9Lت ا`وS ا دR*)ب( أآ طP اN ه. x i +1 = Axi , i = 0,1, 2,… State Space 9ء اi= N/! " # T* وأآ9; ا .i8 *;ا ه ا3:8 a 9 L " # ; وx0 = 1, y0 = 1 9و0 ; ا9Lت ا`وS ا د9#2 J N Excel *)ام+ ()ج . a = 0.3 وi = 1, 2,… , 20 :3^ال اB ا Drilling رة2` ر اb )اN ب دا+ دادG8 *ولN; 3= x (Wells رbد ا#) d
(Drilling Fraction ` ا9 ) 3 # ر وbد ا# 3 # *K 3* واxɺ ( Wells
YfK) وn (Normal Drilling Fraction 98 ا` اد9 ) بU N% 2ي ه6وا ا6 وهe ( Effict of Reserves on Drilling Fraction ` ا9 " # 3D*Aا 3;* اr
(Petroleum Reserves ا*ول3D* )إ9H2 9L# 3:8 0ا 151
3D*A 3ى ا2* ا9 # رة# 2 ا*ول ه3D* إ3;* ء اGJ ا.` ا9 و ; واi
(Initial Petroleum Reserves ا*ول3D*A 9و0 ا9 ;ا*ول ا" )ا
/ ا*ول ا9 آX L ا*ول آ3D* إNL ;ار أ3;K ان آ38 9 ا56 ه3= .;K ` ا9 ن+= ا6/ وهTH إ ا3= 98*دLA ا9 ; اX L وآTHإ *)ا 3 ه3* وا− xɺ ( Closing Wells 9; Z ر اb )ا2 ب رج ه+ L*8 رbد ا# *)اجA اYfK) وc (Normal Closing Fraction 98\ق ادA ا9 ) بU N% *)اجA اYfK . f (Effect of Extraction on Closing Fraction \قA ا9 3 # (Extraction per Well N/ *)اجA )ا3 # Yf* اK 9 دا2\ق هA ا9 3 # m (Maximum Extraction per Well N/ O#0 *)اج اA )اQ 9 ;ر
p
3K N/ O#0 *)اج اA إ" اN/ *)اجA ا9 3= ;f= . \قA ا9 3 # دادGK ا6T و98*دLA ا5واH ا مN`L Ni=0 ا$ وا9 2% ^ أآg% *)اج اAأن ا YfK) وN/ O#0 *)اج اAب اU N% 2 هN/ *)اجA ا.\قA ا9 ( Effect of Reserves on Extraction per Well N/ *)اجA " ا# 3D*Aا 9 XLK /= .3D*A ا*ول ا *)ج وا9H2 9L# ي8 0ا ا6 وهh .L*8 N/ *)اجAن ا+= 3* و9 2% ^ اآg8 *)اجAن ا+= ا*ول3D*إ .N إ ب دا$ c ا*ول3D*ى إ2* ن+= دJ* \ ر2ن ا*ول ه0أ ا و g ( Extraction *)اجA )ا9 / ا9 / ا2 ا*ول ه3D* إrɺ ب ا)رجAا
. رbد ا# وN/ *)اجAب اU N% 3 ه3* وا982 ا 9Lت ا`وS ا دR*م وأآOا ا6T (9 تL# ) ى وإ ب2* d:) )أ( أر .$= /*K 3* اDifference Equations Vensim *)ام+ *)اج ا*ولA ذج2 ن2)ب( آ
:V^ال اB ا . Jاء اGH اQ H " إ9@ اJ آS 3Z ام اN;K 9)i ن آA اR L *ر# إ/ 8 *دد8 مO آR ; ا. Oscillator ا( آ *ددH 38;K N/P ) نA اR L 9H6 / 8 آ . ;K 9 أيSystole )ء و إ;ضK إر9 أيDiastole إ ط:م# N/P * Electro-Chemical
3@ وآT آ6` R ;ت اi# إ;ض وإ ط3= R *8
9 آv وR ; ا3= Muscle Fiber 9 i# 9` ل2D 2 هx أنU*=ذا إ+= . Stimulus
152
R ; ا9 i# 9` ل2D 3= ;ضA ط واA أن ل ا9 رب اJ* اH =; و،6` ا 9` اJ V Y ا`قQ ( µ > 0 R K X ^ ) * ;8 و6` ا9 آQ دادG8 .9` ل ا2D Q L*K 6` ا9 أن ل آH آ و.T2D( و98 د * و0* ا#)أ 9 i# 9` 9ذج آ2 Difference Equations 9 U`*ت اSن ا د2 آ:Sأو .9; ت ا: " ا# * اR ;ا :a@* اL و9* ; اVensim ذج2 ن2 آ:Y µ = 2 cm / sec, x ( 0 ) = 2 cm
v ( 0 ) = 1 microgrm, t = 0 ( 0.1) 100 sec
:Q ال ااB ا Logistic
9* H2 ا9 اT* آl%2K 3* ا9:اه ا2O^ ا/هك ا
3 ه ا2 g8 Y ( c f* ا9 ) : 2 أK هةlK 9 اا56 ه.Function ه2 lL2*8 Y (ai ا9 ) ه2 BD*8 Y (زدهرA ا9 ) `*ةExponential " إGK n0 ن+= n ( t ) G t G ا# هةO ا9 ; Gذا ر+= .( *;ارA ا9 ) 2 )وهM 2هة هO ا56T 2 اl; وإذا آنInitial Value 9@ او ا9و0 ا9 ;ا (98TS " إGول اB8 # $ K 9;; ا3= – هةO ا9 L $ اNK ان/ 8 "Lأ :3 ه9ز0 اN آ# هةO اlK 3* اDynamic Model 9 اآ9ن ا د+= n (t ) =
M , t≥0 1 + ( M − n0 ) n0 e − ct
.هةO ا56T Vensim d:) ن2)أ( آ :9*)ب( ات ا Year
Volume
Year
Volume
1984
0.0
1992
15.0
1985
1.5
1993
18.5
1986
2.0
1994
20.0
1987
2.5
1995
22.5
1988
4.0
1996
23.5
1989
5.0
1997
23.0
153
1990
8.5
1991
11.5
1998 1999
27.0 27.5
. c وn0 وM ر اL أ *)مy =
a N/P " ا# 9 ا د6 وExcel أوCurve Expert أ *)م:9O 1 + be − cx
.9 هJK ا ىL أو أيa0 = 30, b0 = 30, c0 = 0.5 9آ; او .a@* اLم وO ي آةH ا`;ة )أ( وأ3= d:) ا; ا ;رة آ6 ()ج . ب
dn ( t ) M − n (t ) = c× × n ( t ) 9L أ *)م ا:9O dt M
154
ا ا ا هـ1422/1421 3^ اN` 3@T *ر ا9 * تHإ V 203 ا دة :ول0ال اB 9 Hإ ( )أ a x xɺ −0.5 yɺ = 1 −0.5 y a −0.5 A= −0.5 1 −0.5 − λ det ( A − Iλ ) = det 1
( −0.5 − λ )
2
=0 −0.5 − λ a
−a =0
λ1 = − a − 0.5 λ2 = a − 0.5
. *;اNن ا2/8 !وط أن56 وهλ2 < 1 وλ1 < 1 أن آJ −0.4 < a < 0.4 ; ()ب xi +1 = −0.5 xi + ayi ,
a < 0.4, i = 0,1, 2,...
yi +1 = xi − 0.5 yi a xi xi +1 −0.5 , i = 0,1, 2,... y = 1 −0.5 yi i +1
.i8 ه ا:K 9; ا`;ة ا3= 9 HAن ا+= $ # وT ` 3 هState Matrix 9 ا9=2` ()ج x
y
1
1
=-0.5*A2+0.3*B2
=A2-0.5*B2
=-0.5*A3+0.3*B3
=A3-0.5*B3
=-0.5*A4+0.3*B4
=A4-0.5*B4
=-0.5*A5+0.3*B5
=A5-0.5*B5
=-0.5*A6+0.3*B6
=A6-0.5*B6
=-0.5*A7+0.3*B7
=A7-0.5*B7
=-0.5*A8+0.3*B8
=A8-0.5*B8
=-0.5*A9+0.3*B9
=A9-0.5*B9
155
=-0.5*A10+0.3*B10
=A10-0.5*B10
=-0.5*A11+0.3*B11
=A11-0.5*B11
=-0.5*A12+0.3*B12
=A12-0.5*B12
=-0.5*A13+0.3*B13
=A13-0.5*B13
=-0.5*A14+0.3*B14
=A14-0.5*B14
=-0.5*A15+0.3*B15
=A15-0.5*B15
=-0.5*A16+0.3*B16
=A16-0.5*B16
=-0.5*A17+0.3*B17
=A17-0.5*B17
=-0.5*A18+0.3*B18
=A18-0.5*B18
=-0.5*A19+0.3*B19
=A19-0.5*B19
x
y
1
1
-0.2
0.5
0.25
-0.45
-0.26
0.475
0.2725
-0.4975
-0.2855
0.52125
0.299125
-0.546125
-0.3134
0.5721875
0.32835625
-0.59949375
-0.34402625 0.628103125 0.360444063 -0.65807781 -0.377645375 0.689482969 0.395667578 -0.72238686 -0.414549847 0.756861008 0.434333226 -0.79298035 -0.455060718 0.830823401 0.476777379 -0.87047242 -0.499530415 0.912013589 0.523369284 -0.95553721
:3^ال اB 9 Hإ
156
x ( t ) = x ( t − dt ) + (u − v )dt , u = xɺ , v =< − xɺ > y ( t ) = y ( t − dt ) − wdt , w = yɺ
xɺ = bx,
yɺ = px, < − xɺ >= gx, g = cf
b = ne,
p = mh,
f = p m, e = y a, h = y a
()ب
c
n dx/dt
<-dx/dt> x
m
u
v f
g
p
b dy/dt y w
h
e a
b u (x) x g v (x)
157
x
w
y
u
(x)
v
(x)
p w
y
e
b
h
p
x
y
a h y
p m
c g m
f
158
:V^ال اB 9 Hإ ;8 و6` ا9 آQ دادG8 R ; ا9 i# 9` ل2D 3= ;ضA ط واA)أ( أن ل ا إذاT2D و9` اJ V Y ا`قQ ( µ > 0 R K X ^ ) * dx ( t ) = v ( t ) − µ x 3 ( t ) 3 − x ( t ) dt
إذا9` ل ا2D Q L*K 6` ا9 و أن ل آ dv ( t ) = − x (t ) dt
d N/P T*/و dx = v − µ ( x3 3 − x ) dt
dv = −x dt
Vensim a N^ ذج2 )ب( ا x dx
mu
v dv
(01) dv= -x Units: microgrm/sec
(02) dx= v-mu*(((x^3)/3)-x) Units: cm/sec
(03) FINAL TIME = 100
159
Units: Second The final time for the simulation.
(04) INITIAL TIME = 0 Units: Second The initial time for the simulation.
(05) mu= 2 Units: cm/sec
(06) SAVEPER = TIME STEP Units: Second The frequency with which output is stored.
(07) TIME STEP = 0.1 Units: Second The time step for the simulation.
(08) v= INTEG ( dv, 1) Units: microgrm
(09) x= INTEG ( dx, 2) Units: cm
160
Current 1 x 4 2 1 0 -2
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
-4 dx 1
4 2 0 -2 -4
1
1 1
1
1
1
1
1
1
1
1
1
1
1 1
0
25
50 Time (Second)
75
100
(1) N/! 1
Current v 4 2 0 -2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
-4 dv 4 2 0 -2 -4
1
1
0
1
1
1
25
1
1
1
1
50 Time (Second)
1
1
1
1
75
100
(2) N/!
161
Heart Rate v Fiber Length 4
2
0
-2
-4 -3
-2
dx : Current
-1
1
1
1
0 x 1
1
1
1 1
1
2
1
1
1
3 cm/sec
1
(3) N/! Heart Rate v Stimulus 4
2 1 1
1
1
1
0
-2
-4 -3 dx : Current
-2 1
-1 1
1
0 v 1
1
1
1 1
1
1
2 1
1
1
3 cm/sec
(4) N/!
162
Stimulus v Fiber Length 4
2
0
-2
-4 -3 v : Current
-2
-1
1
1
1
0 x 1
1
1
1 1
1
2
1
1
1
3 microgrm
1
(5) N/!
Stimulus v Heart Rate 4 1
1
1
1
1
1
1
2 1
0
-2
1
-4 -4 v : Current
-3
-2 1
1
-1 1
1
0 dx 1
1
1 1
1
2 1
1
3 1
1
4
microgrm
(6) N/! :a@* ا9PL Systole ;ضA( إ" ا9 82D )أفDiastole طA*;ل اA( أن ا1) N/! 8
ي6 وا3` إ ب3= R *8S 3* ) ول0 ا3= ءd ;ضA اT= ثK (ةL )أف 163
" ام إQ= Q8 N/P j;K ف0ن ا+= 6` ا9# 9 آ# /( وR ;ذي اB8 (3) ل/!0 ا.5ZK ول6` ي ا2* 3= Oscilation ا*دد8 (2) N/! .ا)رج :ت82* =" ا98 ا*ددات اور9D gU2K (6) "إ (3) N/! 9 iل ا2D U R ; ا9 i# تi ل-1 (4) N/! 6` ا9 آU R ; ا9 i# تi ل-2 (5) N/! 9 iل ا2D U 6` ا9 آ-3 (6) N/! R ; ا9 i# تi لU 6` ا9 آ-4 :Q ال ااB 9 Hإ ( )أ
M-n(t)
n(t) dn(t)/dt M
n(0) c
ا8;K ()ب Excel *)ام+ Sاو
A
B
C
D
c=
0.473488084598268
M=
27.218780487999
n0=
0.691567114138888
E
Year
Delta Year
Value
Logistic
Sq Error
1980
=A5-$A$5
0
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B5))
=(D5-C5)^2
1981
=A6-$A$5
1.5
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B6))
=(D6-C6)^2
1982
=A7-$A$5
2
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B7))
=(D7-C7)^2
1983
=A8-$A$5
2.5
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B8))
=(D8-C8)^2
1984
=A9-$A$5
4
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B9))
=(D9-C9)^2
164
1985
=A10-$A$5
5
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B10))
=(D10-C10)^2
1986
=A11-$A$5
8.5
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B11))
=(D11-C11)^2
1987
=A12-$A$5
11.5
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B12))
=(D12-C12)^2
1988
=A13-$A$5
15
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B13))
=(D13-C13)^2
1989
=A14-$A$5
18.5
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B14))
=(D14-C14)^2
1990
=A15-$A$5
20
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B15))
=(D15-C15)^2
1991
=A16-$A$5
22.5
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B16))
=(D16-C16)^2
1992
=A17-$A$5
23.5
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B17))
=(D17-C17)^2
1993
=A18-$A$5
23
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B18))
=(D18-C18)^2
1994
=A19-$A$5
27
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B19))
=(D19-C19)^2
1995
=A20-$A$5
27.5
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B20))
=(D20-C20)^2
Sum=
=SUM(E5:E20)
A
B
C
D
c=
0.473488085
M=
27.21878049
n0=
0.691567114
E
Year
Delta Year
Value
Logistic
Sq Error
1980
0
0
0.691567114
0.478265073
1981
1
1.5
1.093543839
0.165206611
1982
2
2
1.714073963
0.081753699
1983
3
2.5
2.650992755
0.022798812
1984
4
4
4.019328309
0.000373584
1985
5
5
5.923649251
0.85312794
1986
6
8.5
8.403403286
0.009330925
1987
7
11.5
11.36711781
0.017657676
1988
8
15
14.56684923
0.187619589
1989
9
18.5
17.66362481
0.699523461
1990
10
20
20.35934383
0.129127985
1991
11
22.5
22.49780667
4.81069E-06
1992
12
23.5
24.07261592
0.327888994
1993
13
23
25.16994472
4.708660072
1994
14
27
25.90542485
1.198094761
1995
15
27.5
26.38562558
1.241830348
Sum=
10.12126434
Curve Expert *)ام+ Y 165
166
167
3 ا; ا;رة هJو User-Defined Model: y=a/(1+((a-b)/b)*exp(-c*x)) 168
Coefficient Data: a=
27.218854
b=
0.69156932
c=
0.47348677 M = 27.218854, n0 = 0.69156932, c = 0.47348677
أي
()ج
M-n(t)
n(t) dn(t)/dt M
n(0) c
(01) c = 0.47
(02) dn(t)/dt = c*(M-n(t)/M)*n(t)
(03) FINAL TIME = 15
(04) INITIAL TIME = 0
(05) M = 27.22
(06) M-n(t) = INTEG ( -dn(t)/dt, 30)
169
(07) n(0) = 0.69
(08) n(t) = INTEG ( dn(t)/dt, 1)
(09) SAVEPER = TIME STEP
(10) TIME STEP = 1 Current "n(t)" 40 30 20 10 0 "dn(t)/dt" 6 4.5 3 1.5 0
0
3.8
7.5 11.3 Time (Month)
15
Current "M-n(t)" 40 30 20 10 0 "dn(t)/dt" 6 4.5 3 1.5 0
0
7.5 Time (Month)
15
170
M-n(t) (n(t)) dn(t)/dt
n(t)
c M
(M-n(t)) n(t) dn(t)/dt
M-n(t)
c M
M-n(t) n(t)
dn(t)/dt (n(t))
(M-n(t)) M-n(t)
dn(t)/dt n(t) :a@* ا9PL
أن ا *عV الB ا3= ف2%2 ا3* H2 ا*ف ا9JK ا9ت ا2 ا dn ( t ) dt Flow بA ا.lL2*8 Y 3 ا2 " إ52 ل2*8 Y ءd 2 8 n ( t ) Stock
L* اfD*8 وQ8 LKدة و8G ا3= lL2K Y 98 دة8 ا" ز9:دة ا8G اZ*8 .lL2*8 "* ا*طء3= * 8و
171
ا ا ا
ﻗﺴﻢ ﺍﻹﺣﺼﺎﺀ ﻭﺑﺤﻮﺙ ﺍﻟﻌﻤﻠﻴﺎﺕ ﺍﻟﻤﺎﺩﺓ :ﺑﻨﺎﺀ ﺍﻟﻨﻤﺎﺫﺝ 203ﺑﺤﺚ ا* Sر اSول #ل ا` 9 ا` Nا^ 1423/1422 3هـ ا*# : G أ Q H 3 # RHا 9 0ا*:9 ا rال اول: أ( #ف ا*:3 اOم ، Systemا ، Entity @/ا` ، Attribute 9اPط 9 ، ActivityاOم System State
ب( أذآ أ2اع ا.9 O0 ج( آ N^ 8 lاOم = 3ا*` /ا.3O ا rال ا: )* +ام N Excelا دSت ا`و 9Lا* 9وار ر ا ; Nا :ة: 1) xn = 0.3 xn −1 + 10, x0 = 0, n = 1,...,50
2) xn = xn −1 (1 + xn −1 ) , x0 = 10, n = 1,...,50
3) xn = xn2−1 + 0.7 xn −1 + 0.2, x0 = 1, n = 1,...,10
ا rال ا: Dا 2ذج " # y = a + bx cات ا*Excel Solver )* 9 9
8
7
6
5
4
3
2
1
0
x
690
509
312
275
190
132
92
65
47
32
y
ار ا; ا :ة وا; ا .9;:
172
ﺑﺴﻢ ﺍﷲ ﺍﻟﺮﺣﻤﻦ ﺍﻟﺮﺣﻴﻢ ﺟﺎﻣﻌﺔ ﺍﻟﻤﻠﻚ ﺳﻌﻮﺩ ﻗﺴﻢ ﺍﻹﺣﺼﺎﺀ ﻭﺑﺤﻮﺙ ﺍﻟﻌﻤﻠﻴﺎﺕ
( ذج#$ء ا$) 203 دة ﻫـ1423/1422 ﺍﻹﺨﺘﺒﺎﺭ ﺍﻝﻨﻬﺎﺌﻲ ﻝﻠﻔﺼل ﺍﻝﺜﺎﻨﻲ ﺴﺎﻋﺎﺕ3 ﺍﻝﺯﻤﻥ : % ا2[& ا%#! 324 k!أ :ال اولr ا gPة ا#L ( أ او2) Occam's Razor مTس اوآ2
(1) : *ر+ ف و# Parsimony Principle
(5) Endogenous Activities 9 اا9:P0( ا4) System State مO ا9 (3) Stochastic
9@ا2P ا9:P0( ا6)
Exogenous Activities 9H ا)ر9:P0ا
System مO ا9H6 (8) Discrete Systems 9 ` ا9 O0( ا7) Activities Flow ب0( ا10) Stock ( ا *ع9) Modeling
(12) Auxiliary Variables ة# ات اZ* أو اConverters تS2 ( ا11) Information Link ت2 اd أو رواConnectors ت%2 ا
:ال اr ا ا98*دLA وا9# *HA وا9H2 وا982اه ا2O اN^ 9:اه ا2O^ ا/هك ا # Z* ع آ ت ا2 J ويK t G ا# 3/8م دO 3= Z* 9 آT= ن2/K =*ةN^ K T V t G ا# Z* ا9 آN^ K xt , t ∈ T Xذا آ+= . t − 2 وt − 1 9ز0ا .9 9ز 9 ز9O أي# مO ا9/8 دlK 3* وا9^ ا9H ار9L ا`و9( إ!* ا د1 . t ∈T
173
9 ءi= N/! 3= 9; ا9L اQU (2State Space Form N/P ا3 # xt = Axt −1 , t = 0,1,..., T
V 1 1 A= 0 1
. A 9=2` Eigenvalues ةG ور ا6J اH( أو3 T # X 3* ا9L ا`و9 ا دN EXCEL *)ام+ وx1 = 1, x2 = 1 9 أوL 6 f (4 .( A1, A2,..., A30 8) ا3= N اQU) . t = 1, 2,...,30 ; (1) 3= ;رن اL C1 = A2 A1,..., C 29 = A30 A29 وB1 = A1 A2,..., B 29 = A29 A30 QU2 (5 . A 9=2` ةG ور ا6J اQ 9JKا :ال اr ا Markovian 9=2 ا آ9%) اT 9 98*دL*د أن هة إLAل أ اء ا2;8 ()أ
؟38 = ذا9^ ا9H ارProperty R 9^ ا9H ار9=2 ا آ9%) اQ*8 9 !آT 32`ل اLA)ب( ا 9L ا`و9ا د xt − axt −1 − bxt −2 = ε t , ∀t ∈ T , ε t ~ N ( 0, 4 )
(Second Order one State
وا9 Z* 9Y 9H در9 ا د56ل ه2
]( وذFirst Order two State Variables) 9 ىZ* " أو9H إ" درVariable) . yt −1 = xt −2 QU2 ; (Steady State) 5م وأ * إ *;ارOا ا6T ذج2 ن2 آVensim *)ام+ ()ج وy0 = 1 وx0 = 1 1) a = 1.2, b = −0.7 2) a = −1.2, b = −0.7
9* إ *)م ا; ا:9O RANDOM NORMAL(-3.99,3.99 , 0 ,2 ,seed=19 ) INITIAL TIME = 0
FINAL TIME = 2
SAVEPER = TIME STEP
TIME STEP = 0.01
174
ﺑﺴﻢ ﺍﷲ ﺍﻟﺮﺣﻤﻦ ﺍﻟﺮﺣﻴﻢ هـ1423/1422 ا2 R;$ر ا9\t 2# إ!ت ( ذج#$ء ا$ ) 203 دة :ال اولr 2 !إ !ء0 اd * :ل2;8 3 # 3` = أ2 وهOccam’s Razor مTس اوآ2 (1) N%`*ه اf أ8 إNL0 " ا# أو98ورi \ اN%`* اN( إ * آNآP )ا (9 /P )اPا ا6د هK 3*ا إذا: ل2;8 ي6 وا3 اV ا3= Parsimony Principle gPة ا#L ( أ او2) NLي ا28 ي6ذج ا2 = * مa@* اc` 3:8 ذج وا2 ^آن هك أآ .Ni=0ذج ا2 ا2ات وا هZ* د ا# TK`%@ت و/ اN آlK اتZ* 3 وهSystem State
مO ا9 (3)
3= اتZ* اQ** مOر ا2:K رس8 و.9 9O # مO ا3= 9:P0وا .$* .مO اN دا9:P0 اlK وEndogenous Activities 9 اا9:P0( ا4) مO ا9 3= 9:P0 اlK وExogenous Activities 9H ا)ر9:P0( ا5) Z مO 3 8 9H ر9:Pf Yf*8S ي6م اO ا.مOا ا6 ه3 # YBK 3*وا .ح2*` مO $f l%28 ي6 وا9H ا)ر9:P0 Yf*8 ي6م اO اc/ N/P هYfK Z*8 3* ا3 وهStochastic Activities 9@ا2P ا9:P0( ا6) ي6 اXL2 = ^ ا3 * إQ8ز2* l%2K ت *دة/ إTJ@* ن2/K و3@ا2P# Z*8 9& ل:# إG آ ان ا3 * إQ8ز2* l%28 Q J* 9& $LZ* K .3@ا2P# N/P Q:;* N/P مO ا9 T= Z*K 3* واDiscrete Systems 9 ` ا9 O0( ا7) 3 # R D ل2% و،Q:;* N/P ث8 Q ا3= 9#i = ^ إآ لG اQ . اQ:;* N/P ث8 Q@iا
175
) 9H6 (8اOم System Modelingرا O 9م RJ8ان 2/ن او 2 3ذج l%2 Modelه6ا اOم Zض إHاء ا*Jرب 3 # 9 Hأ 9وإ=*اUت / 8SإHا@ 3 # TاOم !ة وذ] *:i8S 3ب اOم ا 3 %0و8ث إرKك =B8 $ # 3دي ا ZK 3اOم و=;ا2) $ا $%ا 9 %0آ ان درا 9 ا 2ذج SاOم # R8JK / Kة 2ارات 2ل اOم وذ] #+دة ا 2ذج ا 3ا 9ا # 9 %0إHاء آ2 Nار c/اOم ا 3 %0ا6ي إذا / 8S ZKإ#د 9Kة ا ى * 9ا ^ = 9 %0را O 9م إ*Lدي *Z ت اض واB8 L R :دي ا .T /# / 8S a@* 3آ أن ا 2ذج / 8 ان 8رس = 3أز 9إ=*ا / 8 ^ = 9UإHاء آة Oم )* +ام ا 2ذج و==K j 9ت اOم `*ات #ة ا! Tاو = 3د .9 L @Lوآ]6 8D # / 8ا 2ذج درا 9اOم NLإ $@Pوو2Hد 5ا 8 ^ = %ء Qو# 8ة رات ء = * 8اي ر ا=2/ Niن 2ذج N/ر وآK 3ف ا XK Qه 56ا)رات. ) (9ا *ع Stockأي ! Q Z*8ا) Gد* ،3/8ك( G8داد و ;8و" 8 ا2* i8ى Levelاو * ، State Variable 9 Zا *ع R#2* 8ا!0ء `*ة ز 9وه 56ا!0ء ا * TOKS 9#2أو O 3`*)Kإذ اiK Tف أو (N;K) R Kل =*ة ز.9 ) (10ا Aب Flowه 2ل 9 Z8 Rateا *ع 8Gُ8 2T= ،أو ِ;ُ8ا *ع. ا Aب ا6ي 8G8ا *ع " 8أ Aب اا Nأو ا ر ،Sourceوا6ي ;8ا *ع " 8 أ Aب ا)رج أو ا2Zر . Sink ) (11ا S2ت Convertersأو ا *Zات ا #ة 2K Auxiliary Variablesى أرLم2;K ،م ت 9أو 98Hأو 9;:و *Kا * 3= Controllers 9 /ا 2ذج. ) (12ا %2ت Connectorsأو روا dا 2ت Information Linkوه 3ا*3 N Kأو d K ا 2ت GHء إ" & = 3ا 2ذج ،ا Tأن ان ا %2ت N;KS آ ت دN^) 98 176
ءGJ N;*K ذج2 ء اGH 9 ;= d;= ت2 N;K 3 هN (د2; او ا9L: او ا5 ا . & :ال اr 2 !إ xt −1 3 هt − 1 G ا# وxt 3 هt G ا# Z* ا9 ال آB ت ا: (1
إذاxt −2 3 هt − 2 G ا#و xt = xt −1 + xt − 2 , t ∈ T
:9L =و9 دN/! " # أو xt − xt −1 − xt − 2 = 0, t ∈ T
:N/P ا3 # 9L ا`و9 ا دQi (2 xt = xt −1 + xt − 2 , t ∈ T
J yt −1 = xt −2 QU2 و xt = xt −1 + yt −1 yt = xt −1
N/P " ا# 3وه xt 1 1 xt −1 y = t 1 0 yt −1
.ب2 : اN/P ا2وه 9L *)ج اK ةG ور ا6J( ا3 A − λI = 0 1 1 1 0 −λ =0 1 0 0 1 1− λ
1
1
−λ
=0
(1 − λ )( −λ ) − 1 = 0 λ 2 − λ −1 = 0 1+ 5 1− 5 , λ2 = 2 2 λ1 = 1.618034
∴ λ1 =
λ2 = 0.618034
177
(4 1
1
1
2
0.5
1
1.5
0.666667
2
1.666667
0.6
3
1.6
0.625
5
1.625
0.615385
8
1.615385
0.619048
13
1.619048
0.617647
21
1.617647
0.618182
34
1.618182
0.617978
55
1.617978
0.618056
89
1.618056
0.618026
144
1.618026
0.618037
233
1.618037
0.618033
377
1.618033
0.618034
610
1.618034
0.618034
987
1.618034
0.618034
1597
1.618034
0.618034
2584
1.618034
0.618034
4181
1.618034
0.618034
6765
1.618034
0.618034
10946
1.618034
0.618034
17711
1.618034
0.618034
28657
1.618034
0.618034
46368
1.618034
0.618034
75025
1.618034
0.618034
121393
1.618034
0.618034
196418
1.618034
0.618034
317811
1.618034
0.618034
514229 832040
أن ا; BK Bi 3= 9ول إ" λ1 = 1.618034و ا; BK Ci 3= 9ول إ" λ2 = 0.618034
إ! r 2ال ا:
178
" ان# ل2;K Markovian Property of order
k 9H ار9=2 ا رآ9%) ا: ()ا
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(
)
P yt < s | yt −1 , yt −2 ,..., yt −k , yt −( k +1) ,... = P ( yt < s | yt −1 , yt −2 ,..., yt −k )
(Second Order one State وا9 Z* 9^ ا9H ار9L ا`و9)ب( ا د Variable) xt − axt −1 − bxt −2 = ε t , ∀t ∈ T , ε t ~ N ( 0, 4 )
(First Order two State Variables) 9 ىZ* " أو9H درN/! " # QU2K xt = axt −1 + byt − 2 + ε t , ε t ~ N ( 0, 4 ) yt = xt −1
Vensim *)ام+ مO ذج2 )ج( ا a
epsil on
x(t) dx(t) b y(t) dy(t) a = 1.2, b = −0.7 ; -1
(01)
a= 1.2
Units: **undefined**
(02)
b= -0.7
Units: **undefined**
179
(03)
"dx(t)"= a*"x(t)"+b*"y(t)"+epsilon
Units: **undefined**
(04)
"dy(t)"= "x(t)"
Units: **undefined**
(05)
epsilon= RANDOM NORMAL(-3.99, 3.99 , 0 , 2 , 19 )
Units: **undefined**
(06)
FINAL TIME = 2 Units: Day
The final time for the simulation.
(07)
INITIAL TIME = 0
Units: Day
The initial time for the simulation.
(08)
SAVEPER = TIME STEP
Units: Day The frequency with which output is stored.
(09)
TIME STEP = 0.01
Units: Day
The time step for the simulation.
(10)
"x(t)"= INTEG ("dx(t)", 1)
Units: **undefined**
(11)
"y(t)"= INTEG ("dy(t)", 1)
Units: **undefined**
180
Graph for epsilon 4
2
0
-2
-4 0
0.50
1 Time (Day)
1.50
epsilon : Current
Current "x(t)" 2 1.7 1.4 1.1 0.8 "dx(t)" 6 3 0 -3 -6
0
0.50
1 Time (Day)
181
1.50
2
2
Current "y(t)" 4 3 2 1 0 "dy(t)" 2 1.7 1.4 1.1 0.8
0
0.50
1 Time (Day)
1.50
2
6
3
0
-3
-6 0.80
0.95
1.10 "x(t)"
1.25
"dx(t)" : Current
a = 1.2, b = −0.7 ; ;* ذج2 ان ا0 اN/P اgUوا a = −1.2, b = −0.7 ; -2
(01)
a= -1.2
Units: **undefined**
(02)
b= -0.7
Units: **undefined** 182
1.40
(03)
"dx(t)"= a*"x(t)"+b*"y(t)"+epsilon
Units: **undefined**
(04)
"dy(t)"= "x(t)"
Units: **undefined**
(05)
epsilon= RANDOM NORMAL(-3.99, 3.99 , 0 , 2 , 19 )
Units: **undefined**
(06)
FINAL TIME = 2
Units: Day
The final time for the simulation.
(07)
INITIAL TIME = 0
Units: Day
The initial time for the simulation.
(08)
SAVEPER =
TIME STEP
Units: Day
The frequency with which output is stored.
(09)
TIME STEP = 0.01
Units: Day
The time step for the simulation.
(10)
"x(t)"= INTEG ("dx(t)", 1)
Units: **undefined**
(11)
"y(t)"= INTEG ("dy(t)", 1)
Units: **undefined**
183
Current "x(t)" 1 0.5 0 -0.5 -1 "dx(t)" 6 3 0 -3 -6
0
0.50
1 Time (Day)
0.50
1 Time (Day)
1.50
2
Current "y(t)" 2 1.7 1.4 1.1 0.8 "dy(t)" 1 0.5 0 -0.5 -1
0
184
1.50
2
2
1.7
1.4
1.1
0.8 -1
-0.50
0 "x(t)"
0.50
1
-0.50
0 "x(t)"
0.50
1
"y(t)" : Current
6
3
0
-3
-6 -1 "dx(t)" : Current
a = −1.2, b = −0.7 ; ;* \ ذج2 ان ا0 اN/P اgUوا
185
:QHا ا 1) A Course in Mathematical Modeling By: Douglas D. Mooney and Randall J. Swift Published and Distributed by: The Mathematical Association of America.
2) Mathematics for Dynamic Modeling, 2nd ed. By: Edwaed Beltrami Published by: Academic Press
3) An Introduction to Mathematical Modeling By: Edward A. Bender Published by: Dover Publications, inc.
4) Mathematical Modelling Techniques By: Rutherford Aris Published by: Dover Publications, inc.
5) Introduction to Difference Equations By: Samuel Goldberg Published by: Dover Publications, inc.
6) Matrix Computations By: Gene H. Golub & Charles F. Van Loan Published by: North Oxford Academic
7) The State Space Equations and Their Time Domain Solution Lecture Notes By: Dr. J. R. White, UMass-Lowell 186
ﺑﻨﺎﺀﺍﻟﻨﻤﺎﺫﺝ ﺒﺈﺴﺘﺨﺩﺍﻡ
Excel and Vensim
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9*/ اء اYA و8/ ا$TH2 N ا ا6ز هJ إ3= 3;=28 ا ا ان2Hا وار6ه .*ب/ا ا6 هN^ " ا`;ة إ9 ا 3= X*A ا9/! " # Hا2* ن2/ و# RD ي0 3J *ب/ا ا6ن ه2/ http://www.abarry.ws/books/ModelBuildingBook.pdf QL2 ا . =2 وا ا lB ا ا ي# H ن# .د د2 ] ا9H هـ1423 RHر م2002 *
4
ت اب ;...................................................................................... 9
3
ا اول ..............................................................................
9
أ ت ا :9H6اZض و ا2U2ح و ا 2ارد ..................................
9
#ا;; 9و #ا 2ذج .............................................................
10
ا*` /ا 3Oوآ 9اOم ........................................................
14
Kر l8أ ........................................................................... 9
14
l8Kا 2ذج و O 9THا*` /ا.................................... 3O
15
N^ KاOم = 3ا*` /ا 3Oوآ 9اOم ...................................
19
ا ا .............................................................................
20
ا دSت ا*`.................................................................... 9 U
20
ا دSت ا*` 9 Uار 9Hا0و" ا)..................................... 9:
20
ا دSت ا*` 9 UارHت ا ..............................................
23
ا دSت ا) 9:ت ..................................................... 9* Y
23
ا 9ا *.......................................................................... 9 J
23
ا \ 9ا *.................................................................... 9 J
25
2اص N#ا*`............................................................. D NU
26
98Oا* 2 Nا................................................................... N
26
ا6Jور ا ` ......................................................................... 9
27
ا6Jور ا /رة ..........................................................................
28
ا6Jور ا آ 9ا *;ر............................................................... 9
28
ا ا .............................................................................
32
ا دSت ا`و....................................................................... 9L
32
ا دSت ا`و 9Lا) 9:ار 9Hا0و" ......................................
32
Nا دSت ا`و 9Lا) 9:ار 9Hا0و" )* +ام ........... Excel
34
R8;Kد 9 U`K 9د= 9و............................................... 9L
40
ا دSت ا`و 9Lا) 9:ت .......................................... 9* Y
43
5
ا Nام د 9ا *.......................................................... 9 J
44
ا Nا)ص د 9ا.......................................................... 9 /
47
ا*ف ا2 3@Tل )* +ام .......................................... Excel
51
ا اا .............................................................................
56
Hو ب ا `=2ت ...............................................................
56
ا ت ا " # 9 0ا `=2ت ..................................................
56
jا)2اص ا `ة دات ......................................................
57
ا; ا Gة .......................................................... Eigenvalues
58
ا *TJت ا Gة .................................................. Eigenvectors
58
ر=2; 9 9=2` Qة ...............................................................
60
ر=2; e Qة `............................................................ 9 9=2
61
]/`Kا; 9اPذة .............................................................. SVD
61
ب ا `=2ت ......................................................................
61
9;*Pا `..........................................................................9=2
61
N/Kا `.......................................................................... 9=2
61
9;*Pا 2/س ..........................................................................
61
ا ا ...........................................................................
63
i= N^ Kء ا....................................................................... 9
63
دSت =iء ا 9و 3= Tا Jل ا.................................... 3G
63
ا* N82إ" ! N/ا................................................................ 9
64
N82Kا دSت ا*` 9 Uار........................................... n 9H
64
O N82Kم ار 3 9Hإ" ! N/ا.......................................... 9
67
9ا دSت ا*` 9 Uوا 98Jا )* ....................................... 9:
67
N82Kا \ 9 O0ا) 9:إ" ............................................... 9:
68
Nا 9 O0ا) 9:ا *;ة ..........................................................
69
ا 9 O0ا) 9:ا *;ة ا *.................................................. 9 J
69
ا 9 O0ا) 9:ا *;ة \ ا *........................................... 9 J
70
i= N^ Kء ا 9دSت ا`و............................................... 9L
71
6
ا)* + Nام .................................................................. Excel
72
i= N^ Kء ا 9دSت ا`و2 9Lا ....................... Vensim 9:
72
ا ا دس ...........................................................................
73
jا ذج ا................................................................. 9 0
73
2ذج ا ) 2أو اAل( ا)................................................... 3:
73
2ذج ا ) 2أو اAل( ا.................................................... 3 0
74
2ذج ا 2واAل ا)........................................................ 3:
75
2ذج ا 2وا0ل ا......................................................... 3 0
75
ا 2ذج ا* ......................................................................... 3
76
2ذج اS2دة وا 2ت ..................................................................
77
2ذج " ا 2ا ......................................................... 3* H2
78
2ذج ا 2ا ود Pوط ) ........................................... (3* H2
79
2ذج l%28دSت ا*`Z* 9 Uي ............................... 9
80
2ذج l%28دSت ا*`^ 9 Uث *Zات ....................... 9
81
ا ا .............................................................................
83
Sت درا 9ء jا ذج ....................................................
83
(1ء 2ذج آ2 9ق ...........................................................
83
Kا 2ذج )* +ام ................................... Curve Expert
91
Kا 2ذج )* +ام .................................... Excel Solver
91
2 (2ذج )Gون ;2ة ......................................................... 9 #
96
اLت ا 3= 9ا 2ذج ...........................................................
97
دورات ا* 986Zا) `....................................... Feedback Loops 9
99
دSت ا 2ذج ........................................................................
100
2 (3ذج ** ........................................................... 3P*2= 9
105
2ذج !i= N/ء ا)* + 3P*2= 9 ** 9ام ............... Vensim
105
(4ا ذج اآ 9ا2Pا@......................................................... 9
107
ا) 9%ا رآ..................................................................... 9=2
107
N^ Kا ذج اآ 9ا2Pا@ 9ا رآ)* + 9=2ام .............. Vensim
107
7
112
..................................................... bifurcation RP*ذج ا2 (5
112
............................................ VenSim d:) RP*ذج ا2 N^ K
117
..................................................... R L 3= 9 i 38;K ذج2 (6
117
.................................................... Vensim a N^ ذج2 ا
123
................. Lorenz Attractors Models c*8ر2 ذ تH ذج2 (7
123
..................................................... Vensim 9: ا2 ذج2 اN^ K
129
............... Prey and Predator Model 9iذج ا `*س وا2 (8
129
.................................................... Vensim *)ام+ ذج2 اN^ K
132
......................................................................... 9 ت اLأ
132
.......................................................................... 9` )اورات ا
134
....................................... 9; *رات اA ت اH وإ9 ( أ1)
186
.................................................................................... QHا ا
8
ا اول: أ&&%ت ا : !"#$ا+ض و ا ) ح و ا #ارد 8ف ا #$ذج " #ا# N 2 ^ = ، /%/ %#0 : $ة أ!)ص ه 2ا 2ذج RJ8 L اه :ه 2د 9ر ، 9U8و : & RJ8 Lه:L 2ر 9و& 2/Kن إ : $* Hه2 2ذج " 8اد إ $@Pو : & RJ8دKK 9ي 2Yب = 3أ ت ا).9D fK 2ا Nا *Pك = 3إ H2 TK Hا = ، "9;; N^ K " $دSت اN^ K 9U8 ) 9//آ XK Q *J 2 (9ارا ^ 9و :Lر ا :L N^ 8 9ر ;; 3و 2ذج ا " N^ 8ا N/Pا6ي و $ #ا " ا;; 3إ ، $@Pا N/! N^ K 9ا cوآl # TOرKا@.$ ا* l8ا 2ذج \ آ= ;K / 8 l/= 3و;ر 9ا ذج ا )* `9؟ = N/ا9 وا;:ر ا 9;; N^ K 9و /هك إ *ف آ Tو; / 8Sر*d:) ^ = .T !2ارع 98و)D d:ق ا ` = 3ا ة ا* T= Q;K 3ا 98آ9;; N^ K T N و8وان * T Pو / 8S /إ *)ام أه /ن ا ، bإذا هك ! 2Hهي *; ا 2ذج أ Sوه 2ا+ض Purposeو l8K g8 3Kا #$ذج " #ا %#0 : $ذا 1ض d:) = ، /%/ 2ا2Pارع ه 9;; N^ K 2ا*K 3و 2! Tارع ] Kا Z 98ض اS* Aل 2! " #ار T#وهS* g 8S D 2ل D " #ق ا ` = ] K 3ا ة وg 8 "*S * ;ط إزدم ا ور = ] K 3ا 98و /ه6ا NJ8Sا 2ذج .هك أ2اع )* `9 :Lرات ا0ب Ti ،ص Rا`D0ل و Q H 3 Tiذج ا;:رات )^N Q Hا2:ا (Qوه6/ا \ = ،اض ا )* ` a*K 9ذج )* `.9 هك ! 2Hهي & = 3ا 2ذج و 2ف *^ $Uل ا* 2 :3د#ك أ اL%0ء اJد إ" 0 $Gول ة واKS D Xف ا ];8 / = ، 8:أن 9# ] d)8 )2P d:ارع ا*BK 3دي إ" 2/8 L ، $Gن ) ;* \ d d:أو )8 Lج ] )d: ه 9;: O 3ا* T= / 8 3أو ) N P8 d:ا T / 98و، $G QL2 ] !B8 ه 56ا )::ت 2/8 Lن = TدرHت )* ` 9ا*` N%و ; c8ر )* ` ، 9وه6ا B8دي إ" " 8ر 9Hا2U2ح أو ا ) ح ^ = ، ResolutionاN%`K 8K R ا!0ء 2%رة ا 6ت 8 82K 9ل ه اBري ) ون =2آ (Focus cو =9 ة و2/* /ن ا*` N%واH 9Uا ودة إذا ا 6ت 2 82K 9ع Hو !) *) = 3ا* 82و = .Rذج ا T i8در 9Hو2Uح وه6ا *8
9
;ار ا*` N%اiور 98وا 9Tا* 3= N K 3ء ا 2ذجT= .ك ذج 9 Lا2U2ح Low resolutionو ذج 9#ا2U2ح . High resolution ا Pا2Jهي ا 3= 0ء 2ذج ه 2ا #ارد Resourcesوا/Aت و N PKا 2اد ا :ة وا Gا *ح وا)ة ا *=2ة أو ا /إآ* U Tا Gا *ح وا #اPي وا L " = ، 3ر ا 2ارد وا/Aت * Q:ان 2 3ذج Zض ا 2 :ب و 2U2ح ا ^ = ، Rإذا ا X:#ورi 9Lء و Lر%ص و :ة =; dو ] R Dر )d: 2Pارع وا ا= 2ل T)* 8 3/ ]*Hا:ب ا * 8Jوا Gا *ح ] c د2 = ، @Lارد ا :ة 2ف 2:) 9:8 Kط *; 9أ 2د " #ا jوj liK ات ا*K 3ل " #ا ا= ا ، 9 Tاbن 2ا ] 3:#أLم * ]+= 92ا:ق ا@ 2 9ن وا`2 9#ن & وا ا= 2fان Kل ^ = T*#2 " #ا 2 Hن وا 2 #:ن وا 2 R#ن & وه6/ا 8GKا*` N%وا2U2ح ;ر اTف ا 2 :ب وا 2ارد ا * ^ = 9ر أر 9`%أو أ!Jر أو زل )KSم اTف ا 2 :ب و K L ا) N ;K N%`* 9:8و T2Uآ ا*K Tج ا" و XLا2Dل ا *ح. ا دئ ا 3= $ Tء ا ذج وا6ي * JT 3= $ه89 2أ ا 6 7%#ا& 3245 Bottom up designوه 2اء 2ذج N L dا2U2ح Yز8دة ا*` N%ا ; 9ر g Kا 2ارد و*" ;*8اZض ا 2ذج .ه6ا c/#أ ا* اN` " #0 Top down designوا6ي 8أ *` N%آ^ة و;ة T N8G8 Yا*` \ N%ا ، 9 T وا6ي * 5ه. = 3درا * 2ف *:ق =;#$2 dذج ا ) %وه 3ذج )* + "Kام ا:ق ا9U8 )L#ت L# ، 98Hت 2K ، 9;:ز8ت إ* 2 ، 9ارزت و ا 9 2 aا(. و 2ف `f Tن ا 9H6ه 9 # 3ء 2ذج ر.3U8
74ا /%/و 74ا #$ذج : Real World and Model World ا ا;; 3ا6ي =H ; $ا " # 3* 8 N/Pا Sن = Tا Pآ Nا;;N/P 9 آ Nاو *" 3@GHو6Tا fJ 8ا Sن ا" ا*; R8وا* P dح 9Dه 56ا Pآ3/ N 8ول N^ K / 8 ، Tا ا;; 9 U 9 N/P 3آ = 3ا N/Pا*:3
10
Occam's razor
Real Worl
Model World
Interpreting and Testing Formulating model world problem
Model results
Model
Mathematical analysis
(مTس اوآ2 5 وود$=اDى أ2 K) d 3 هN/! " # ب6T8 أو3;;;ب ا ا8و و*;ل ا3;; اd K وR8;K 2ذج ه2 ا# و، ذج2 ا# N^ 8 ي6وا 3 # 3` = أ2 وه، Occam's Razor *)م س اوآ;م8 ذج2 ا# " إ3;;ا 7هD أ8 إF ا324 ور أوG ا%1 %H آ ا8I9&آ( إA#ء )ا%>? ا% 9 :ل2;8 KAة ا84F أ او$ # 3 ا ا أ6 ه، ( 2A# )اJ%Aد ه"ا ا8 0 ا%Hا ذج# 6 ك أآ$ إذا آن ه: ل2;8 ي6 وا3 اV ا3= Parsimony Principle
ه7I#ات وا%+# ا6 د84 F ذج ا"ي ي ا#$P QR$ ا9 /0 SI 8Tوا 2ي ه6ر وا2TP اE = mc 2 *P8ن ا2L ا ا أ6 " ه# أدلS و.GP ذج ا#$ا و\هmxɺɺ = F K2 9ن اآ2L ]6 وآ، قDA " ا# T وأه9U8ا ا2; اd ا .92 ا9 O اR ا*آ9: ا9U8ا ا2; ا ا2L و% QU و3= l /K ^ ا ء واj $ م2;8 ي انO 9TH و 52:* أl%2 (^ او اآ9`% Ti ) ر2: ة# و;ة92: 9U8ر م# وT P= ا2:\ وg *ط اA ا اN وا آL ، TU2 \*ه *;ه و ه ب6 f8 j وا9` /* ا وا ا2; ا56 هN^ 9;; ا9 /P T * إ و ط0 ا3= * وS2; ن2/8 $D* *; ان إ8 N 5\ N # آN 8( و9U2 )ا
11
ا 9ون دSت ر; 9U8ة و2K 9 82Dي ا ^/ا *Zات وا \ اiور .98وآهن " #ذ] = 3اAار ا *د GK R2 9 Lداد آ NأN;* Z* `U إ" ا د2 "* 9آن ه6ا ا *f X 8S Zي 9 9`%أو 9;:إ" ا * Zا* N^ Q د U 9= Nط ا Jو أ2Dال ا`D0ل 5إ" c` 3= 9 15ا ) Q *JآT N G8داد وه6ا K 3:8ا 2L dي و * 3ز8دة = . ( R2 3وه6/ا ى ا ^/ا^ GKد د ^/ TKSا*` \ N%اiور 98وه 2 3Kf8س أوآTم :إ * آ Nا*`N% \ اiورH 98ا وا ; 3ا 2ذج = 3ا dا!.$/ N82K 9 #ا ا;; 3إ" #ا 2ذج ه 3= 3ا;; 9أه وأ2: R%ة = 3ء ا ذج وا*2; 3م 8T* Tأ 2س اوآTم =` 9 # 3ء ا 2ذج * آ Nا*`\ N% ا Z 8!* $ Tض وا2U2ح او ا*2 8!* T*J / 8S 3ارد ا *.9 = ; Q:ا ا;; 3ا" / 8 Jا* $ Nآ 2 Q:L 2س .أ 2س اوآTم أداة 9و :ة ،إذا ا X:*Lا+= ^/ن ا Nا ** aا 2ذج 2/8ن $اي 9 % 9 /Pا;; 9وإذا ا X:*Lا; += Nن ا 2ذج 2 9 Rارد ا *2 .9ف Pح ا N/Pا و2س اوآTم ^ل ا*:3 ; در 3= XاU8ت =2U2 3ع ا*` NUوا*2U2 N/ع ا;6ا@ Projectiles lوا*3 Q*Kر ، =/ Q:Lا;*!Aق 8أ ل g X Y NJKو!وط او 9= 9وا s 0 9#
و
v0و N/ا د9
" # N K a (t ) = gاAزا 9آاG 9
2U " # . s (t ) = 12 gt 2 + v0t + s 0ء ا N/Pا و2س اوآTم =+ن ا 3= 9 /Pا ا;;2;K 3م * # Bآ 9`86L 9أو ا!ء 9 Yآ J 3او ! 9\ L ، $T ا 9 /Pا;; 9و; Tإ" #ا 2ذج *) 2س اوآTم ا6ي ا=*ض ان ا ا;;3 g:و ل NJKاو Hذ X Y 9و= 3ا`اغ )#م و2Hد \ف 2Hي( # Qم و2Hد اي 2Lى YBة ا ى U ،ه 56ا`وض او ه6ا ا =+ن 2ذج ا ر=/ Q:L N/! " # ا 9 /PK /و Y ، $ا * = 3ا Nو ت ا آ6T ، ^ 9ة ا\0اض وا ا 2ذج Hا ،ا) :ه 3= TO8ا`f 9Uن ا 2ذج f*8ان Q Hا;6ا@ T lآ9 Q*Kر 3= =/ Q:L N/! " #ا ا;; 3ذا 2ر ر 9P8أو آة Dو(a2 a ) 9 = 3و2Hد ر8ح؟ .ا # T S i8ا;6ا@ lان = 9Uل ا* NJا^ Xوإام ;و 9ا2Tاء \ 9%و 3:Kأ :ء N`K Lا وا .9 8GTوZ /ض ا*;+= R8ن رJ 3رة أو ا!ء 28 3= 9 ;Yم = $ا8ح ``+= 9ن 2ذج ا ر ا;=/ Q:
12
H Rا .ا H Tا اYء آ* * *; N a@* #8ا 2ذج ان Qiا`Uت ا*3 T # 3ا 2ذج =`^ 3ل ا;6ا@ lأ=* Uان ا ا;;\ H28S g: 3ف 2Hي )أي = 3ا`اغ( وHذ 9* Y 9و2L H2KSى ا ى .ه 56ا`Uت هGH 3ء Tو2ي ا 2ذج = )! 38ر 9P8و2;8ل ] :ا2 !Oذ* = ، f: ]Hد ! : $ # ا 9P8آ Q** Xر 2 =/ Q:L N/! " #ر 3= Xا`اغ 8 L .د ] #ان 2ذ]H \ وا= 3= S 0 3Lاغ و 2/ن رد ! : $ #ء ا^; 9وا*= "K 3ت Lة =+ن #YfKم و2Hد =اغ ه / 8 YfK 2إه $وه $*^K 2ا*Jرب ا .9 ا`Uت وا* Aرات 2ذج ه 3اه 2:ط ا=ع Uا //Pوا2 8Lذج. Pوع g8) :ا N R:ه6ا ا Pوع H T $0ا * Aب ا دة( `*ض ا 2ف ;2م Pوع 2 9ز N%2* 8ا 98 @ Gا" اLب :ر. ا=*ض ان ا = 9ا 98وا :ر 50آ 5JK *2ا2ا 8 .ان 9` /K R ا6T 8Gة ا) R 3/ 9ا ا U; Rة ا2 Gن .إذا ا=* XUان ا رة T * d 2إ *Tك 25آ 2H/ن*#A 6 .ر :3 8 ا ر ا 8GاZ*K L 9 اوLت ا*2H2 Yf*K N%2د زم وري أو $# 9ا ) 2Jا:ر\ ،ر ،ر8ح 982Lا( إ *ف ا =ت دا Nا ) 98إذا آن ا :ر ^ = ! 3ل ا += 98ن ا 3= @ Gا2JبZ* Kق ر * 9= Tا2Dل( و2Hد /ن =رغ GAال ا2 Gن #ا2%2ل :ر ام اO*Aر `اغ /ن * Q:إJ8د إ*#رات ا ى آ^ة و /ذا 2 #س اوآTم؟ ه Nآ Nا*` N%ا ;9 9 T؟ ذا T 6 fواي Q:*; T؟ Zض ا 2 :ب و Uا 2ارد ا *9؟
13
مO ا9 وآ3O ا/`*ا Cause Yf* واR ا9L# " # م2;8 ا*; ي5JKA اR 3;; ا اNآP Q N*ا .j اTi # لG N/P 9 /P اء اGH أ6 f8 و9: اand Effect " إ9 /P اQّ:;8S System Approach/ Thinking 3O ا/`* أو ا3O ا5JKAا S 3* *دة واd رواH2K م إذO اT= 3* ا9 ا8* N/ آT O8 N 9G اءGHأ T JK دة# إY " ة# ءGH N إ" آO انS T=P/* *ر وإ#A ا3= ه6 ا م لO =تK وRاآK T= ا*ف و9 " و# Gآ8 3O ا/`*= .;S 9و0 اR ا*اآ56 " ه# مOف اK # اتB*" اK Y و9 أوRاآK إ *)ام 9 ا*آ$%ا2 9 مO ا" اO *K مO ا9 آ ان آ.T # ا *ف اءH+ g 8 ا6 وه9 اراXK مOف اP/* A ء ذجR :*K "* واG اQ ةZ* ا او/`* " ا*; ا# # K وره3*م واO ا9H6 9 ;* إ *رات R *فKS T*: 9 O0د إ" ان ا28 3O ا5JKA ا3K 3= R ا.3O ا5JKAا 9H6 ا. ه ا اg 8S ا*; ي5JKAن ا+= ا6T و$T8 = *فK L N 3T8 ا/`K /`* ا3= ت8ورi ا ا3 هModeling or System dynamics مO ا9أو آ .ل2 ت واU`*; اK T0 3Oا
: %&& اV رI0 : System مW$ا .j اTi " # *K وN#`*K !ء0 ا9#2 J : Entity 6Rآ . إه* م ص$= مO ا3= ! 2وه : Attribute H .@/ 9`% او9% 2وه : Activity طA .مO ا9 3= ZK R K 9 # أي : State of the System or State Variables ات ا%+ م اوW$ اT ر2:K رس8 و.9 9O # مO ا3= 9:P0 واTK`%@ت و/ اN آlK اتZ* 3وه .$* 3= اتZ* اQ** مOا
14
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15
:مW$[ ا% اتZ* ا56 هN^ $2 d ا3 # YB8 $ آ ا$Hث رK 3*ات اZ* مO اYf*8 ا6 وه$* م وO اود اG م انO ا9H6 # اH T = ا.مO ا9 3 # YBK .مOا ا6 ه9 هاف وراء درا0 ا9= *د8 ا6Tم وO اYfK * رجK تJ* ا3 # ت: /*K 3* اNا2 اQ ^ل ا3`= *ر إذ ان#0 ] ذ6 اRJ= R : ا3 # YfK إذا آن ض/م وO ا9 ءGH 3T= .مO ا9:Pط اP 9Lة ا6* هK ت و:ل ا2% ووQ ت اH) 9L# هك :Endogenous Activities %2\ا8 اSAا .مO اN دا9:P0 اlKو :Exogenous Activities %! ارSAا 9H ر9:Pf Yf*8S ي6م اO ا.مOا ا6 ه3 # YBK 3*م واO ا9 3= 9:P0 اlKو .ح2*` مO $f l%28 ي6 وا9H ا)ر9:P0 Yf*8 ي6م اO اc/ Z مO 3 8 :Deterministic Activities دة8 # اSAا .TK مK N/P TJ@* 8K / 8 3* ا3وه : Stochastic Activities %R اAI اSAا 3 * إQ8ز2* l%2K ت *دة/ إTJ@* ن2/K و3@ا2P# N/P هYfK Z*8 3* ا3وه 9& ل:# إG آ ان ا3 * إQ8ز2* l%28 Q J* 9& $LZ* K ي6 اXL2= ^ ا .3@ا2P# N/P Z*8 :Continuous Systems ]2# ا#Wا ي2H ر3= @ةD 9 آ5JK و * = ^ إN* N/P مO ا9 T= Z*K 3* ا3وه .5JKA اg* Smooth # N/P ثK 3bر ا: ا/K XK :Discrete Systems 2$# ا#Wا N/P ث8 Q ا3= 9#i = ^ إآ لG اQ Q:;* N/P مO ا9 T= Z*K 3*وا . اQ:;* N/P ث8 Q@i ا3 # R D ل2% و،Q:;* :System Modeling مW$"! ا# اءHض إZ مOا ا6 هl%2 Model ذج2 3 ن او2/ انRJ8 مO 9 را 3* ]م !ة وذO ا3 # T@اH إ/ 8S تU وإ=*ا9 أ3 # 9 H ربJ*ا $%ا2) $م و=;اO اZK 3دي اB8 $ # 3= كKث إر8 و3 %0م اOب ا:i8S 16
ا 9 %0آ ان درا 9ا 2ذج SاOم # R8JK / Kة 2ارات 2ل اOم وذ] #+دة ا 2ذج ا 3ا 9ا # 9 %0إHاء آ2 Nار c/اOم ا 3 %0ا6ي إذا ZK / 8Sإ#د 9Kة ا ى * 9ا ^ = 9 %0را O 9م إ*Lدي * Zت اض واB8 L R :دي ا .T /# / 8S a@* 3آ أن ا 2ذج / 8ان 8رس = 3أز 9إ=*ا9U = ^ / 8إHاء آة Oم )* +ام ا 2ذج و==K j 9ت اOم `*ات #ة ا! Tاو = 3د .9 L @Lوآ 8D # / 8 ]6ا 2ذج درا 9اOم NLإ $@Pوو2Hد5 ا 8 ^ = %ء Qو# 8ة رات ء = * 8اي ر ا=2/ Niن 2ذج N/ ر وآK 3ف ا XK Qه 56ا)رات. أ اع ا#$ذج: ) %R _%Pد ( :Physical Models وه 3ا*2 3K 3اد N^ 9ء 2ذج @:ة = 9 3ا* وذ] * Aر هT / XKوف .9 ر ) %2%2 0 ) %او :Mathematical Models ( 8 Z0 وا*)* 3م @L# Tت ر2K ) 9U8ز8ت إ* ،9دوالH ،اول ،ر 2ت ا( هك ا lK i8ذج اJة Static Modelsوا* Q T* Z*KS 3ا Gوا ذج اآ 9أو ا Dynamic Models 9/8وا* Q T* Z*K 3ا.G وآ ^ل 3 #ا ذج ا` 9@8GاJة 2ذج ء ا Jاام2 ،ذج ` 9او @Dة =3 =6ة 3 R*/ا وآ ^ل 2ذج = 3@8Gآ 3ذج & 9ا*Aاق اا 3= 3ور!9 آ 9ا 9 Tوآ^ ا ذج ا` 9@8Gاآ L N 3= 9ا`8Gء 9 /ا 2م. :8 Kأذآ أ^ j 9ا ذج ا 9U8اJة وآ ]6اآ ).9أ8ذ :ا2Oاه ا*3 l%2KدSت 9 U`Kأو =و2/K 9Lن آ(9 2ف رس 9;Kء ا ذج S 8D #ت درا 3= 9` *) 9 O0 Case Study 9 SJت ا #0ل وا*LAد وا 9وا`8Gء وا R:وا Q *Jا
17
N^ KاOم = 3ا*` /ا 3Oوآ 9اOم: Kر l8أ :9 ا#ع : Stock أي ! Q Z*8ا) Gد* ،3/8ك( G8داد و ;8و " 8ا2* i8ى Levelاو * ، State Variable 9 Zا *ع R#2* 8ا!0ء `*ة ز 9وه 56ا!0ء ا *9#2 TOKSأو O 3`*)Kإذ اiK Tف أو (N;K) R Kل =*ة ز.9 أ^ :9آ 9ا G 3= 5ان# ،د ا T 0ا* ، ^* T/ * 8 3ا 9L:ا ): 3= 9Gر# ،98د ا@/ت = ،Q *J 3آ 9ا;2د = 3ب ،3/ا)2ف ،ا ،RiZا/اه ،9ا* Aب ،أ ا %ب : Flow ه 2ل 9 Z8 Rateا *ع 8Gُ8 2T= ،أو ِ;ُ8ا *ع .ا Aب ا6ي 8G8ا *ع " 8أ Aب اا Nأو ا ر ،Sourceوا6ي ;8ا *ع " 8أ Aب ا)رج أو ا2Zر . Sink أ^ :9آ 9ا 5ا G) 986Zان )إ ب دا Nأو ر( او ا ) $ 9 2إ ب رج أو \2ر(# ،د ا T 0ا *Pاة )إ ب دا Nأو ر( أو ا ) 9#إ ب رج أو \2ر(، ا:ر8ت ا T 9=Jأ ب رج =; dوه 3ا 9L:ا 9@ T/ا *)# ،9د ا@/ت =3 S2 8GK Q *Jدة )إ ب دا Nأو ر( و N;Kأو 2 ;Kت )إ ب رج أو \2ر(، آ 9ا;2د = 3ب GK 3/داد Aد ر )إ ب دا Nأو ر( و ;Kف )إ ب رج أو \2ر( ،ا)2ف G8داد م ا2Pر 0ن )إ ب دا Nأو ر( و;8 ) 9 :إ ب رج أو \2ر( ،اG8 RiZداد *ش )إ ب دا Nأو ر( وN;8 * Nا) R:إ ب رج أو \2ر( ،ا/اه) 9إ ب دا Nأو ر( GKداد *=c و* N;Kون )إ ب رج أو \2ر( ،ا* Aب DA 8G8ع )إ ب دا Nأو ر( وA N;8ه ل )إ ب رج أو \2ر( ،ا / 8 .ان Oا" ا Aب " #ا "N=" $إذ ا$ 2;8م ` Nاو 3= N #ا 2ذج " " #أ ء" )ا *ع( ا` #ت Convertersأو ا%+#ات ا84 #ة : Auxiliary Variables 2Kى أرLم2;K ،م ت 9أو 98Hأو 9;:و *Kا *3= Controllers 9 / ا 2ذج.
18
،9H ا# @اG اN ا، بA ا،TJ ا،T ا،5 ان اG 3= 9=ا2: ا:9 ^أ O8 ا و،عDAت ا# د# ،c=* ا9#2 ، ا*ش9#2 ،9G ا*ض ر ا adverb آل او ف$ا : Information Link ت2I# اS أو راConnectors تaH #ا أن انT ا،ذج2 ا3= & "ء إGH ت2 اd K أوN K 3* ا3وه 9 ;= d;= ت2 N;K 3 هN (د2; او ا9L: او ا5 اN^) 98 آ ت دN;KS ت%2 ا . & ءGJ N;*K ذج2 ء اGH د# 3 # YB8 T ا، ا" ا م5 `ع اK ى إرN;8 انG) ا3= 9=ا2: ذراع ا:9 ^أ 9 آ،بJA ا3 # ث ا;دراتAد ا# ،زTH NZP* TJ ا9 آ،T@ !ا/ 8 3* اT 0ا . ا،ل =*ة ا*ش2D ،9G ] ر اUK ;ار،9H ا# @اG اN ا :م آW$ ا%آT وW$ ا% مW$ ا# و
Stock Source/Sink Flow
Information Link
AuxiliaryVariable
19
:ا ا Differential Equations %2)د`ت اI#ا Z* 9*;ت اP ن وده2/*K 98H تS د# رة# 3 ه9 U`*ت اSا د x = f ( t ) Z* 9 U`K تS د3*ن ا+= x = f ( t ) آن2 ^ = NU`* 9 L يH dx + tx + 3 = 0 dt d 2x dx + 5 − 14 x − 10 = 0 2 dt dt 3 2 d x d x dx − − +x+2=0 dt 3 dt 2 dt
"ق إ:* ف2 وLinearity T*: وDegree T*H ر9 U`*ت اS ا دG *Kو قD ة9 U`*ت اS ا دNK و.9:) وا9^و" وا0ت اHت ارSا د .ا ا ;ر6ى ه2* R 8 T *ض
: %S ا3ر! او8 ا6 %2)د`ت اI# ا:`او N/P " ا# 3وه f (t, x)
dx + h (t, x ) = 0 dt
. x وt 3= 9: دوالh ( t , x ) وf ( t , x ) V :N` 9 L اتZ* ي2K تS دN N/P اTU و/ 8 h ( t , x ) وf ( t , x ) N آXإذا آ f (t, x ) = p (t ) q ( x )
h (t, x ) = r (t ) s ( x )
2 ه9 U`* ا9 ا دN/! ن2/8 أي p (t ) q ( x )
dx + r (t ) s ( x ) = 0 dt
لJ ا3= t L Q J p ( t ) ≠ 0 " و: ل اJ ا3= x L Q J s ( x ) ≠ 0 " !ط ان#و 3*ات آZ* اN= Q:* s ( x ) p ( t ) ≠ 0 " # 9 ا د3=D 9 ; ،": ا q ( x) r (t ) dx + dt = 0 s ( x) p (t )
! اN/* N " ا# Nو 20
q ( x)
r (t )
∫ s ( x ) dx + ∫ p ( t ) dt = C .9و0وط اP " ا# *8 إ *ريX Y C V : ل 9* ا9 U`* ا9 ا دN dx 1 − x − =0 dt 1 + t
: ا " # N= x ≠ 1 " !ط أن# 1 − x " # 9 ا د3=D ; 1 dx 1 − = 0, x ≠ 1 1 − x dt 1 + t
J اودRKK وdt NU`* ا3= =:ب اi dx dt = , x ≠1 1− x 1+ t
فD N/ N/* ا6 أ/ 8 وj اTi # 92` X%ن اbات اZ* ا dx
dt
∫ 1 − x + ℓn C = ∫ 1 + t ,
x ≠1
أو − ℓn 1 − x + ℓ n C = ℓn 1 + t , x ≠ 1
أي
(1 − x )(1 + t ) = C , x ≠ 1 t N;* اZ* ا9S x Q *اZ* Nو x = 1±
C , x ≠1 1+ t
:\b ل 9* ا9 U`* ا9 ا دN t
dy − ( x − t ) = 0, y ≠ 0 dt
21
9;ل أن اا8 ) "و0 ا9H ار9 J* دوال3 هf ( t , x ) = t , h ( t , x ) = x − t اوال 9;; ا56 ( وهg ( kx, ky ) = k n g ( x, y ) 9L اX;; إذاn 9H ار9 J* g ( x, y ) :3* آN ا3= # K J تU`K N/! " # 9 ا دQU ووx = st j82* t ( s − 1) dt − t ( s dt + t ds ) = 0, t ≠ 0
J N/* واt Z* 9 اودRK*
∫ ds = − ∫
dt + C, t ≠ 0 t
أي s = C − ℓn t , t ≠ 0
x Z* ا9S أو x = t ( C − ℓn t ) , t ≠ 0
:"و0 ا9H ار9:) ا9ا د امN/P اTو dx + P (t ) x = Q (t ) dt
9L ":8 Nا − P ( t ) dt − P ( t ) dt ∫ P(t ) dt +e ∫ x = Ce ∫ ∫ Q ( t )e dt
واxCF $ G8 وComplementary Function 9 / ا9 " اا8 ول0 ا اV N/P " ا# ن2/8 N أي اxPS $ G8 وParticular Solution ا)صN ا3^ا . x = xCF + xPS : ل 9 U`* ا9 ا دN dx + tx = t dt
22
N واxCF = Ce − ن2/8 وxPS = e −
1 t2 2
1 t2 2
− P ( t ) dt T أJ= P ( t ) = t 6 f ] وذxCF = Ce ∫ 9 / ا9 ااH2
∫ te
1t2 2
− P ( t ) dt ∫ P( t )dt dt = 1 J= Q ( t ) = t V xPS = e ∫ ∫ Q ( t )e dt ا)ص
General Solution امNا x = Ce
− 12 t 2
+1
*ري *ج إ" !وطA اX ^د اJ8A . C إ *ري واX Y ي28 امN ان اS 0 = C + 1 ⇒ C = −1 J j82*= t = 0 # x = 0 /* Initial Values (IV) 9أو
. x = 1 − e−
1 t2 2
9و0وط اP اXK امNن ا2/8و : ل 9 U`* ا9 ام دN اHأو
d 2 x 1 dx − = te 2t dt 2 t dt
J y =
dx QU2 / و9^ ا9Hو ارK 9ا د dt
dy 1 − y = te 2t dt t
.R: 8 * آT *ك8 . y 3= "و0 ا9H ار3وه
Higher Order Differential :%2Iر!ت ا8 ا6 %2)د`ت اI#ا Equations Linear Equations with Constant c تa I# %Sد`ت اI# ا:`أو Homogeneous Case Z# ا اCoefficients N/P " ا# 3وه dnx d n −1 x + a + ⋯ + an x = f ( t ) 1 dt n dt n −1
م# N/P وɺɺx =
d 2x dx وxɺ = 2 ه9 ا دN/! d * صGK ف *)م2 2 dt dt
N/P " ا# 9; ا9 ا دg*= x( n ) = x( n ) + a1 x ( n −1) + ⋯ + an x = f ( t )
23
dnx dt n
ن2/K \) X ا2Y a1 , a2 ,..., an V ،9* Y 9: ت9 U`K تS " دKو أيf ( t ) = 0 T= ن2/8 3*ت واS ا د56T cJ* اN/P اSف رس او2 .(9;; N/P " ا# تSا د x( n) + a1 x ( n −1) + ⋯ + an x = 0
3= j82* ، C إ *ريX ^ وeλt ≠ 0 V x = Ceλt Nب اJ تS ا د56 هN a*8 9; ا9ا د
(λ
n
+ a1λ n −1 + ⋯ + an ) eλt = 0
.Characteristic Polynomial ةG " آ^ة اود اK P ( λ ) ≡ λ n + a1λ n −1 + ⋯ + an 9ا د
(`ر%أ
أو
ور6H
)أو
ل2
3ه
T
ح2 ا
λ
L
Auxiliary Equation ة# ا9 " ا دK 3* واP ( λ ) ≡ λ n + a1λ n−1 + ⋯ + an = 0 ور6J اn T P ( λ ) = 0 ن+= n 9H آ^ة ود ار3 هP ( λ ) أنVو او ازواج9;; ن2/K ور إ ان6J ا56ن ه+= 9;; a1 , a2 ,..., an X وإذا آλ1 , λ2 ,..., λn xi = Ci eλit , i = 1, 2,..., n * اN آ. Complex Conjugates 9 ا *;ر9اد ا آ#0ا
ن2/8 وCi إ *ريX ^ 9 U`* ا9 دN 2ه x = C1e λ1t + C2 eλ2t + ⋯ + Cn e λnt
تS ا د9 3= وComplementary Function
9 / ا9 اا3 8 وi8أ
.9 / ا9 اا2 ام هNن ا2/8 9 J* ا : ل ɺɺ x + 3 xɺ + 2 x = 0 9 U`* ا9 ا دN
: ا ن2/8 وλ1 = −1, λ2 = −2 3 هP ( λ ) = 0 ور6H وP ( λ ) ≡ λ 2 + 3λ + 2 ة# ا9ا د امNا x = C1e − t + C2 e −2t
.9و0وط اPدان ا8 8 ^ * إ *ر : Repeated Roots ر6Jار ا/K 9 3ر ه6Jا ا6T 9:K ل ا2 ن ا+= ا اتr د# λ = λ1 ^ ر6H ر/K إذا eλ1t , teλ1t , t 2 eλ1t ,..., t r −1eλ1t
24
: ل ɺɺɺ x + 4 ɺɺ x + 5 xɺ + 2 x = 0 9 U`* ا9 ا دN
: ا 3 هP ( λ ) = 0 ور6H وP ( λ ) = λ 3 + 4λ 2 + 5λ + 2 = ( λ + 1) ( λ + 2 ) ة# ا9ا د 2
امNن ا2/= K ر/ −1 ر6J اλ1 = −1, λ2 = −1, λ3 = −2 x = C1e − t + C2te− t + C3e −2t t = 0 # x = 1, xɺ = 0, ɺɺ x = 1 9و0وط اP 98 *رA اX ا2^ اL Hأو
NU`* ا8D # H2 امN ا xɺ = −C1e− t + C2 (1 − t ) e − t − 2C3e−2t
ɺɺ x = C1e − t − C2 ( 2 − t ) e −t + 4C3e −2 t
J 9و0 ; اj82* و C1 + C3 = 1 −C1 + C2 − 2C3 = 0 C1 − 2C2 + 4C3 = 1
ا)صN " ا# N j82* وC1 = −1, C2 = 3, C3 = 2 ل2 اT 3*وا x = ( 3t − 1) e− t + 2e −2 t
Linear Equations with Constant c تa I# %Sد`ت اI# ا:%c : Inhomogeneous Case Z# ا%1 اCoefficients 9 J* ت \ اS ام دNا x( n ) + a1 x ( n −1) + ⋯ + an x = f ( t )
Complementary Function 9 / ا9 " اا8 xCF V x = xCF + xPS N/P " ا# 2ه .Particular Solution ا)صN " ا8 xPS و . ا)صNد اJ8ن إbف *ض ا2 و،9 / ا9د ااJ8 إ9` آ9; ا`;ة ا3= 8; رأ : ا)صNد اJ8ق إD : D N ا9;8D . Dn ≡
dn d2 d 2 و D ≡ ^ = D ≡ 9L ف8 وD NU`* اN# :l8K n 2 dt dt dt
9L P ( D ) ≡ D n + a1 D n −1 + ⋯ + an Polynomial Operator آ^ة اودN# ف
25
P ( D ) f ( t ) ≡ ( D n + a1 D n −1 + ⋯ + an ) f ( t ) = f ( n ) ( t ) + a1 f ( n −1) ( t ) + ⋯ + an f ( t )
أيD 0 x ( t ) = x ( t ) وan D0 2 هan أن اS) ةn NU`* 9 L 9 داf ( t ) V .(ة2 اNU`K N# : ل D NU`* اN# N/! " # 9* ا9 U`* ا9 ا دQU d 2x dx − 3 + 2 x = te − t 2 dt dt ɺɺ x − 3 xɺ + 2 x = te− t
: ا D 2 x − 3Dx + 2 x = te − t
(D
2
− 3D + 2 ) x = te − t
P ( D ) ≡ D 2 − 3D + 2 ه
: D ) ا4 \ اص ل آ^ات# P ( D ) , Q ( D ) , R ( D ) وآنNU`* 9 L دوالf ( x ) , g ( x ) Nإذا آن آ 9*اص ا2) اT ل آ^ات اود# ن+= ود 1) P ( D ) + Q ( D ) f ( x ) = P ( D ) f ( x ) + Q ( D ) f ( x ) 2 ) P ( D ) Q ( D ) f ( x ) = P ( D ) Q ( D ) f ( x )
3) D ( aD r ) f ( x ) ≡ aD r +1 f ( x )
4 ) P ( D ) f ( x ) + g ( x ) ≡ P ( D ) f ( x ) + g ( x ) f ( x ) 5) P ( D ) + Q ( D ) ≡ Q ( D ) + P ( D ) 6 ) P ( D ) Q ( D ) R ( D ) ≡ P ( D ) Q ( D ) R ( D ) 7 ) P ( D ) Q ( D ) + R ( D ) ≡ P ( D ) Q ( D ) + P ( D ) R ( D ) 8 ) ( D − λ ) Q ( D ) ≡ DQ ( D ) − λ Q ( D )
The Factorization Theorem اI %2 اW
( λ − λ1 ) , ( λ − λ2 ) ,..., ( λ − λn ) Nا2# T λ n + a1λ n −1 + ⋯ + an `*ض أن آ^ة اود 6@# D n + a1 D n −1 + ⋯ + an ≡ ( D − λ1 )( D − λ2 )⋯ ( D − λn )
.$ Permutaion N8دK يf *ل8 ان/ 8 80ف ا: ان اS 26
: ل ɺɺ x + 5 xɺ + 6 x = cos t
(D
2
+ 5 D + 6 ) x = cos t
( D + 2 )( D + 3) x = cos t ( D + 3)( D + 2 ) x = cos t Distinct Roots 2$#"ور اZ ا: ل T ا2# N/! 3= 9^ ا9H ار9 ا9 ا" ا دO
( D − λ1 )( D − λ2 ) x = 0, λ1 ≠ λ2 g*= 9; ا9L ا3= ( D − λ2 ) x = u ض2#
( D − λ1 ) u = 0 أو du − λ1u = 0 dt
N اT وu 3= "و0 ا9H ار9 J* 9 U`K 9 د3وه u = C1eλ1t
J ( D − λ2 ) x = u # j82* و
( D − λ2 ) x = C1eλ t 1
dx − λ2 x = C1eλ1t dt
N/P " ا# وx 3= "و0 ا9H ار9 J*\ 9 U`K 9 د3وه dx + P (t ) x = Q (t ) dt
2 هT و − P ( t ) dt − P ( t ) dt ∫ P (t ) dt x = Ce ∫ +e ∫ ∫ Q ( t )e dt
(ب: 8 * آN%`**ك اK) Nن ا2/8 وQ ( t ) = C1eλ t وP ( t ) = −λ2 V 1
x=
C1 eλ1t + C2 eλ2t λ − λ ( 1 2)
x = C1′eλ1t + C2 eλ2t
27
Repeated Roots رة#"ور اZ ا: ل 9^^ ا9H ار9 J* ا9* ا د
( D − λ1 ) ( D − λ2 ) x = 0 2
g8 وu = C1eλ t T 3* ( واD − λ2 ) u = 0 a*8 ( D − λ1 ) x = u QU ووNا2 اN8* 2
2
( D − λ1 )
2
x = C1eλ2t
9; ا9 ا دgK ( D − λ1 ) x = v 9 */ و
( D − λ1 ) v = C1eλ t 2
N اT 3*وا v = C1′eλ2t + C2 e λ1t
وا ا
( D − λ1 ) x = C1′eλ t + C2eλ t 2
1
" # N Tو x = C1′′eλ2t + ( C2 x + C3 ) eλ1t
.N N%`* إآ ل اR: " ا# :9O Complex Conjugate Roots ر/# ا9آ#"ور اZ ا: ل N/P " ا# 9;; تT 3* اP ( λ ) = 0 9 دRر آ6H د2H و9 3= ا)صN اN/! ن2/8 وλ = µ − iν N/P " ا# & ر6H ف ان هك+= λ = µ + iν e µ t ( C1 cosν t + C2 sin ν t )
N/P ا$ ن2/8 ا)صNن ا+= ةm ور6J ا98دK 9 3=و e µ t Pm−1 ( t ) cosν t + Qm−1 ( t ) sin ν t
فiK و98 إ *رX ا2Y T وm − 1 9H ارt 3= آ^ات ودQm−1 ( t ) وPm−1 ( t ) V 9 اا3:K 3/ P ( λ ) = 0 9 د9;; ور6H a*K 3* ي ا0 اود إ" اود ا56ه . xCF 9 / ا : ل 9 إ" ا دO
(D
5
− 5 D 4 + 12 D 3 − 16 D 2 + 12 D − 4 ) x = 0
28
Nا2# N/! 3=و
( D − 1)( D − 1 − i ) ( D − 1 + i ) 2
2
x=0
3 ه9 / ا9 اا. 2 98دK T ν = 1 وµ = 1 9 ا *;ر9 ا آNا2ا xCF = et C1 + ( C2 + C3t ) cos t + ( C4 + C5t ) sin t
/ و9 / ا9د ااJ8A gU واN/P تS ا دN d 8 D N ا:K آ أن . ا)صNد اJ8 إ3= ^ اآgi*K ف2 D Nة ا2L *) اN/P 9 J* \ 9 د9 */ P ( D ) x = f (t )
9L ":8 ا)صNوا xPS = P −1 ( D ) f ( t )
:3*` آ8س و2/ اN اP −1 ( D ) V / xɺ =
dx =t dt
أو Dx = t
J x Z* N 1 x = D −1t = t 2 + C 2
]6 وآ. ة واةN/* اN^ K D −1 أي D2 x = t
J س2/ د اJ8+ x = D −2t = D −1 ( D −1t )
J ر/* اN/* و 1 1 x = D −2t = D −1 t 2 + C1 = t 3 + C1t + C2 2 6
. ةr ر/* اN/* ا3K D − r أي أن
29
س2/ ن2/K وD −1 3= ن آ^ة ود2/K P −1 ( D ) ن+= D 3= آ^ة ودP ( D ) أن أيP ( D ) P −1 ( D ) P ( D ) ≡ 1 xPS = P −1 ( D ) f ( t ) أيf ( t ) " # P ( D ) س2/ :K 9D 2 ا)ص هNد اJ8إذا إ
.N/* 9 L f ( t ) ن2/K " أن# Inverse Operator سI# اI ا:V I0 9L 3:8 f ( t ) 9 " دا# N 8 ي6 ( واD − λ ) س2/ اN اYfK −1
(D − λ)
−1
f ( t ) = eλt ∫ f ( t )e − λt dt
: ل 9 ا)ص دN اHأو
( D − 1)( D − 2 ) x = et ن2/8 ( وD − 2 ) ( D − 1) 2س ه2/ اNا −1
x = ( D − 2)
−1
( D − 1)
−1
−1
et
a*8 ( D − 1) et 3 # اl8* ا:* −1
( D − 1)
−1
et = tet
a*8 ( D − 2 ) tet " # l8* ا:K Y −1
xPS = − ( t + 1) et
.ب2 : ا2وه :\b ل 9 ا)ص دN اHأو
(D
2
− 2 D + 2 ) x = et
:Nا Nا2# N/! " # 9 ا دQU2
( D − 1 − i )( D − 1 + i ) x = et :3* آN اQ *و
30
( D −1+ i ) x = ( D −1− i ) = e(
1+ i )t
−1
et
t −(1+ i )t
∫e e
dt = iet
ن2/8 3* و x = ( D − 1 + i ) iet −1
= e(
1−i )t
t −(1−i )t
∫ ie e
dt = et
2 ا)ص هNأي ان ا xPS = et
.ب2 : ا2وه
31
:ا ا Difference Equations %Fد`ت اوI#ا يH Z* تLن وده =و2/*K 98H تS د# رة# 3 ه9Lت ا`وSا د Z* 9Lت =وS د3*ن ا+= xk = x ( tk ) , k = 0,1, 2,..., n آن2 ^ = 9 ` L 6 f8 xk , k = 0,1, 2,..., n xk − xk −1 − xk − 2 = 0 xk + 2 xk −1 = 5 xk − 3 = 0 xk − 3k − 7 = 0
9*اص ا2) ا$ وB 3` ) ا9زاA اN# :V I0 1) Bxk = xk −1 2 ) B 2 xk = B ( Bxk ) = B ( xk −1 ) = xk − 2 3) B m xk = xk − m 4 ) Bc = c
5 ) B ( cxk ) = cxk −1
9*اص ا2) ا$ ≡ ∇ و1 − B 8`* اN# :V I0 1) ∆xk = xk − xk −1 2 ) ∆ 2 xk = ∆ ( ∆xk ) = ∆ ( xk − xk −1 ) = xk − 2 xk −1 + xk − 2 3) ∆ m xk = ∆ m −1 ( ∆xk ) 4 ) ∆c = 0
5 ) ∆cxk = c∆xk
6 ) ∆ ( xk + yk ) = ∆xk + ∆yk
:ة84F m m m− j ∆ m xk = ∑ ( −1) xk − j j =0 j
S ≡ ∆ −1 3@Tb اQ J* اN# وF ≡ B −1 30 ا9زاA اN# ه8 & # هك
.9H اXL وT)* ف2 3*وا :3ر! او8 ا6 %S ا%Fد اوI#ا N/P " ا# 3وه a0,k xk + a1,k xk −1 = ck , k = 1, 2,..., n
32
N/P " ا# i8 اR*/Kو xk = −
a1,k a0, k
xk −1 +
ck , k = 1, 2,..., n a0,k
xk = Axk −1 + C , k = 1, 2,..., n
.9* Y وA ≠ 0 V J k = 1 QU2 ،ة: x0 " =ض ان# xk = Axk −1 + C , k = 1, 2,..., n 9 ا دN x1 = Ax0 + C k = 2 9 ;و x2 = Ax1 + C
= A ( Ax0 + C ) + C = A2 x0 + (1 + A ) C k = 3 9 ;و
x3 = Ax2 + C
= A ( A2 x0 + ( A + 1) C ) + C = A3 x0 + (1 + A + A2 ) C
م# N/P و xk = Ak x0 + C (1 + A + A2 + ⋯ + Ak −1 ) , k = 1, 2,...
انS 1+ A + A +⋯ + A 2
k −1
1 − Ak , if = 1− A k, if
A ≠1 A =1
N اg8و k 1 − Ak , if A ≠ 1 A x0 + C , xk = 1− A x0 + Ck , if A = 1
k = 0,1, 2,...
:1 ل Xإذا آ xk = 2 xk −1 + 1, k = 1, 2,...
J x0 = 5 9 او9 ;و
33
, k = 0,1, 2,...
1 − 2k xk = 5 ( 2k ) + 1 1− 2
= 6 ( 2 k ) − 1, k = 0,1, 2,...
و :#+ء kا; J 0,1, 2,...ان ا2/8 Nن ا ** . 5,11, 23, 47,... 9 TاI#د`ت او %Fا 6 %Sا8ر! او8&f 3ام : Excel Nا دSت ا`و/K 9Lار / 8 8ا;م 2 9ا Excel 9:آ*:3 أد Nا*) 3ا : 0ا0ول و*" ا^ Y Vا ا :ا^ "* Vا ى ا :(R
ا*:9J
و Tا N/Pا*:3
34
x(k) 7000 6000 5000 4000 3000 2000 1000 0 0
2
4
6
8
10
12
:2 ل Xإذا آ 2 xk − xk −1 = 4, k = 1, 2,...
J x0 = 3 9 او9 ;و xk =
1 xk −1 + 2, k = 1, 2,... 2
=(
)
1 k 2
3+ 2
1 − ( 12 )
k
1 − ( 12 )
, k = 0,1, 2,...
= 4 − ( 12 ) , k = 0,1, 2,... k
. 3,3 12 ,3 43 , 3 87 ,3 1615 ,... 9 ** ن ا2/8 N ان اJ 0,1, 2,... ; اk ء:#+ و : Excel ام8&f ا :(R *" ا ى اV^ ا: ا اY V^ول و*" ا0 ا: 0 )ا3* اN أد
9J*ا
35
:3* اN/P اTو 4.5 4 3.5
x(k)
3 2.5 2 1.5 1 0.5 0 0
1
2
3
4
5
k
36
6
7
8
:3 ل Xإذا آ xk + xk −1 = 1, k = 1, 2,...
J x0 = 1 9 او9 ;و xk = − xk −1 + 1, k = 1, 2,... = ( −1) + k
=
1 − ( −1)
k
1 − ( −1)
, k = 0,1, 2,...
1 k 1 + ( −1) , k = 0,1, 2,... 2
. 1, 0,1, 0,1, 0,... 9 ** ن ا2/8 N ان اJ 0,1, 2,... ; اk ء:#+ و : Excel ام8&f ا :(R *" ا ى اV^ ا: ا اY V^ول و*" ا0 ا: 0 )ا3* اN أد
9J*ا
:N/P اTو
37
1.2 1 0.8 0.6 0.4 0.2 0 30
25
20
15
10
5
0
k
: AF$ رأ 3= 8ا 9 ^0ا ; 9YY 9ا دSت ا`و 9Lآن Nا0و" ; 9ا0و 9ا :ة ** 9ا0رLم *#ة ،و Nا^ 9 ; 9ا0و 9ا :ة ** 9ا#0اد ا*;*K 3رب إ 3اد ، 4و Nا^^ 9 ; 9ا0و 9ا :ة ** 9ر 0 Lو 1ا */رة ) أي ا; 6 6*Kب 0و .(1 :8 Kأو L Hاو 9دSت ا`و 9Lا ;K 3:K V 9ف \ ا6ي OSة ;. " ه 3ا; 9ا0و 9ا* 9 ** 3:K 3ا0رLم أو ا#0اد ا *;ر 3= 9ا ^ل 1وه6/ا. YfKا; ا0و " # 9ا ** 9ا:9JK 2ف *ض YfKا; 9ا0وK " # 9ف ا ** 9ا 9JKا^ Nل ا*:3 ل:4 Nا د 9ا`و 9Lا*9 1 1 xk −1 + , k = 1, 2,... 2 2
; ا0و. x0 = 0,1, 2 9 ا ا&: Excel S
38
= xk
ان هك 3= ;* Nآ NاSت إذ ;*Kرب ** 9ا Nإ" .1 ل:5 Nا د 9ا`و 9Lا*9 xk = 3 xk −1 − 1, k = 1, 2,...
; ا0و. x0 = 0,1, 2 9 39
: Excel S&ا ا
.N ا9 ** #*K إذ9 اي3= ;* N هكc ان
: %FوP دI# %2)0 دI k /0 *; ف2 .T 9 K 9L =و9 دN 98T آ9 U`K 9 دN دJ8 إ/ ا ; 9= 9 داx /* .9* Y و" ت0 ا9H ار9:)ت اS " ا د# 9PL ا 9 U`* ا9; ا دK 3* واa ≤ t ≤ b =*ة3= t 9;; Dx ( t ) =
d x ( t ) = Ax ( t ) + C , a ≤ t ≤ b dt
40
.ة: x0 = x ( a ) 9و0 ا9 ; و`ض أن ا98 إ *رX ا2Y C وA ≠ 0 V 9 ` ; a ≤ t ≤ b إ *ال ا`*ةS اوR :*K 9L =و9 د9 U`* ا9 ا د56 هR8;* " إa و; ا`*ةn RH2 اg اL ا6 f .T # 9L ا`و9 ا دl8K / 8 ;* ;ط اh = ( b − a ) n ل2: ذات ا98اء ا * وGH0 اn " إb t0 = a, t1 = a + h, t2 = a + 2h,..., tn = a + nh = b
. xk = x ( tk ) = x ( t0 + kh ) 9 */ ز2 اd #د انNU`* ا وف ;ر ا ∆xk h →0 h
Dx ( tk ) = lim
Dx ( t ) =
d x ( t ) = Ax ( t ) + C , a ≤ t ≤ b 9 U`* ا9وا إ *ال ا د8 ا6Tو dt
9L ا`و9 د ∆xk = Axk + C , k = 0,1, 2,..., n − 1 h
أو xk +1 = (1 + Ah ) xk + Ch, k = 0,1, 2,..., n − 1
و" ت0 ا9H ار9L =و9: 3 ة ه0 ا9 ا د.": x0 "و0ط اP اQ آT و9* Y C C k xk = (1 + Ah ) x0 + − , k = 0,1, 2,..., n − 1 A A C = 0 # :9%) ا9ا
9 U`* ا9 ا د.9 T اTK;:* ] وذ9 اهT 9 3وه Dx ( t ) =
d x ( t ) = Ax ( t ) , a ≤ t ≤ b dt
. x 9 ااQ R *8 t ـ9 x 9 ا3O اZ*أو ل ا 3 هT 9 * ا98;* ا9L ا`و9ا د xk +1 = (1 + Ah ) xk , k = 0,1, 2,..., n − 1
T 3*وا xk = x0 (1 + Ah )
k
41
2 ه9 U`* ا9 ا دN انS x ( t ) = x0 e
A( t − t0 )
ث8 A < 0 # وx 9 اExponential Growth 3 ا2 a*8 A > 0 9 3= . Exponential Decay 3 ش ا/إ
42
Linear Difference Equations c تa I# %S ا%Fد`ت اوI#ا : with Constant Coefficients N/P " ا# X إذا آ9: T اk = 0,1, 2,... ; " ا# 9= 9L =و9;ل ان د8 f 0 ( k ) xk + n + f1 ( k ) xk + n −1 + ⋯ + f n −1 ( k ) xk +1 + f n ( k ) xk = g ( k )
ا تX إذا آ. k = 0,1, 2,... " # 9= k دوال ـf 0 , f1 ,..., f n−1 , f n , g V 9* Y ن2/K م انG 8S g 9 )ا9* Y ت9: ن2/K 9ن ا د+= X ا2Y T آf 0 , f1 ,..., f n . n 9Hن ار2/K 9ن ا د+= f n ≠ 0 وf 0 ≠ 0 N آX إذا آ.(9: او :9 ^أ 9* Y ت9: 9Lت =وS د3*ا 2 xk +1 − xk = 6 3 xk + 2 + 2 xk +1 + xk = 3k
xk +3 − xk = k
9 ا د3=D ;K Ni=0 = اf 0 ≠ 1 ن2/K # .RK* " ا# 3 و2 و1 تH درT 9 ا دN/! g8ا و2 8 وxk +n N NJ 3/ T # xk + n + a1 xk + n −1 + ⋯ + an −1 xk +1 + an xk = r ( k )
. ai = fi f 0 , i = 1,..., n; r ( k ) = g ( k ) f 0 V ف `*ض ان2 .9* Y ت9:) ا9Lت ا`وS د0 اN/Pف *)م ا2 ; " ا# 9= 98 إ *ر9 داr ( k ) , k = 0,1, 2,... 9 وااan ≠ 0 i8 واX ا2Y a1 , a2 ,..., an 3 و2 و1 تH ار9* Y ت9: 9Lت =وS د3 ه3*= ا6 ه3 # و.ة: ا 3ا2* " ا# xk +1 + a1 xk = r ( k )
xk + 2 + a1 xk +1 + a2 xk = r ( k )
xk +3 + a1 xk + 2 + a2 xk +1 + a3 xk = r ( k )
9ا د xk + n + a1 xk + n −1 + ⋯ + an −1 xk +1 + an xk = 0
9:) ا9L ا`و9 د9 * وا9* Y ت9 J* ا9:) ا9L ا`و9 " ا دK . xk + n + a1 xk + n −1 + ⋯ + an −1 xk +1 + an xk = r ( k ) 9* Y ت9 J* \ا
43
: W ن+=
xk + n + a1 xk + n −1 + ⋯ + an −1 xk +1 + an xk = 0
9 د
x( 2)
و
x(1)
إذا آن
. C2 وC1 98 إ *رX ا2^ T N i8 ا2 هC1 x(1) + C2 x( 2) :98O N xPS وآنxk + n + a1 xk + n−1 + ⋯ + an −1 xk +1 + an xk = 0 9 J* ا9 دN xCF إذا آن N xCF + xPS ن+= xk + n + a1 xk + n −1 + ⋯ + an −1 xk +1 + an xk = r ( k ) 9 J* \ ا9 د . xk + n + a1 xk + n −1 + ⋯ + an −1 xk +1 + an xk = r ( k ) 9 J* \ ا9 د General Solution of the Homogeneous Z#د اI#2 مIا ا : Equation xk + n + a1 xk + n −1 + ⋯ + an −1 xk +1 + an xk = 0 9 J* ا9 ام دNد اJ8ف *ض إ2
:3* آ9 ^f 9 J* ا9 ام دN اH( او1 xk − xk − 2 = 0
Auxiliary Equation ة# ا9 " ا د# N= xk = λ k , λ ≠ 0 Nب اJ λ k − λ k −1 = 0 λ k −1 ( λ − 1) = 0
امNن ا2/8 وλ1 = λ2 = 1 8 * و8رGH T ة# ا9ن ا د+= λ ≠ 0 و ان 2 ه9 J* ا9 د xk = ( C1 + C2 k )1k = C1 + C2 k
.9و0وط اPه ا8K *8 C2 وC1 8 * إ *رY Tو 9 J* ا9 ام دN اH( او2 xk − 3 xk −1 + 2 xk − 2 = 0
Auxiliary Equation ة# ا9 " ا د# N= xk = λ k , λ ≠ 0 Nب اJ λ k − 3λ k −1 + 2λ k − 2 = 0
λ k − 2 ( λ 2 − 3λ + 2 ) = 0
44
λ1 = 1, λ2 = 2 ورGJ اT ( وλ 2 − 3λ + 2 ) = 0 gK ة# ا9ن ا د+= λ ≠ 0 و ان
2 ه9 J* ا9 ام دNن ا2/8و xk = C1λ1k + C2λ2k = C1 + C2 2k
.9و0وط اPه ا8K *8 C2 وC1 8 * إ *رY Tو 9 J* ا9 ام دN اH( او3 xk + 3 xk −1 + xk − 2 = 0
Auxiliary Equation ة# ا9 " ا د# N= xk = λ k , λ ≠ 0 Nب اJ λ k + 3λ k −1 + λ k − 2 = 0
λ k − 2 ( λ 2 + 3λ + 1) = 0
ورGJ اTو
(λ
2
+ 3λ + 1) = 0
gK ة# ا9ن ا د+=
2 ه9 J* ا9 ام دNن ا2/8 وλ1 =
λ≠0
و ان
−3 + 5 −3 − 5 , λ2 = 2 2
xk = C1λ1k + C2λ2k k
−3 + 5 −3 − 5 = C1 + C2 2 2
k
.9و0وط اPه ا8K *8 C2 وC1 8 * إ *رY Tو 9 J* ا9 ام دN اH( او4 xk + xk −1 + 14 xk − 2 = 0
Auxiliary Equation ة# ا9 " ا د# N= xk = λ k , λ ≠ 0 Nب اJ λ k + λ k −1 + 14 λ k − 2 = 0
λ k − 2 ( λ 2 + λ + 14 ) = 0
λ1 = λ2 = − 12 ورGJ اT ( وλ 2 + λ + 14 ) = 0 gK ة# ا9ن ا د+= λ ≠ 0 و ان
2 ه9 J* ا9 ام دNن ا2/8و xk = C1λ1k + C2λ2k = ( C1 + C2 k ) ( − 12 )
k
.9و0وط اPه ا8K *8 C2 وC1 8 * إ *رY Tو 9 J* ا9 ام دN اH( او5 45
xk + xk − 2 = 0
Auxiliary Equation ة# ا9 " ا د# N= xk = λ k , λ ≠ 0 Nب اJ λ k + λ k −2 = 0
λ k − 2 ( λ 2 + 1) = 0
ورGJا
Tو
(λ
2
+ 1) = 0
gK
ة# ا
9ا د
ن+=
λ≠0
ان
و
2 ه9 J* ا9 ام دNن ا2/8 وλ1 = i = −1, λ2 = −i = − −1 xk = C1λ1k + C2λ2k
πk = C1 cos + C2 2
.9و0وط اPه ا8K *8 C2 وC1 8 * إ *رY Tو Polar Form 3:L N/! " إ9اد ا آ#0 اN82K :9O 3* اj82* r ( cos θ + i sin θ ) ":; اN/P إ" اa + bi Rل اد ا آ28 r = a 2 + b2 a b cosθ = , sin θ = , −π < θ ≤ π 2 2 2 a +b a + b2
R*/K λ1 = i = 0 + 1i
= r ( cos θ + i sin θ )
r = 02 + 12 = 1 cosθ =
0 1 π = 0, sin θ = = 1 ⇒ θ = 1 1 2
λ2 N^ و
2 ه9 J* ا9 ام دNن ا2/8و xk = Ar k cos ( kθ + C )
N/P " ا# T J xk = Ar k cos ( kθ + C ) 9 ا د3= j82* ا8 * آR: *ك8 . xk = C1 cos
πk
+ C2 2
. اN xk = sin
kπ ه أنx0 = 0, x1 = 1 9و0 ; ا:8 K 2
46
إذا آن:9J* xk + a1 xk −1 + a2 xk − 2 = 0
ة# ا9ري ا د6H λ2 وλ1 وآنa2 ≠ 0 وX ا2Y a2 وa1 V λ 2 + a1λ + a2 = 0
تL ":8 xk + a1 xk −1 + a2 xk −2 = 0 9L ا`و9 ام دNن ا+= xk = C1λ1k + C2 λ2k
;; و)* ` وλ2 وλ1 وذ] إذا آن xk = ( C1 + C2 ) λ1k
و8 ;; و* وλ2 وλ1 وذ] إذا آن xk = Ar k cos ( kθ + C )
N/P " ا# آ *;ر أيλ2 وλ1 وذ] إذا آن λ1 = a + bi = r ( cosθ + i sin θ )
λ2 = a − bi = r ( cos θ − i sin θ )
: Particular Solution of the Complete Equation 2 د اI#2 ا اص 9 / ا9 ا)ص دNد اJ8ف *ض إ2 xk + n + a1 xk + n −1 + ⋯ + an −1 xk +1 + an xk = r ( k )
:3* آ9 ^f 9( * ا د1 xk + 2 − 3 xk +1 + 2 xk = 3k
Method of Undetermined Coefficients ا ت \ ا دة9;8: T ف2 :3*آ 9 L دJ8 ول إ. د8 A X ^ اN اV xk* = A3k N/P " ا# N دJ8ول ا ض2 ، xk* T= ن2/8 (تH )إن وA ـ
47
x*k +2 − 3 xk* +1 + 2 xk* = A3k + 2 − 3 A3k +1 + 2 A3k = A3k ( 9 − 9 + 2 ) = 2 A3k ∴A=
1 2 1 2
9 / ا9 اا2 وه9 J* ا9 ام دN اH ان و. ا)صN اxk* = 3k ن2/8و 2 ه9 / ا9 ام دNن ا2/8 و، 9 / ا9 د xk = C1 + C2 2k + 12 3k
.9و0وط اPه ا8K *8 C2 وC1 8 * إ *رY Tو K ^آ0 ا9( * ا د2 xk + 2 − 3xk +1 + 2 xk = a k
ا ^ل ا3= آ.( a = 3 6 f ]ا وذ6 ه9% 9 ) ا ^ل اX Y a V ض2 وxk* = Aa k 38J* اN ا6 f x*k +2 − 3 xk* +1 + 2 xk* = Aa k + 2 − 3 Aa k +1 + 2 Aa k = Aa k ( a 2 − 3a + 2 )
= A3k ( a − 1)( a − 2 ) ∴A =
1
( a − 1)( a − 2 )
, a ≠ 1, 2
3ة ه# ا9ور ا د6H انS . ا)صN اxk* =
1
( a − 1)( a − 2 )
a k , a ≠ 1, 2 ن2/8و
/* ^ أيa = 2 أوa = 1 Xآ2 ذا. 2 و1 xk + 2 − 3xk +1 + 2 xk = 1
ض2 وxk* = Ak ن2/8 a = 1 QU2 وxk* = Aka k Nب اJ x*k +2 − 3xk* +1 + 2 xk* = A ( k + 2 ) − 3 A ( k + 1) + 2 Ak = −A ∴ A = −1
. xk* = −k ا)صNن ا2/8و صxk* = Ak 2k ه انxk + 2 − 3xk +1 + 2 xk = a k 9 ا د3= a = 2 9 3= :8 K امN اT xk + 2 − 3xk +1 + 2 xk = 2k 9L ا`و9 وان ا دA = 1 2 #
48
xk = C1 + C2 2 k + k 2 k −1
9 ا)ص دN اH( او3 8 xk + 2 − 6 xk +1 + xk = 5sin
kπ 2
Nب اJ xk* = A sin
kπ kπ + C cos 2 2
38J* اN ض2 ،ه8K * 9* Y تC وA V ( k + 2 ) π + C cos ( k + 2 ) π − 6 A sin ( k + 1) π + C cos ( k + 1) π 8 xk*+ 2 − 6 xk*+1 + xk* = 8 A sin 2 2 2 2 kπ kπ + A sin + C cos 2 2
9^ ^ ت اL *)ام ا+ و sin
( k + 2)π
cos
( k + 2)π
sin
( k + 1) π
cos
( k + 1) π
2 2 2 2
kπ kπ = sin + π = − sin 2 2 kπ kπ = cos + π = − cos 2 2 kπ kπ π = sin + = cos 2 2 2 kπ kπ π = cos + = − sin 2 2 2
a*8 8 xk*+ 2 − 6 xk*+1 + xk* = ( −7 A + 6C ) sin
kπ kπ + ( −6 A − 7C ) cos 2 2
انJ 8 xk + 2 − 6 xk +1 + xk = 5sin
kπ 9 %0 ا9 ا دQ 56 ه9 ا د9; : 2
−7 A + 6C = 5 −6 A − 7C = 0 7 6 A=− , C = 17 17
ا)صNن ا2/8و xk* =
1 kπ kπ + 6 cos −7 sin 17 2 2
49
: AF$ 9;8:= ، r ( k ) 9 ااN/! $D* إ/ 8 ا)صN اN/! انa** 9; ا9 ^0 ا بU N% ع او2 J r ( k ) 9 ااXح إذا آJ T إ *)ا/ 8 ا ت \ ا دة د# n وX ا2Y ايb وa V a k , sin bk , cos bk , k n N/P وال ا9:) اRاآK .RH2 g% ا ;*ح38J* اN واr ( k ) 9 اN/! 3:8 3*ول اJا r (k )
xk*
ak sin b k / co s b k
Aa k A sin b k + C co s b k A 0 + A1 k + A 2 k 2 + ⋯ + A n k n
kn k nak
a k ( A0 + A1 k + A 2 k 2 + ⋯ + A n k n )
a k sin b k / a k co s b k
a k ( A sin b k + C co s b k )
: ل 9 ام دN اHأو xk + 2 − 4 xk +1 + 4 xk = 3k + 2k
9ن اا2/K 9 # وλ1 = λ2 = 2 رة/ ا8ر6J اT λ 2 − 4λ + 4 = 0 ة# ا9ا د 9 ا د80ف ا: ا" اO اa** 8 ا)صN ا. xCF = ( C1 + C2 k ) 2k 9%)ا A0 , A1 , A L ودxk* = A0 + A1k + Ak 2 2 k N/P " ا# $ اJ= ولJ 9* Aوا
j82* xk* + 2 − 4 xk* +1 + 4 xk* = ( A0 − 2 A1 ) + A1k + 8 A2k
J xk + 2 − 4 xk +1 + 4 xk = 3k + 2k Q 9 ;ر A0 − 2 A1 = 0,
A1 = 3, 8 A = 1
امNن ا2/8 وA0 = 6, A1 = 3, A = 18 أي xk = ( C1 + C2 k ) 2k + 6 + 3k + 18 k 2 2k
50
اف ا 2 2 R;$ل 8&fام : Excel 2ف رس Kف ا 2ل ا N 9JKد= 9و ; 9Lأو: 9ة ورس ا*ف ا ** 3@Tت اد " # 98أ^ Excel )* 9آ*:3 (1أدرس Kف ا د 9ا`و 9Lا*9 xk + 2 − 3 xk +1 + 2 xk = 0 ; أو x0 = 0, x1 = 1 9و x0 = −1, x1 = −2و x0 = x1 = 2 ا ; :Nا0وx0 = 0, x1 = 1 9
ان ا #*8 Nإ" ∞ . و ; ا0وx0 = −1, x1 = −2 9
ان ا #*8 Nإ" ∞ . − و ; ا0وx0 = x1 = 2 9
51
ان ا 9* Y 9 L Nوه6ا Q H " # :8ا; ا0و. x0 = x1 9 (2أدرس Kف ا د 9ا`و 9Lا*9 2 xk + 2 + 3 xk +1 − 2 xk = 0
3 1 ; أو x0 = 1, x1 = 9و x0 = 1, x1 = −2و 2 2 1 ا ; :Nا0و9 2
= x0 = 1, x1
ان ا;*8 Nرب إ" .0 و ; ا0وx0 = 1, x1 = −2 9
52
. x0 = 2, x1 = −
ان ا* #*8 Nددا 9H2 Lو .9 3 و ; ا0و9 2
x0 = 2, x1 = −
` cا*ف ا 3@Tا . (3أدرس Kف ا د 9ا`و 9Lا*9 xk + 2 + xk = 0
; أو. x0 = 1, x1 = 0 9
53
.*دد ودا8 Nا أنS 1, if k = 0, 4,8,12,... kπ cos = −1, if k = 2, 6,10,14,... 2 0, if k = 1,3,5, 7,...
9* ا9L ا`و9ف ا دK ( أدرس4 4 xk + 2 + xk = 0
. x0 = 1, x1 = 0 9 ; أو
.0 " إH )*8 Nا
54
1 kπ 9 و; أوxk = A cos + C ; وآن9 ا د56T امN اH; و 2 2 k
kπ 1 kπ انS . xk = cos امN اg8 x0 = 1, x1 = 0 2 2 2 k
ا*دد وR 8 cos
.)* اR 8 (1 2 )
k
: AF$ أ إذا.H )*K ء اوd L*K او8اG*K او9* Y NOK L 9 *;رN ا9 ** X( إذا آ1 أو− ∞ " * إN/P L*K أو98TS ∞ " * إN/P 8اG*K L T+= ة#* Xآ .3@TS اJ` او3@T * دداK يK .9و0 وا; ا9L ا`و9 ا دN " آ# N ا9 ** فK *8 (2 .N 3@T ا*ف ا8K 3= 9 T اNا2* اK ة# ا9ور ا د6H (3 : هWTa N واxCF 9 / ا9ن اا2/*8 9L وا`و9 U`*ت اS دxGS امN; ذآ أن ا وxCF Nف آK " # *8 N 3@T ا*ف ا. xGS = xCF + xPS أيxPS ا)ص N i8ة وL `*ةTransient 3;* او إ3 xCF فK ن2/8 \ وxPS . xPS Yf* ا3= * 8 و9 82D *!" =*ة8و
55
:ا اا :Matrix Algebra and Calculus تP # ب اT و9! ف2 ا6T و، و ء ا ذج3U8 اN * ا3= اH 9 T دوات ا0=ت ا2` * اK .ا ا ;ر6)م ه8 =ت2` و ب اH N ;ق إ" ا:* ص2) " ا# و. ة#ر وأ2: N/! 3= !ء0 اRآK T " ا# 9=2` ف اK :3* آA 9=2` ف اK aij ∈ ℝ 9;;اد ا# a11 ⋯ a1n A = aij = ⋮ ⋮ ⋮ am1 ⋯ amn
:تP # ا324 %&&ت ا%2#Iا =ت2` Q H -1 1- C = A + B ⇒ cij = aij + bij ب دi ا-2 2- C = α A ⇒ cij = α aij 9=2` بi ا-3 n
3- C = AB ⇒ cij = ∑ aik bkj k =1
9=2` (ل2;) س2/ -4 4- C = AT ⇒ cij = a ji 9 ا9=2` ا-5 5- A = aij , i = 1,..., n; j = 1,..., n ة2 ا9=2` -6 6- I n = ek( n ) , ek( n) = ( 0,..., 0,1( k th position ) , 0,..., 0 )
T
9=2` ب2 ; -7 7- AB = I ⇒ B = A−1 Singular !ذةA ن2/KS " !ط ا# ب2 ; س ا2/ -8
56
8- ( A−1 ) = ( AT ) = A−T −1
T
9=2` دة ا-9 det ( A ) = ∑ ( −1) a1 j det ( A1 j ) n
j +1
j =1
j د2 ول وا0 ا: ف ا6 T # N ( n − 1) × ( n − 1) 9=2` A1 j V . A 9=2` ا :دات8 #2 ة8%# ا اص اYI
( i ) det ( AB ) = det ( A ) det ( B ) ( ii ) det ( AT ) = det ( A ) ( iii ) det ( cA ) = c n det ( A) ( iv ) det ( A ) ≠ 0 ⇔ A is nonsingular . y = [ yi ] , i = 1, 2,..., m أيm × 1 9=2` # رة# 2د ه2 ا$J* :l8K 38 y = Ax ن+= $J* x و9=2` A إذا آن n
yi = ∑ aij x j , i = 1, 2,..., m j =1
":8 y = y j , j = 1,..., n وx = [ xi ] , i = 1,..., m TJ* 3Hب ا)رi ا:l8K 9L x1 y1 ⋯ x1 yn xy = ⋮ ⋮ ⋮ xm y1 ⋯ xm yn T
":8 y = y j , j = 1,..., n وx = x j , j = 1,..., n TJ* 3 ب ااi ا:l8K 9L n
xT y = ∑ xi yi = y T x i =1
. A 3= k د2 اck V A = [ c1 ,..., cn ] 2 هA 9=2` دي2 اN^ * ا:l8K r1T . A 3= k : اrkT V A = ⋮ 2 هA 9=2` ي: اN^ * ا:l8K rmT
. aij = a ji , ∀i, j = 1,..., n , i ≠ j *ة إذا آنA 9 ا9=2` ;ل ان ا8 :l8K 57
;K *ة إذاA ا *ة9 ا9=2` ;ل ان ا8 :l8K AT A = AAT = I
أو A−1 = AT
أي ckT c j = 0, , k = 1,..., n ; j = 1,..., n, k ≠ j
أو rk rjT = 0, k = 1,..., n ; j = 1,..., n, k ≠ j
9=2` ا
ن+=
1 ≤ j1 < ⋯ < js ≤ n
9@GH 9=2` 3 هbpq = ai
p jq
و
1 ≤ i1 < ⋯ < ir ≤ m
Xآ
إذا
:l8K
9L 9= اB = bpq , p = 1,..., r ; q = 1,..., s
9@GH 9=2` B ن+= p = 1,..., r ; i p = j p وr = s X وإذا آ، A Submatrix ن+= p = 1,..., r ; i p = j p = p ] " ذ# 9=U وإذا آن إPrincipal Submatrix 9 @ر .Leading Principal Submatrix 9;* 9 @ ر9@GH 9=2` B T= ن2/8 9 9=2` 3 وهDiagonal Matrix 98:; ا9=2` ا:l8K أيaij = 0, ∀i ≠ j a11 0 ⋯ ⋮ ⋮ ⋮ A = 0 ⋮ aii ⋮ ⋮ ⋮ 0 0 ⋯
0 0 ⋮ ⋮ ⋮ 0 ⋮ ⋮ 0 ann
. A = diag ( aii ) , i = 1,..., n T G8و
{λ1 ,..., λn } ; ا9#2 J 3ه
A 9 9=2` Eigenvalues _ة%## ا7%/ ا:V I0
9; ا دK 3*وا det ( A − λi I ) = 0, ∀i = 1,..., n
تTJ* ا9#2 J 3 هA 9 9=2` Eigenvectors _ة%##;ت اZ# ا:V I0 ;K 3*{ واλ1 ,..., λn } ةG ; ا9 *ا Avi = λi vi , ∀i = 1,..., n
أو 58
( A − λi I ) vi = 0,
∀i = 1,..., n
.G وج اG ن ا8 T Q K G 9J*ة وG ا9 ; ا:l8K 3* ا9=2` ا3 هA 9 ا9=2` 9 * اModal Matrix ا;س9=2` :l8K ;K 3* واA 9=2` ةG تTJ* TK #ن ا2/*K A = M ΛM −1
. A 9=2` ةG ه ا; ا%# 92/ ا98:; ا9=2` ا3 هΛ = diag ( λi ) V . AM = M Λ أنS 9=2` سL 9=2`ة وG تTJ*ة وG ا; اH أو:^ل −2 −3 4 A = 0 1 0 −2 −2 4
:Nا −2 − λ −3 det ( A − λ I ) = 0 1− λ −2 −2
(1 − λ )
−2 − λ
4
−2
4−λ
4 0 =0 4−λ
= (1 − λ ) ( λ 2 − 2λ ) = 0
∴ λ = [1 2 0] T
1 0 0 Λ = 0 2 0 0 0 0
3* ا;س آ9=2`ة وG ت اTJ* اH2 −3 4 vi1 −2 − λi ( A − λi I ) vi = 0 1 − λi 0 vi 2 = 0 −2 −2 4 − λi vi 3 −3 −3 4 v11 1 λ1 = 1 ⇒ 0 0 0 v12 = 0 ⇒ v1 = −1 −2 −2 3 v13 0
59
−4 λ2 = 2 ⇒ 0 −2 −2 λ3 = 0 ⇒ 0 −2
−3 4 v21 1 −1 0 v22 = 0 ⇒ v1 = 0 1 −2 2 v23 −3 4 v31 2 1 0 v32 = 0 ⇒ v1 = 0 1 −2 4 v33
1 1 2 M = −1 0 0 0 1 1
. A = M ΛM −1 وAM = M Λ ; أنK :8 K :3* آR 8 R \ g% د# k V Ak ة/ I P P ر:V I0 A k = M Λ k M −1
. Λ k = diag ( λik ) و 2 هA 9 ا9=2` اTrace Y أ:l8K n
tr ( A ) = ∑ aii i =1
: أن9 ه/ 8 n
tr ( A ) = ∑ λi i =1
و n
det ( A ) = ∏ λi i =1
:تO N/ ( A − λi I ) vi = 0 ;8 vi و$J* H28S ا6T !ذة و9=2` ( A − λ I ) أنS (1 انgU28 ا6 وهα ≠ 0 V α vi ]6/= λi ةG ا9 ; Q K G $J* vi ذا آن+= λi .5 ;ارc و$هJK إ3= ن2/K G ا$J* ا9 أه Right and Left Eigenvectors 98 وا ر9 ة اG ت اTJ* ا#2 ( هك2 9L 3 اG ا$J* = ا# أن، {λ1 ,..., λn } ةG ; ا9 *ا ف8 ا ريG ا$J* ا، AM = M Λ T # ي6 واAvi = λi vi , ∀i = 1,..., n N ي6 واuiT ( A − λi I ) = 0, ∀i = 1,..., n أوuiT Avi = uiT λi , ∀i = 1,..., n 9L ا 60
ن2/*K M −1 ر2: ن+= vi ن2/*K M ة# أXذا آ+= M −1 A = ΛM −1 9L ا$ # .ة وةG أن ا; اS . uiT I P ة/ e P ر:V I0 e
At
( At ) = I + At +
2
2!
( At ) + 3!
( )
3
+⋯
= M diag eλi t M −1
: (SVD) Singular Value Decomposition ذةA ا#%/ اm%0 K* *=2` H28 $+= A ∈ ℝ m×n Xإذا آ U = [u1 ,..., um ] ∈ ℝ m×m
و V = [ v1 ,..., vn ] ∈ ℝ n×n
V U T AV = diag (σ 1 ,..., σ p ) ,
p = min ( m, n )
vi وui تTJ* واA 9=2` ذةP ا; اσ i 3 K . σ 1 ≥ σ 2 ≥ ⋯ ≥ σ p ≥ 0 V و
.3ا2* " ا# ذP ا80 اi $J* ذ واP ا80 اi $J* ا : Matrix Calculus تP # ب اT ه%# V t Z* " ا# *K =ت2` B ( t ) = bij ( t ) وA ( t ) = aij ( t ) /* . !*;ق9 L ن دوال2/K bij ( t ) وaij ( t ) P # ا/A :V I0 d d A ( t ) = aij ( t ) dt dt
P # ا0 :V I0
∫ A ( t ) dt = ∫ a ( t ) dt ij
بiة ا#L :l8K d d d A (t ) B (t ) = A (t ) B (t ) + A (t ) B (t ) dt dt dt d
∫ A ( t ) dt B ( t ) dt = A ( t ) B ( t )
limits
d − ∫ A ( t ) B ( t ) dt dt
61
سI# ا/A :V I0 d −1 d A ( t ) = − A−1 ( t ) A ( t ) A−1 ( t ) dt dt
ن+= 9* Y =ت2` C = cij وA = aij Xإذا آ d At e = Ae At dt
∫e
At
dt = A−1e At + C
62
: ا ا : State Space Representation ء اGP %#0 The State Space Equations and $ _ل اZ# اP ;2Tء ا وGP د`تI : their Time Domain Solution :3 ;ط ه9YY " # هGف آ2 .3 ; ا9ء اi= N/! " 8 و9 اآ9 O0 ا3= 9 اZ* امN^ * ا-1 .3 ; اN/P " ا# 9 ا9 O0 اQUق وD -2 .3Gل اJ ا3= 9ت اS دN دJ8ق اD -3 State
9ات اZ* 9: ا2 9 اآ9 O0 ا9J 9ء اi= N^ K *)م8
. Standard State Space Form 3 ; ا9ء اi= N/! 3= TU ووVariables =ت2` واx, y, u, f , g $J* وال اR*/8 * 3م آO 9 اZ* امN/Pا N/P " ا# A, C , D 9* ^ا d x ( t ) = Ax ( t ) + f ( x, t ) + Bu ( t ) dt y ( t ) = Cx ( t ) + Du ( t )
أنRJ8 A مO ا9=2` .9:)ة واود \ اZ* ا ت اNي آ2K f ( x, t ) V .م# N/! ايT ن2/8 ان/ 8 D وC وB =ت2` وا9 ن2/K :1^ل سL N/! 3= 9* ا9^ ا9Hم ارO ا9 دQU d x1 ( t ) = 7 x1 ( t ) + 3 x2 ( t ) + 4tx1 ( t ) + x1 ( t ) x2 ( t ) − u1 ( t ) + 2u3 ( t ) dt d x2 ( t ) = 9 x1 ( t ) − 5 x2 ( t ) − 3 x22 ( t ) + 4u2 ( t ) dt y ( t ) = x1 ( t ) + x2 ( t ) − 2u2 ( t )
:Nا d x ( t ) = Ax ( t ) + f ( x, t ) + Bu ( t ) dt y ( t ) = cT x ( t ) + d T u ( t )
V
63
u1 ( t ) x1 ( t ) x (t ) = , u ( t ) = u2 ( t ) x2 ( t ) u3 ( t ) 4tx ( t ) + x1 ( t ) x2 ( t ) −1 0 2 f ( x, t ) = 1 , B = 2 −3 x2 ( t ) 0 4 0 cT = [1 1] , d T = [ 0 −2 0] 7 3 A= , 9 −5
: > ا3ا ا ءi= N^ K ص2) " ا# و9 اN/! " إ9 O0 اj N82* 9;8D ء:# إ2ه= ه ه \ 9 O0 اN82K و9: *) ا98J وا9 U`*ت اS وا دn 9H ار9 O0 9ا .9: " إ9: : n !ر8 ا6 %S ا%2)د`ت اI# ا0 -1 n 9H ار3*م اO" ا: dn d n −1 d y t + a y ( t ) + ⋯ + an −1 y ( t ) + an y ( t ) = ( ) 1 n n −1 dt dt dt n n −1 d d d b0 n u ( t ) + b1 n −1 u ( t ) + ⋯ + bn −1 u ( t ) + bnu ( t ) dt dt dt
Qi x1 ( t ) = y ( t ) − β 0u ( t ) d d d y ( t ) − β 0 u ( t ) − β1u ( t ) = x1 ( t ) − β1u ( t ) dt dt dt 2 2 d d d d x3 ( t ) = 2 y ( t ) − β 0 2 u ( t ) − β1 u ( t ) − β 2u ( t ) = x2 ( t ) − β 2u ( t ) dt dt dt dt ⋮ x2 ( t ) =
x j (t ) =
d j −1 d j −1 d j −2 d β β y t − u t − u ( t ) − ⋯ − β j −1u ( t ) = x j −1 ( t ) − β j −1u ( t ) ( ) ( ) 0 1 j −1 j −1 j −2 dt dt dt dt
V β 0 = b0 β1 = b1 − a1β 0 β 2 = b2 − a1β1 − a2 β 0 ⋮
β j = b j − a1β j −1 − ⋯ − a j −1β1 − a j β 0
" # N ا6/وه
64
d x ( t ) = Ax ( t ) + bu ( t ) dt y ( t ) = cT x ( t ) + du ( t )
V x1 ( t ) β1 β x2 ( t ) x (t ) = , b = 2 , cT = [1 0 0 ⋯] , d = β 0 = b0 ⋮ ⋮ β n xn ( t ) 0 0 A= ⋮ 0 −an
1
0
⋯
0
1
⋯
⋮
⋮
⋯
0
0
⋯
−an −1
− an − 2 ⋯
0 0 ⋮ 1 −a1
:2^ل 9ء اi= N^ K " إ9* ا9ل ا د2 d3 d2 d d y ( t ) + 6 2 y ( t ) − 8 y ( t ) + 4 y ( t ) = 2 u ( t ) + 7u ( t ) 3 dt dt dt dt
j82* 9; ا9;8: *) ا x j (t ) =
d x j −1 ( t ) − β j −1u ( t ) dt
J x1 ( t ) = y ( t ) − β 0u ( t ) = y ( t ) d d d y ( t ) − β 0 u ( t ) − β1u ( t ) = y ( t ) dt dt dt 2 2 d d d d2 x3 ( t ) = 2 y ( t ) − β 0 2 u ( t ) − β1 u ( t ) − β 2u ( t ) = 2 y ( t ) − 2u ( t ) dt dt dt dt x2 ( t ) =
]6وآ β j = b j − a1β j −1 − ⋯ − a1β 0 β 0 = b0 = 0 β1 = b1 − a1β 0 = 0 β 2 = b2 − a1β1 − a2 β 0 = 2
β 3 = b3 − a1β 2 − a2 β1 − a3 β 0 = 7 − 6 ( 2 ) − 5
J 9 U`* ا9 ا د3= j82* و 65
x1 ( t ) 0 1 0 x1 ( t ) 0 d x2 ( t ) = 0 0 1 x2 ( t ) + 2 u ( t ) dt x3 ( t ) −4 8 −6 x3 ( t ) −5 x1 ( t ) y ( t ) = [1 0 0] x2 ( t ) + [ 0] u ( t ) x3 ( t )
: > ا3 إ3 !ر8 ا6 مW 0 ": d3 d2 d y t + a y ( t ) + a2 y ( t ) + a3 y ( t ) = u ( t ) ( ) 1 3 2 dt dt dt
انi8 ا. u ( t ) *;تP ى28S 80ف ا: ان اO ،9 N/! "م اOا ا6ل ه2 N 9JKة ا# ا9ور ا د6H Q 9; :* 9JKم اO ا9=2` ةG ا; ا .ة: ا9 د9 * ا9 J* ا9ا د ت/ض ا2 9 ا56T :Nا x1 ( t ) = y ( t ) x2 ( t ) = y′ ( t ) =
d x1 ( t ) = x1′ ( t ) dt
م# N/P و x j (t ) =
d x j −1 ( t ) dt
J y′′′ ( ' t ) =
d x3 ( t ) = x3′ ( t ) QU2 9= ا9ن ا دb اx1 ( t ) = y ( t ) Q dt
d x3 ( t ) = −a1 x3 ( t ) − a2 x2 ( t ) − a3 x1 ( t ) + u ( t ) dt
3* آ3=2` N/! " # QU2K x3′ ( t ) وx2′ ( t ) وx1′ ( t ) 3= تSن ا د+= ا6/وه x1 ( t ) 0 d x2 ( t ) = 0 dt x3 ( t ) − a3
1 0 − a2
0 x1 ( t ) 0 1 x2 ( t ) + 0 u ( t ) −a1 x3 ( t ) 1
و x1 ( t ) y ( t ) = [1 0 0] x2 ( t ) + [ 0] u ( t ) x3 ( t )
66
3 ه9 J* ا9ة د# ا9ا د λ 3 + a1λ 2 + a2λ + a3 = 0
N ":K 9 ا9=2` ةG وا; ا det ( A − λ I ) = 0 −λ
1
0
0
−λ
1
− a3
− a2
−a1 − λ
=0
J ول0 ا: و `] ا ة −λ
−λ − a2
0 1 −1 − a3 −a1 − λ
1 −a1 − λ
= −λ ( λ 2 + a1λ + a2 ) − 1( a3 ) = λ 3 + a1λ 2 + a2λ + a3 = 0
3: مO 9 ا9=2` ةG ; أي ان ا; اT # 3* ا9J* اc` 3وه .9 J* ا9ة د# ا9ور ا د6H Q :*K ;* : S2# ا9Z وا%2)د`ت اI# اT -2 3 ; ا9ء اi= N/! " # 3*م اO اQU d d x1 ( t ) + x2 ( t ) = −4 x1 ( t ) + x4 ( t ) + x5 ( t ) + u ( t ) dt dt d x2 ( t ) = x2 ( t ) − 5 x3 ( t ) + 3u ( t ) dt 0 = x3 ( t ) + x4 ( t ) = x1 ( t ) − 3x2 ( t ) + x3 ( t )
0
+ 7u ( t )
= x1 ( t ) − x2 ( t )
0
+ x5 ( t )
3=2` اN/P " ا# تS ا د56 هR*/
1 0 0 0 0
1 0 0 1 0 0 0 0 0 0 0 0 0 0 0
d dt x1 ( t ) 0 d x ( t ) −4 0 0 2 0 dt 0 1 −5 d 0 x3 ( t ) = 0 0 1 dt 0 1 −3 1 d x4 ( t ) 1 −1 0 0 dt d x5 ( t ) dt
67
1 1 x1 ( t ) 1 0 0 x2 ( t ) 3 1 0 x3 ( t ) + 0 u ( t ) 0 0 x4 ( t ) 7 0 1 x5 ( t ) 0
3*=ت آ2` ; ا
1 0 0 0 0
1 1 0 0 0
0 0 0 0 0
0 0 0 0 0
d dt x1 ( t ) 0 d x ( t ) −4 2 0 dt 0 d 0 x3 ( t ) = 0 dt 1 0 d 0 x4 ( t ) 1 dt d x5 ( t ) dt
0 1 0 −3 −1
0 −5 1 1 0
1 1 x1 ( t ) 1 0 0 x2 ( t ) 3 1 0 x3 ( t ) + 0 u ( t ) 0 0 x4 ( t ) 7 0 1 x5 ( t ) 0
N/P " ا# X%; ا E11 0
d xd ( t ) 0 dt C11 C12 xd ( t ) bd u (t ) = + C21 C22 xa ( t ) ba 0 d x t dt a ( )
N اT 3*وا d xd ( t ) = Axd ( t ) + wu ( t ) dt
V A = E11−1 ( C11 − C12C22−1C21 )
w = E11−1 ( bd − C12C22−1ba )
: %S\ 3 إ%S\ %1 #W ا0 -3 مO 3: R8;K Hأو −1 2 A= , −1 −3
x2 (t ) 1 f = 2 , bu ( t ) = u ( t ) 0 0
Equilibrium Point ازن2* ا9:; Reference State QH ا9 6 + ف2 . u0 ( t ) Q dx0 ( t ) dt = 0 :Nا 3 وهQH ا9 ود اu0 ( t ) وx0 ( t ) ل2 مOف `] ا2 d x0 ( t ) = 0 dt Ax0 ( t ) + f ( x0 ( t ) , u0 ( t ) ) + 0 = 0
68
أو 2 0 = − x10 ( t ) + 2 x20 ( t ) + x20 (t )
0 = − x10 ( t ) − 3 x20 ( t )
J RK*دة ا#+ و x10 ( t ) = −3 x20 ( t )
2 3 x20 ( t ) + 2 x20 ( t ) + x20 ( t ) = ( 5 + x20 ( t ) ) x20 ( t ) = 0
3 *;ار هA ا3* ا6/وه 0 15 x10 ( t ) = , x20 ( t ) = 0 −5
نf # $# * ا/ 8 9 ا$J* 9 * اJacobian Matrix 9 2آJ ا9=2` ا J u (t ) = 0
J x(t )
∂ ∂x ( t ) f1 ( x ( t ) , u ( t ) ) 1 = ∂ f2 ( x (t ) , u (t )) ∂x1 ( t )
∂
f1 ( x ( t ) , u ( t ) ) 0 2 x2 ( t ) = ∂ 0 0 f 2 ( x ( t ) , u ( t ) ) ∂x2 ( t ) x ( t ), u (t ) 0 0 ∂x2 ( t )
x0 ( t ),u0 ( t )
2 ( ه: NH ي6 ) ا3:)م اO اx0T ( t ) = [ 0 0] Q QH ا9 −1 2 1 d δ x (t ) = δ x (t ) + δ u (t ) dt −1 3 0
2 ه3:)م اO اx0T ( t ) = [15 −5] QH ا9 و −1 −8 1 d δ x (t ) = δ x (t ) + δ u (t ) dt −1 −3 0
.9` *) 9ت اوS ل2 3 %0م اO 3: R8;K N^ K ا *;ة9:) ا9 O0 ا56ه :ة/ # ا%S ا#W اT : Z#ة ا/ # ا%S ا#Wا (9J*) 3=2` Z* 9 دScalar دي# Z* 3= 9 دNف ;رن ا2 :98 اد9ا d x ( t ) = ax ( t ) , x0 ( 0 ) = x0 dt
N اJ 9 U`*ت اS ا دN 9; ق ا:
69
x ( t ) = x0e at
:9=2` ا9ا d x ( t ) = Ax ( t ) , x0 ( 0 ) = x0 dt
2 هN ا9;8: اc` x ( t ) = e At x0
State Transition Matrix 9 إ*;ل ا9=2` " K e At 9=2` ا :9 إ*;ل ا9=2` اص2 1.
d At e = Ae At = e At A dt
2.
∫e
3. e
At
dt = A−1e At = e At A−1
A ( t +τ )
= e At e Aτ
4. let τ = −t , e 5. e(
A+ B)t
A ( t +τ )
= e At e − At = I
= e At e Bt , if
or
−1
e At = e − At
AB = BA
: Z#ا%1 ة/ # ا%S ا#Wا :98 اد9ا d x ( t ) − ax ( t ) = bu ( t ) dt
N اJ 9 U`*ت اS ا دN 9; ق ا: x ( t ) = x0e at + ∫ e a ( t −τ )bu (τ ) dτ t
0
:9=2` ا9ا d x ( t ) − Ax ( t ) = Bu ( t ) dt
2 هN ا9;8: اc` x ( t ) = e At x0 + ∫ e A( t −τ ) Bu (τ ) dτ t
0
70
: %Fد`ت اوI#2 ء اGP %#0 N آ ان9 U`*ت اS د5O ^/ NT أ9Lت ا`وS د9ء اi= N^ K آNد اJ8A Excel N^ a *)م8م وO اري/K N 8D # *8 aKم اOا .;S ى :3* آ9ء اi= N^ K N/P 9H أي در9L =و9 أي دQU2K xk +1 = Gxk + Huk
\ ود9=2` H ( و9 ) 9 ا9=2` G وk ار/* ا# 9 ا$J* xk V .cJ*ا :1^ل "و0 ا9H إ" د* ار9 Z* و9^ ا9H ار9* ا9L ا`و9ل ا د2 .9 يZ*و xk + 2 + 5 xk +1 − 7 xk = 2k
:Nا :N/P " ا# 9 ا دS أوQi xk + 2 = −5 xk +1 + 7 xk + 2k
a*= xk +1 = yk QU xk +1 = yk yk +1 = −5 yk + 7 xk + 2k
أي xk +1 1 0 xk 0 y = −5 7 y + 2 k k k +1
V xk +1 = Gxk + Huk N/P " ا# 3وه 1 0 G= −5 7
و 0 H = 2
م اO N H أوx0 = 1, y0 = 1 9; أو
71
:Excel ام8&f ا :3* اN أد
J
N^ 9 اآ9 O0 وآة اN K a ا3= T*J 92T 2 ه9ء اi= N^ K 9 أه . و\هSTELLA وVensim a :Vensim 9: ا2 9Lت ا`وS د9ء اi= N^ K : Vensim 9: ا2 م اO اN^ K x dx
y dy
k
dx = y dy = -5*y+7*x+2*k k = LOOKUP( [(0,0)-(10,10)],(0,0),(1,1),(10,10)) x = INTEG ( dx, 1) y = INTEG ( dy, 1)
72
:ا ا دس : %&&ذج ا#$ اYI ;اK ^ ء ذج أآ9آت اوK *K 3* ا9 0 ا ذج اj ورس3 ف2 3;;ب إ" ا اLوا
: Sل( اa ` )ا#$ ذج ا# -1 # ى او ا *ع2* ان اU= 2 = ،X Y لL*8 داد أوG8 مO RآK d ا ا6ه 9L ":8 (a < 0) ;( أو اa > 0) دة8Gن ل ا+= x (t ) ;ارN^ 8 t Gا dx (t ) = xɺ (t ) = a dt
:N/P N^ 8و x(t) dx(t)/dt
a
N/P ":K (a > 0) و ـ
Graph for x(t) 40
30
20
10
0 0
10
20
30
40
50 60 Time (Month)
"x(t)" : Current
73
70
80
90
100
xɺ (t ) G
dx (t ) دا@ ـG ف2 :9O dt
:&`ل( اa ` )ا#$ ذج ا# -2 2 = ،دة2H2 ا9 / اQ R *8 ; ل8 أو8اG*8 ى أو *ع2* *8 ذج2 ا ا6ه ( اوa > 0) دة8Gن ل ا+= x (t ) ;ارN^ 8 t G ا# ى او ا *ع2* ان اU= 9L ":8 (a < 0) ;ا xɺ (t ) = ax (t )
:N/P N^ 8و x(t) dx(t)/dt
a
N/P ":K (a > 0) و ـ
Graph for x(t) 200
150
100
50
0 0
10
20
30
40
50 60 Time (Month)
70
80
90
100
"x(t)" : Current
(a < 0) 9 ; ; ا/P أر ا:8 K
74
: Sل اa ` وا#$ ذج ا# -3 & X Y لL*8 XL2 اc` 3=( وa > 0) X Y ل8اG*8 ى2* *ع اوN^ 8و :3* آN^ 8( وb < 0) xɺ (t ) = a − b
x(t) dx(t)/dt
-dx(t)/dt
a
b
:&`ل اa ` وا#$ ذج ا# -4 2 = ،دة2H2 ا9 / اQ R *8 ; ل8 و8اG*8 ى أو *ع2* *8 ذج2 ا ا6ه ( وa > 0) دة8Gن ل ا+= x (t ) ;ارN^ 8 t G ا# ى او ا *ع2* ان اU= 9L ":8 (b < 0) ;ا xɺ (t ) = (a − b ) x (t )
:N/P N^ 8و x(t) dx(t)/dt
-dx(t)/dt
a
b
75
:I% ذج ا#$ ا-5 QL2 N^ 8 x (t ) آن2 ^ = ، 9 آ9J* 3:8 3/ *: * # Q J8 ذج2 ا ا6ه أنS) v (t ) G 9# G ر2 و9* ^ ا9;=S اT*# N^ 8 xɺ (t ) ن+= 9آ N/P 9ت اآS دR*/Kو
dv (t ) = vɺ (t ) = xɺɺ (t ) ن2/8 NJ*ن ا+= ( v (t ) = xɺ (t ) dt
9ا د xɺɺ (t ) = a
; NJK a < 0 دة و8 زNJK a > 0 V
x(t) dx(t)/dt
v(t) dv(t)/dt a
"و0 ا9H ار9 U`K K داN/! " # 9^ ا9H ار9 U`* ا9 ا دR*/Kو :3* ( آState Equations 9ت اS)د xɺ (t ) = v (t ) vɺ (t ) = a
9و; او x ( 0) = 0 k v (0) = 0 k/sec xɺ (t ) = 5 k/sec vɺ (t ) = 3 k/sec 2
76
Graph for x(t) 80,000
60,000
40,000
20,000
0 0
10
20
30
40 50 60 Time (Second)
70
80
90
"x(t)" : Current
100 k
Graph for v(t) 400
300
200
100
0 0
10
20
30
40 50 60 Time (Second)
"v(t)" : Current
70
80
90
100
k/Second
77
Birth-Death Model ت2 دة واS2ذج ا2 (3) 9 U`* ا9 دl%28و xɺ = ( b − d ) x
9Lأو ا`و xt = xt −1 + ( b − d ) xt −1
x
dx/dt
b
d
dx/dt=(b-d)x
Logistic Growth Model 3* H2 ا2 ذج " ا2 (1) 9 U`* ا9 دl%28و xɺ = k ( H − x )
9Lأو ا`و xt = xt −1 + k ( H − xt −1 )
78
Inventory x dx/dt
k
Sales rate
H
Market limit
dx/dt=k(H-x)
Product Limit Growth (3* H2) وطP ا ود2 ذج ا2 (5) 9 U`* ا9 دl%28و xɺ = kx ( h − x )
9Lاو ا`و xt = xt −1 + kxt −1 ( h − xt −1 )
79
Product sold x dx/dt Sales rate
k z
Residual market z=H-x
H Market limit dx/dt=kx(h-x)
:9 يZ* 9 U`*ت اS دl%28 ذج2 (6) yɺ = k1 ( h − y )
xɺ = k2 ( y − x )
9Lأو ا`و yt = yt −1 + k1 ( h − yt −1 )
xt = xt −1 + k2 ( yt − xt −1 )
80
Houses sold y= Number of houses sold Houses supply h= Number of housholds
y dy/dt
x= Number of airconditions
k1 h dy/dt=k1(h-y)
Airconditions sold
Airconditions supply
x dx/dt
k2 dx/dt=k2(y-x)
:9 اتZ* ^ث9 U`*ت اS دl%28 ذج2 (7) hɺ = k4 h
yɺ = k1 ( h − y )
xɺ = k2 ( y − x ) − k3 x
9Lأو ا`و ht = ht −1 + k4 ht −1
yt = yt −1 + k1 ( ht − yt −1 )
xt = xt −1 + k 2 ( yt − xt −1 ) − k3 xt −1
81
Hous holds h
dy/dt=k1(h-y)
dh/dt dx/dt=k2(y-x)-k3x
k4
dh/dt=k4h Houses sold y dy/dt k1 Airconditions sold Broken aircond.
x
dx/dt
k3
k2
82
: ا ا :ذج#$ اYI ء$9 %&`ت دراT :ق2 ا9ذج آ2 " ء# ^ل-1 ق2 ا2 gU28 ذج2 ا.@ داa* 82 K 9ذج آ2 ف *ض ء2 ا ا ^ل6 ه3= 9= ر اP*ف * ان ا2 .ن2 زN/ ة واة$@* !ا8 V ر2D 8H a* 3 /ا *8 a* ن ل !اء ا+= ا6T وa* ا ا6 ا!*وا ه86 @ اG " ا# *K a* ا ا6 ه# :" # a* ا!*وا ا86 @ اGد ا# (1) a* *وا اP8 انQL2*8 86 اL @ اGد ا# (2) Tآ/* إ# a* إ!*وا ا86 @ اGء اU L2* @ اG*ع اL إ9H( در3) j اTi : ف2 ،(3@TS) اH آL2* @ اGد ا# T= ن2/8 3* ا9 اL ف2 :Sأو انQL2*8 86 " ا#( و8*P) ActualCustomers = a* ا!*وا ا86 @ اG " ا# اآL2* د ا# T= 3* ا9ف * ا2 ( وL2*) PotentialCustomers *واP8 .8*P دة ا8 ز# O اjZ X Y 8;K NO8 و8*P ^ ا/ 3 إL2* اc∆t ل28 8*P وا اN∆ آt ةL 9 `*ة ز$`*ض ا 3:8 ي6( واconversion coefficient N82* اN) RH2 X Y c V ،8*P .8*P ا28 3/ L2* ع اLA 8*P ا9= ; ى L2* اc∆t ل2 T واN آ8*P اn ( t ) H28 t G ا# انU*=إذا إ g8 8*P د ا# t + ∆t G∆ = اt 9G ا`*ة ا3= 8*P "إ n ( t + ∆ t ) = n ( t ) + n ( t ) c∆ t
أو n ( t + ∆t ) − n ( t ) dn ( t ) = = cn ( t ) ∆ t →0 ∆t dt lim
(1) 9 /K 9 دN/! " # أو
t
n ( t ) = n0 + ∫ cn (τ ) dτ
(2)
0
2 ه9; ا9 ا دN ، t = 0 G ا# 8*P د ا# 2 هn0 V 83
n ( t ) = n0e ct , t ≥ 0
( 3)
:9L =و9 دN/! 3 # (3)-(1) 9; ت اLف ا2 :9O n ( t + ∆ t ) − n ( t ) = n ( t ) c∆ t for ∆t = 1 (one time unit) nt +1 = nt + nt c nt +1 = (1 + c ) nt
when t = 0, nt = n0
n1 = (1 + c ) n0
n2 = (1 + c ) n1 = (1 + c )(1 + c ) n0 = (1 + c ) n0 2
n3 = (1 + c ) n2 = n0 (1 + c )
3
or nt = n0 (1 + c ) , t ≥ 0 t
Vensim 9Z ذج2 اN^ 8 3* اN/Pا
n(t)
n(0)
dn(t)/dt
c
(1)
c = 1 Units: **undefined**
(2)
"dn(t)/dt" = c*"n(t)" Units: **undefined**
(3)
FINAL TIME = 100 Units: Month The final time for the simulation.
(4)
INITIAL TIME = 0 Units: Month The initial time for the simulation.
84
(5)
"n(0)" = 1
Units: **undefined**
(6)
"n(t)"= INTEG ( "dn(t)/dt", "n(0)") Units: **undefined**
(7)
SAVEPER = TIME STEP Units: Month The frequency with which output is stored.
(8)
TIME STEP = 1 Units: Month
The time step for the simulation.
:أو
Actual Customers
InitialActualCustomers
ConversionFlow
ConversionConstant
(1)
ActualCustomers = INTEG ( ConversionFlow, InitialActualCustomers) Units: **undefined**
(2)
ConversionConstant = 1 Units: **undefined**
(3)
ConversionFlow = ConversionConstant*ActualCustomers Units: **undefined**
(4)
FINAL TIME = 10
Units: Month The final time for the simulation.
(5)
INITIAL TIME = 0 Units: Month The initial time for the simulation.
(6)
InitialActualCustomers = 1 Units: **undefined**
(7)
SAVEPER = TIME STEP Units: Month The frequency with which output is stored.
(8)
TIME STEP = 1 Units: Month
The time step for the simulation.
85
1 Current 1 ActualCustomers 2,000
1
1
1
1
1
1
1
1,500 1
1,000 1
500 1 1 0 ConversionFlow 2,000
1
1
1
1
1
1,500 1
1,000 1
500 1
0
1
0
1
1
2.5
1
1
1
5 7.5 Time (Month)
10
.8*P " إL2* و ل إ ب ا8*P د ا# N/ 3 S ا2 اd S 2 ا9;; ا3= ن0 ،(3@T) ودL2* د ا# T= ن2/8 3* ا9 اL نb ا:Y ده# إذا آن8*P " إT ا آ22* L2* ن ا0 ] * وذ8 ان/ 8S 3 Sا .ود 2 هt G ا# ;* @ اGد ا# ن2/= M 2 هL2* 3 /`*ض ان اد ا N/ N82* ل ا2 هc و `*ض انt G ا# 8*P د ا# 2 هn ( t ) V M − n ( t ) 3 / اد اQ : R *8 N82*ف `*ض ان ل ا2 ،L2* M H28 # *يP N82*ن ل ا+= L2* اl ";K آن2 ^ = ،9 ز9O اي# ;* اL2* .ا6/ وهc / 4 2*ي هP N/ N82*ن ل ا+= L2* اQ ر3;K وإذاc / 2 2*ي هP N/ 2∆ هt 9G ا`*ة ا3= *ى واP 2 اL2* د ا# ن+= تU` ا56 هXK
{ M − n ( t ) M } × c∆t 2*ى هP n ( t ) N/ 2 اL2* د ا# ن2/8و
86
{
}
n ( t ) × M − n ( t ) M × c∆t
g8 8*P د ا# t + ∆t G ا#
{
}
n ( t + ∆t ) = n ( t ) + n ( t ) × M − n ( t ) M × c∆t n ( t + ∆t ) − n ( t ) = c × M − n ( t ) M × n ( t ) ∆t
{
}
أو dn ( t ) M − n (t ) = c× × n (t ) dt M
(4) 9 /K 9 دN/P أو
t
n ( t ) = n0 + ∫ c × 0
M − n (τ ) × n (τ ) dτ M
( 5)
2 ه9; ا9 ا دN ، t = 0 G ا# 8*P د ا# 2 هn0 V n (t ) =
M , t≥0 1 + ( M − n0 ) n0 e − ct
(6)
logistic curve 3* H2 " ا " ا8 " (6) 9ا د :9L =و9 دN/! 3 # (6)-(4) 9; ت اLف ا2 :9O M − nt × nt M n nt +1 = nt + c 1 − t nt M nt +1 = nt + c (1 − α nt ) nt nt +1 − nt = c ×
where α = 1 M
د# 2 هn0 = 1 أنU= ذا+= ار/* ا9;8: T ف2 9: \ 9L =و9 د56وه M = 100 2 هL2* 3 / واد اc = 1 N82* ول اt = 0 G ا# 8*P ا
ن+= α = 1 M = 0.01 nt +1 = nt + c (1 − α nt ) nt n1 = n0 + c × (1 − α n0 ) n0
= 1 + 1 × (1 − 0.01 × 1) × 1 = 1.99
87
n2 = n1 + c × (1 − α n1 ) n1
= 1.99 + 1 × (1 − 0.01 × 1.99 ) × 1.99 = 3.94
n3 = n2 + c × (1 − α n2 ) n2
= 3.94 + 1 × (1 − 0.01 × 3.94 ) × 3.94 = 7.73
n4 = 14.85 n5 = 27.50 n6 = 47.44 n7 = 72.37 n8 = 92.37 n9 = 99.42 n10 = 99.996
:3* اN/P اTو
Graph for ActualCustomers 100
1
1
9
10
1
75
1
50
1
1
25 1
0 0
1
1
1
2
1
3
ActualCustomers : Current
4 1
5 6 Time (Month)
1
1
1
1
7 1
8 1
1
1
1
1
:3* آVensim *)ام+ ذج ا2 ا6`
88
TotalMarket Potential Customers
Actual Customers
InitialActualCustomers
ConversionFlow
ConversionConstant
(01)
ActualCustomers = INTEG ( ConversionFlow, InitialActualCustomers) Units: **undefined**
(02)
ConversionConstant = 1 Units: **undefined**
(03)
ConversionFlow = ConversionConstant*(PotentialCustomers/TotalMarket)*ActualCustomers Units: **undefined**
(04)
FINAL TIME = 10
Units: Month The final time for the simulation.
(05)
INITIAL TIME = 0 Units: Month The initial time for the simulation.
(06)
InitialActualCustomers = 1 Units: **undefined**
(07)
PotentialCustomers = INTEG ( -ConversionFlow, TotalMarket-InitialActualCustomers) Units: **undefined**
(08)
SAVEPER = TIME STEP Units: Month The frequency with which output is stored.
(09)
TIME STEP = 1 Units: Month The time step for the simulation.
(10)
TotalMarket = 100 Units: **undefined**
89
n(0)
M
M-n(t)
n(t) dn(t)/dt
c
(01)
c = 1 Units: **undefined**
(02)
"dn(t)/dt" = c*("M-n(t)"/M)*"n(t)"
(03)
FINAL TIME = 10
(04)
INITIAL TIME = 0 Units: Month The initial time for the simulation.
(05)
M = 100
(06)
"M-n(t)" = INTEG ( -"dn(t)/dt", M-"n(0)") Units: **undefined**
(07)
"n(0)" = 1 Units: **undefined**
(08)
"n(t)" = INTEG ( "dn(t)/dt", "n(0)")
(09)
SAVEPER = TIME STEP Units: Month The frequency with which output
Units: **undefined**
Units: Month The final time for the simulation.
Units: **undefined**
Units: **undefined**
is stored. (10)
TIME STEP = 1 Units: Month The time step for the simulation.
90
1 Current 1 ActualCustomers 100
1
1
1
1
1
1
1
1
1 1
75
1
50
1 1
25 1 1
1
1
0 ConversionFlow 40 30
1 1
20 1
10 0
1
1 1
0
1
1 1
2.5
5 7.5 Time (Month)
10
9* H2 ا2 ت اdK اn ( t ) 8*P ا*; يS فd 8 " #0 اN/Pا i8 ا2ي ه6ب( وا2 ; ة ا2T;ن اJ= س )اوJ اN/! $P8 ي6 اN82* ل ا3 N/!و .3* H2 ا2 ا3= N82*ت اS @ ا : Determining Model Parameters ذج2 اK وريi ا3* H2 = " ا،ذج2 *)ام ا3/ اL K اH وريi ا :3* ا ^ل ا56 ه8* 9;8D ف *ض2 . M وn0 وc 8K 9 "* و1984 9 N* و إX=2 و/ تJ* ق2 ا9 L 9 3 ه9*ات ا 1994 Year
1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994
Market 3.0
2.5
4.0
7.5
7.0
13.0
17.0
29.0
46.5
50.0
49.5
Value
3= ق2 ا9 L وx XK د2 ا3= 9 ت اN K Curve Expert a *)ام+ Sاو :3* آUser-Defined Model *ر+ ذج2 ا8 وy XK د2 ا
91
User-Defined Model: y=a/(1+b*exp(-c*x))
9 اوL د Coefficient Data: a=
57.760559
b=
286.77098
c=
0.72892383 S = 3.46006530 r = 0.98725204 75 54.
Y Axis (units)
67 45. 58 36. 50 27. 42 18. 3 9.3 5 0.2
0.0
2.0
4.0
6.0
X Axis (units) User-Defined Model: y=a/(1+b*exp(-c*x)) Coefficient Data: a=
57.760567
b=
138.3462
c=
0.72892364
92
8.0
10.0
12.0
S = 3.46006530 r = 0.98725204 75 54.
Y Axis (units)
67 45. 58 36. 50 27. 42 18. 3 9.3 5 0.2
0.0
1.8
3.7
5.5
7.3
9.2
11.0
X Axis (units)
Excel ام8&f ذج#$ ا7I 6%%I0 ة8H 9`% 3= 3* اN أد
3*دوات أ *ر ا0 ا:
93
Solver ة6= TOK
Equal To 3= و5ZK ع ا ت ا اد2 J 9 L ي2K Set Target Cell أنS 3* اBy Changing Cells ; اZ* ع ا ت2 J ZK *8 وZ* Min * أ " # N Solve " # dZU و$D$1:$D$3 9و0 ا; ا8 3=
94
Sان ا) D1 8و D2و 2K D3ى اbن ا; ا 9@Tا ;رة و 2K E16ى اNL 2 J 9 Lع ت ا: 0ء.
95
The Workforce Inventory Example : 2 4 ة/ ذج _ون# -2 8 2T= 3/8ذج د2 ء9// # ةH ة/= ء:#A اH T d ذج ا2 ا ا6ه م# "دي إBK L 9 ة ا2; اl2K ب2 ون واG) ت إدارة ا3= N#`* ا9`آ l2K 3= =عA ان ا3 وه9;: \ 9J* 8 i8 ا9 آ ا.*جA ا3= *;ارAا 9/8 دgU28 3* اN/P ا. اآ^ إ *;ارا9 # ة2L 3دي اB8 ان/ 8 ا لg8 Kو :ذج2 ا
Inventory Production
Sales
InventoryCoverage
Productivity
TargetInventory InventoryCorrection TargetProduction
Workforce NetHireRate
TimeToCorrectInventory
TargeWorkforce TimeToAdjustWorkforce
" # ي2*8 ذج2 ا ( Level ى2* ) أو اStocks ا *ع#2 (ا Inventory ونG) ( ا1 Workforce 9 ة ا2;( ا2 Flows بAاع ا2 أ9 ب( أر ( Inflow N ون ) إ ب داG) *ع ا8GK وProduction *جA( ا1 ( Outflow 3H) أ ب ر
ونG) *ع اN ;K وSales ( ا ت2
9 ة ا2; *ع ا8G8 أوN ;8 2 وهNetHireRate *)امA ل ا3=% (4 ( و3 Auxiliary Variables ة# اتZ* (ج TargetInventory فT* ون اG) ( ا1 InventoryCorrection ونG) اgK (2 TargetProduction فT* *ج اA( ا3 TargetWorkforce 9=T* ا9 ة ا2;( ا4 Constants ذج2 اX ا2Y (د Productivity 9H*A( ا1 96
InventoryCoverage ونG) ا9:ZK (2 TimeToCorrectInventory ونG) اg* ب2 : اG( ا3 TimeToAdjustWorkforce 9 ة ا2; اN8* ب2 : اG( ا4
Causal Relationships between Variables :ذج2 ا3= 9 ت اLا :ونG) *ع ا:Sأو Workforce Production Productivity
Inventory Sales
Inventory
InventoryCorrection
TargetProduction
:9 ة ا2; *ع ا:Y (Workforce) (TargeWorkforce)
NetHireRate
TimeToAdjustWorkforce
Workforce
Productivity TargeWorkforce TargetProduction
NetHireRate
(Workforce)
Workforce Production
Inventory فT* ون اG) ا# اZ* ا:^Y
InventoryCoverage TargetInventory Sales
97
TargetInventory
InventoryCorrection
TargetProduction
ونG) اgK # اZ* ا:f را Production (Sales)
Inventory
InventoryCoverage Sales
InventoryCorrection TargetInventory
TimeToCorrectInventory
InventoryCorrection
TargetProduction
TargeWorkforce
فT* *ج اA ا# اZ* ا: Inventory TargetInventory
InventoryCorrection TargetProduction
TimeToCorrectInventory Sales
Workforce TargetProduction
TargeWorkforce NetHireRate
9=T* ا9 ة ا2; ا: د Productivity InventoryCorrection Sales
TargeWorkforce TargetProduction
98
(NetHireRate) Workforce TargeWorkforce
Production NetHireRate
(Workforce)
Feedback Loops %2" ا+دورات ا :ونG) *ع ا9` ) ا986Z*دورة ا Loop Number 1 of length 6 Inventory InventoryCorrection TargetProduction TargeWorkforce NetHireRate Workforce Production
:9 ة ا2; *ع ا9` ) ا986Z*دورة ا Loop Number 1 of length 1 Workforce NetHireRate Loop Number 2 of length 6 Workforce Production Inventory InventoryCorrection TargetProduction TargeWorkforce NetHireRate
:فT* *ج اA ا# اZ* 9` ) ا986Z*دورة ا Loop Number 1 of length 6 TargetProduction TargeWorkforce NetHireRate Workforce
99
Production Inventory InventoryCorrection
:ونG) اgK # اZ* 9` ) ا986Z*دورة ا Loop Number 1 of length 6 TargetProduction TargeWorkforce NetHireRate Workforce Production Inventory InventoryCorrection
:9=T* ا9 ة ا2; ا# اZ* 9` ) ا986Z*دورة ا Loop Number 1 of length 6 TargetProduction TargeWorkforce NetHireRate Workforce Production Inventory InventoryCorrection
:ذج2 ت اSد (01)
FINAL TIME = 100
Units: Month
The final time for the simulation. (02)
INITIAL TIME = 0
Units: Month
The initial time for the simulation. (03)
Inventory = INTEG(Production-Sales ,300) Units: Widget
(04)
InventoryCorrection = (TargetInventory - Inventory)/ TimeToCorrectInventory Units: Widget/Month
(05)
InventoryCoverage = 3
Units: Month
(06)
NetHireRate = (TargeWorkforce Workforce)/TimeToAdjustWorkforce Units: Person/Month
100
(07)
Production = Workforce*Productivity Units: Widget/Month
(08)
Productivity = 1 Units: Widget/Month/Person
(09)
Sales = 100 + STEP(50,20)
Units: Widget/Month
(10)
SAVEPER = TIME STEP
Units: Month
The frequency with which output is stored. (11)
TargetInventory = Sales * InventoryCoverage Units: Widget
(12)
TargetProduction = Sales + InventoryCorrection Units: Widget/Month
(13)
TargeWorkforce = TargetProduction/Productivity Units: Person
(14)
TIME STEP = 1 Units: Month The time step for the simulation.
(15)
TimeToAdjustWorkforce = 3
Units: Month
(16)
TimeToCorrectInventory = 2
Units: Month
(17)
Workforce = INTEG(NetHireRate, TargeWorkforce) Units: Person
:9; ا`;ة ا3= ة: ا9و0ذج ; ا2 آت ا9J* 9*ل ا/!0ا runinv01 Inventory 600 450 300 1 150 0 Production 400 300 200 100 1 0 Sales 200 170 140 110 80
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0
50 Time (Month)
100
101
1
runinv01 Workforce 400 300 200 100
1
1
1
1
1
1
1
1
1
1
1
1
0 NetHireRate 60 30 1 0 -30 -60
1
0
50 100 Time (Month)
Graph for Inventory 600 1
1
1
450
1
1
1
1
1
1
1
1
1
300
1
1
1
1
150
0 0
10
20
Inventory : runinv01
30 1
40 50 60 Time (Month) 1
1
1
102
1
1
70 1
80 1
1
90 1
100 Widget
Graph for Workforce 400
300
1
200
1
100
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0 0
10
20
Workforce : runinv01
Time (Month)
30 1
40 50 60 Time (Month) 1
1
1
1
1
70 1
80 1
1
90
100 Person
1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
"Inventory"
Runs:
Inventory
runinv01
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
300
250
241.667 269.444 322.685 388.272 453.215 506.799 541.985 555.977 549.973
528.308 497.203 463.414 433.022 410.524 398.355 396.822 404.408 418.328 435.206 451.737 465.224 473.925 477.189 475.377 469.638 461.582 452.939 445.246 439.628 436.675 436.435 438.496 442.13 446.471 450.676 454.068 456.216 456.971 456.438 454.92 452.836 450.626 448.68 447.279 446.565 446.542 447.099 448.047 449.163 450.232 451.084 451.614 451.786 451.632 451.232 450.693 450.128 449.636 449.287 449.115 449.119 449.269 449.516 449.802 450.074 450.288 450.418 450.457 450.414 450.308 450.169 450.025 449.9
449.813 449.772 449.776 449.816 449.88 449.954
103
Time (Month)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
"Workforce" Workforce 100
Runs:
runinv01
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
141.667 177.778 203.241 215.586 214.943 203.584 185.187 163.991 143.997 128.335 118.894 116.212 119.607 127.502 137.831 148.467 157.585 163.92 166.879 166.531 163.486 158.701 153.264 148.188 144.261 141.944 141.357 142.307 144.382 147.047 149.76 152.061 153.635 154.341 154.205 153.392 152.148 150.754 149.467 148.483 147.916 147.79 148.054 148.598 149.286 149.977 150.557 150.948 151.115 151.069 150.852 150.53 150.172 149.846 149.6
149.461 149.435 149.508 149.651 149.828
150.004 150.15 150.247 150.286 150.272 150.214 150.13 150.039 149.956 149.895 149.861 149.856 149.876 149.913 149.959 150.004 150.04 150.064 150.074 150.069
104
Fibonacci Sequence Model :A 9%P I ذج# -3 وا9 Z* و9^ ا9H ار y ( k + 2 ) = y ( k + 1) + y ( k ) ,
y (1) = 1, y ( 2 ) = 1, k = 1, 2,3,⋯
k = 1 ⇒ y ( 3) = y ( 2 ) + y (1) = 1 + 1 = 2
k = 2 ⇒ y ( 4 ) = y ( 3) + y ( 2 ) = 2 + 1 = 3 k = 3 ⇒ y ( 5 ) = y ( 4 ) + y ( 3) = 3 + 2 = 5 ⋮
:ض2 9 8Z* و" و0 ا9H إ" ارT2 x ( k + 1) = y ( k ) , k = 1,2,3,⋯ ,
x ( 2 ) = y (1) = 1
:9ء اi= N/! 3= j82* ا3P*2= 9 ** 9= ا9 ا دg*= y ( k + 2 ) = y ( k + 1) + x ( k + 1)
x ( k + 2 ) = y ( k + 1) k = 1, 2,3,...
y (1) = 1, y ( 2 ) = 1, x ( 2 ) = y (1) = 1
k = 1 ⇒ y ( 3) = y ( 2 ) + x ( 2 ) = 1 + 1 = 2 x ( 3) = y ( 2 ) = 1
k = 2 ⇒ y ( 4 ) = y ( 3) + x ( 3) = 2 + 1 = 3 x ( 4 ) = y ( 3) = 2
k = 3 ⇒ y ( 5) = y ( 4 ) + x ( 4 ) = 3 + 2 = 5 x ( 5) = y ( 4 ) = 3
⋮
Vensim ام8&f A 9%P I# ء اGP > ذج#
y dy
x dx
105
(1)
dx = y
(2)
dy = y+x
(3)
FINAL TIME = 10
(4)
INITIAL TIME = 1
(5)
SAVEPER = TIME STEP
(6)
TIME STEP = 1
(7)
x= INTEG ( dx,1)
(8)
y= INTEG ( dy,1)
k
1
2
3
4
5
6
7
8
9
y
1
3
8
21
55
144
377
987
2584 6765
x
1
2
5
13
34
89
233
610
1597 4181
10
ي اود2K) x ( و9HوGى اود ا2K) y 9 ; 3P*2= 9 ** انS (98ا`د k
1
2
3
4
5
6
y
1
3
8
21
55
144
x
↑ ց ↑ ց ↑ ց
↑
1
13
2
5
ց
↑
ց
34
106
↑ 89
…
Dynamic Stochastic Models %R اAI ا%% $ 8ذج ا#$ ا-4 Markovian Property of order k 9H ار%P رآ# ا%H ا:l8K
9ز0 ا# T L " # *K t G ا# 9@ا2Pهة اO ا9 L " ان# ل2;K ن+= { yt , −∞ < t < ∞} X = ^ إذا آ.d;= t − 1, t − 2,..., t − k
(
)
P yt < s | yt −1 , yt −2 ,..., yt −k , yt −( k +1) ,... = P ( yt < s | yt −1 , yt −2 ,..., yt −k )
.9=2 ا رآ9%) اQ*K 3* ا98*دLAاه ا2O^ ا/هك ا : Vensim ام8&f %P رآ# ا%R اAI ا%ذج ا آ#$ ا%#0 م28 98T 3= اد2 ا0 `لLAهة وا;س ان اP 8*دLA أ اH و:1 ^ل Q8ز2K $ 3@ا2P# f: Q ; ا2 ا98T 3= `ل دةLA " ا# *8 اولK .هةO ا56 هl8 ذج2 ن2 آ.م2] ا6 σ 2 8K`ي و% d 2* 3D R ن2/= ε t 2 ه3@ا2P اf:) واyt 2م ه2 ا98T 3= `لLA `*ض أن ا:Nا 9=2 ا آ9%)ا yt = φ1 yt −1 + φ2 yt −2 + ε t , ε t ~ N ( 0,σ 2 ) , ∀t
9 Z* 9 J* \ 9^ ا9H ارDifference Equation 9L =و9 د56وه ن2/= xt −1 = yt −2 `*ض،* 3 او9H در3 إT2 ف2 .وا yt = φ1 yt −1 + φ2 xt −1 + ε t , ε t ~ N ( 0, σ 2 ) , ∀t xt = yt −1
9; ت اS ا دN= φ1 = 1.7, φ2 = −0.72, and σ 2 = 1 X ا ^ل ا إذا آ3= :2 ^ل y0 = 0, x0 = 0
9; أو
Vensim *)ام+ :Nا
107
x dx
y dy
phi2
eps phi1
(01) dx = y (02) dy = phi1*y+phi2*x+eps (03) eps= RANDOM NORMAL(-3.99,3.99 ,0 ,1 ,19 ) (04) FINAL TIME = 2
The final time for the simulation.
(05) INITIAL TIME = 0
The initial time for the simulation.
(06) phi1 =
1.7
(07) phi2 =
-0.72
(08) SAVEPER = TIME STEP
The frequency with which output
is stored. (09) TIME STEP = 0.01
The time step for the simulation.
(10) x= INTEG ( dx, 0) (11) y= INTEG ( dy, 0)
108
dy v y 4
2
0
-2
-4 -0.100
-0.050
0 y
0.050
0.100
dy : Current
xvy 0.008
-0.009
-0.026
-0.043
-0.06 -0.100
-0.050
0 y
x : Current
109
0.050
0.100
y 0.2
0.1
0
-0.1
-0.2 0
0.50
1 Time (Day)
1.50
2
1.50
2
y : Current
dy 4
2
0
-2
-4 0
0.50
1 Time (Day)
dy : Current
110
x&y 0.2 0.008
0 -0.026
-0.2 -0.06 0
0.50
1 Time (Day)
y : Current x : Current
111
1.50
2
: bifurcation kIA ذج ا# -5 ^ آa*K T2/ 3U8 رN/P l^/ N/P نbرس اK 3* أه ا ذج اRP* ذج ا . ا ذج56 أ هN^ 8 3*ذج ا2 وا،92/ وا9:اه ا2O ا 9:) ا9 U`*ت اS ا د9#2 J فK ف *ض2 ا ارس6 ه3= xɺ = −0.5 x + ay ,
a < 0.4
yɺ = x − 0.5 y
9ء اi= N/P T*/و a x xɺ −0.5 yɺ = 1 −0.5 y a −0.5 A= −0.5 1 −0.5 − λ det ( A − Iλ ) = det 1
( −0.5 − λ )
2
=0 −0.5 − λ a
−a =0
λ1 = − a − 0.5 λ2 = a − 0.5
:VenSim ?S# kIA ذج ا# %#0 x dx/dt b
a
y dy/dt
c
(01) a = -0.3
Units: **undefined**
(02) b = -0.5
Units: **undefined**
(03) c = -0.5
Units: **undefined**
112
(04) "dx/dt" = b*x+a*y
Units: **undefined**
(05) "dy/dt" = x+c*y Units: **undefined** (06) FINAL TIME = 10
Units: Month
The final time for the
Units: Month
The initial time for the
simulation. (07) INITIAL TIME = 0 simulation. (08) SAVEPER = TIME STEP Units: Month
The frequency with
which output is stored. (09) TIME STEP = 0.0625
Units: Month
The time step for the
simulation. (10) x = INTEG ( "dx/dt",
1)
Units: **undefined**
(11) y= INTEG ( "dy/dt",
1)
Units: **undefined** a = −0.3
1 2 0.06 0.6
3
3
1
3
3
3
3
3
3
3
2 3
2 41
-0.2 -0.08 -1 -0.6
2 2
1
2
4
4
4
2 1
0
4
4
3 4 1
4
4
4
4
1 2
1
3
4
2
5 6 Time (Month)
2
2
7
1
1
1
1
1
2
8
1 2
2
9
10
1 1 1 1 1 1 1 1 1 1 x : Current 2 2 2 2 2 2 2 2 2 2 y : Current 3 3 3 3 3 3 3 3 3 "dx/dt" : Current 3 4 4 4 4 4 4 4 4 4 4 "dy/dt" : Current
a = 0.3
113
2 4 0.06 0.6
3
3
3
3
3
3
3
3
3 3 3
4 3
0.8 0 -0.2 0
2
2
2
2
2
2 4
4
1 1
0
1
1
1
4
2
1
3
41
4 1
4
4
5 6 Time (Month)
4
4
4
4
1
1
1
1
1
2
2
2
2
2
2
7
8
9
10
1 1 1 1 1 1 1 1 1 1 x : Current 2 2 2 2 2 2 2 2 2 2 y : Current 3 3 3 3 3 3 3 3 3 "dx/dt" : Current 3 4 4 4 4 4 4 4 4 4 4 "dy/dt" : Current
a = −0.3, b = −0.5, c = 0.5
20 40 4 8
1
2 2 1
3 4
-20 -40 -4 -8
2 3
12
2 1
34 12
2
3
4
4
1
23
4
1
3
4
34
1
4
1
3
4
1
2
3 3 2
2
1 4
2
3
1
4 3
0
23
46
69
92 115 138 Time (Month)
161
184
207
230
1 1 1 1 1 1 1 1 1 1 x : Current 2 2 2 2 2 2 2 2 2 2 y : Current 3 3 3 3 3 3 3 3 3 "dx/dt" : Current 3 4 4 4 4 4 4 4 4 4 4 "dy/dt" : Current
a = −0.3, b = −0.3, c = 0.5
114
800 B 2e+012 200 B 400 B
4 4 123412341234123412341234123412341234123
3 12
-0.8 Tr -2e+012 -0.2 Tr -0.4 Tr 0
23
46
69
92 115 138 Time (Month)
161
184
207
230
1 1 1 1 1 1 1 1 1 1 x : Current 2 2 2 2 2 2 2 2 2 2 y : Current 3 3 3 3 3 3 3 3 3 "dx/dt" : Current 3 4 4 4 4 4 4 4 4 4 4 "dy/dt" : Current
a,b,c N/ 9` *) ; 9 U`*ت اS ا د9#2 J فK 3 # ربJK H أ:8 K
x dx/dt a
y r
z dz/dt b
115
dy/dt
xɺ = − a ( x − y ) yɺ = − xz + bx − y zɺ = xy − cz
a = −0.1, b = 0.02, c = −0.03
4 1 2 0.4 2 2
2
3 2
5
5
6
6
4 3
0 -1 -0.4 0 -2 -2
6 1
4
1
5
6
5
1
4
12
1
1
4 5
2
6
45 6
2
1 2
4
1
4
5
2
56
4 3
3
2
3
3
3 3
0 x : Current y : Current 2 z : Current 3 "dx/dt" : Current "dy/dt" : Current "dz/dt" : Current
1
2
3
1
1 2 5
5 6
5 6
5 6
116
3 4
5 6
2 3
4
4 5
6
10
1 2
3 4
9
1 2
3 4
8
1 2
3 4
7
1 2
3 4
6
1 2
3 5
4 5 6 Time (Second)
5 6
6
The Pumping Heart Model :k2/ اP 2GI 9 /0 ذج# -6 . Jاء اGH اQ H " إ9@ اJ آS 3Z ام اN;K 9)i ن آA اR L *ر# إ/ 8 *دد8 مO آR ; ا. Oscillator ا( آ *ددH 38;K N/P ) نA اR L 9H6 / 8 آ . ;K 9 أيSystole )ء و إ;ضK إر9 أيDiastole إ ط:م# N/P * Electro-Chemical
3@ وآT آ6` R ;ت اi# إ;ض وإ ط3= R *8
9 آv وR ; ا3= Muscle Fiber 9 i# 9` ل2D 2 هx أنU*=ذا إ+= . Stimulus R ; ا9 i# 9` ل2D 3= ;ضA ط واA أن ل ا9 رب اJ* اH =; و،6` ا 9` اJ V Y ا`قQ ( µ > 0 R K X ^ ) * ;8 و6` ا9 آQ دادG8 .9` ل ا2D Q L*K 6` ا9 أن ل آH آ و.T2Dو ت: " ا# * اR ; ا9 i# 9` 9ذج آ2 9 U`*ت اSن ا د2 آ:Sأو .9; ا :a@* اL و9* ; اVensim ذج2 ن2 آ:Y µ = 2 cm / sec, x ( 0 ) = 2 cm
v ( 0 ) = 1 microgrm, t = 0 ( 0.1) 100 sec
:Nا ;8 و6` ا9 آQ دادG8 R ; ا9 i# 9` ل2D 3= ;ضA ط واA أن ل ا:Sأو إذاT2D و9` اJ V Y ا`قQ ( µ > 0 R K X ^ ) * dx ( t ) = v ( t ) − µ x 3 ( t ) 3 − x ( t ) dt
إذا9` ل ا2D Q L*K 6` ا9 و أن ل آ dv ( t ) = − x (t ) dt
d N/P T*/و dx = v − µ ( x3 3 − x ) dt dv = −x dt
Vensim Q 9 # ذج#$ ا:%c
117
x dx
mu
v dv
(01) dv = -x
Units: microgrm/sec
(02) dx = v-mu*(((x^3)/3)-x) Units: cm/sec (03) FINAL TIME = 100
Units: Second
The final time for the
simulation. (04) INITIAL TIME = 0
Units: Second The initial time for the
simulation. (05) mu = 2
Units: cm/sec
(06) SAVEPER = TIME STEP
Units: Second
The frequency with which output is stored. (07) TIME STEP = 0.1
Units: Second
The time step for the
simulation. (08) v = INTEG (
dv, 1)
Units: microgrm
(09) x = INTEG (
dx, 2)
Units: cm
118
Current 1 x 4 2 1 0 -2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1 1
1
1
1
1
-4 dx 1
4 2 0 -2 -4
1
1 1
1
1
1
1
1
1
1
1
1 1
1
1
0
25
50 Time (Second)
75
100
(1) N/! 1
Current v 4 2 0 -2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
-4 dv 4 2 0 -2 -4
1
1
0
1
1
1
25
1
1
1
1
50 Time (Second)
1
1
1
1
75
100
(2) N/!
119
Heart Rate v Fiber Length 4
2
0
-2
-4 -3 dx : Current
-2
-1
1
1
1
0 x 1
1
1
1 1
1
2
1
1
1
3 cm/sec
1
(3) N/!
Heart Rate v Stimulus 4
2 1 1
1
1
1
0
-2
-4 -3 dx : Current
-2 1
-1 1
1
0 v 1
1
1
1 1
1
1
2 1
1
1
3 cm/sec
(4) N/!
120
Stimulus v Fiber Length 4
2
0
-2
-4 -3
-2
v : Current
1
-1 1
1
0 x 1
1
1
1 1
1
2
1
1
1
3 microgrm
1
(5) N/!
Stimulus v Heart Rate 4 1
1
1
1
1
1
1
2 1
0
-2
1
-4 -4 v : Current
-3
-2 1
1
-1 1
1
0 dx 1
1
1 1
1
2 1
1
3 1
1
4
microgrm
(6) N/! :a@* ا9PL Systole ;ضA( إ" ا9 82D )أفDiastole طA*;ل اA( أن ا1) N/! 8
ي6 وا3` إ ب3= R *8S 3* ) ول0 ا3= ءd ;ضA اT= ثK (ةL )أف 121
B8ذي ا; (Rو # /آ 9# 9ا `+= 6ن ا0ف Q= Q8 N/P j;Kام إ" ا)رج 8 (2) N/! .ا*دد 2* 3= Oscilationي ا ` 6ول .5ZKا/!0ل )(3 إ" ) 9D gU2K (6ا*ددات اور "= 98ا *82ت: -1ل iت 9 i#ا; 2D U Rل ا(3) N/! 9 i -2ل iت 9 i#ا; U Rآ 9ا `(4) N/! 6 -3آ 9ا `2D U 6ل ا(5) N/! 9 i -4آ 9ا ` U 6ل iت 9 i#ا; (6) N/! R :8 K H+ L -1اء JKرب " #ا 2ذج ) ون 2ف وث 2D0 (9 L 9*/ال )* `9 i 9 ا; Rوآ ت )* ` 6` 9وLرن ا*@a 8 3* -2ث إTHد 9 iا; Rو #اي *82ت `6؟ -3ه X Y YfK 2ا* µ R؟ -4د #أي *2ى ا ` 6أو #أي K µ X ^ 9 Lث 9 L 9*/؟ )أي lL2*K 9 i#ا; # Rا*دد 2 Rف او =ح =(3@J
122
:Lorenz Attractors Models $ ذج !ذت ر# -7 ا إدوارد9: ا2 رت2D ت ا دةS ا د9: 9#2 J 3 هc*8ر2 ذ تH ZK أن أي2J ا# B* 9 0 ا9 /P ا.ر/*KS 3* ا2J ط ا0 $* ء دراY أc*8ر2 * د#0 ا# 9# 8 ي6( وا9!ح =اH ` " 8 YfK ) 9و0 ط ا0 ا3= d 2و .9 /# أو9 رi* a@* "دي إB8 9و0وط اP ا3 # ا س \ 9 U`K تSث دY ن2/*8 3م آO رN/! # رة# 2 هc*ر2 9 ذH Z* ا ا6 ه6 إذا ا، واZ* 9TJ* 9 دا2ة ه6ت هS ا دN ،"و0 ا9H ار9: 3= 98 زاوT وG ن2/ ا ار. ر ارQ**8 Nن ا+= G ا$ " أ س ا# :3* آ3ت هS ا د. أ د9YY d x ( t ) = −ax ( t ) + ay ( t ) dt d y ( t ) = bx ( t ) − y ( t ) − z ( t ) x ( t ) dt d z ( t ) = −cz ( t ) + x ( t ) y ( t ) dt
Vensim S& ذج ا#$ ا%#0 x dx/dt a
y
b
dy/dt
z dz/dt
c
dx/dt=a(y-x)
a=10
x(0)=0
dy/dy=bx-y-xz
b=28
y(0)=0. 1
start time=0
c=2.67
z(0)=25
end time=100
dz/dt=xy-cz
dt=0.02
123
(01) a = 10
Units: **undefined**
(02) b = 28
Units: **undefined**
(03) c = 2.67
Units: **undefined**
(04) "dx/dt" = a*(y-x) Units: **undefined** (05) "dy/dt" = b*x-y-x*z
Units: **undefined**
(06) "dz/dt" = x*y-c*z
Units: **undefined**
(07) FINAL TIME = 100
Units: Second
The final time for the
Units: Second
The initial time for the
simulation. (08) INITIAL TIME = 0 simulation. (09) SAVEPER = TIME STEP Units: Second The frequency with which output is stored. (10) TIME STEP = 0.02
Units: Second
The time step for the
simulation. (11) x = INTEG ( "dx/dt",
0)
Units: **undefined**
(12) y = INTEG ( "dy/dt",
0.1) Units: **undefined**
(13) z = INTEG ( "dz/dt",
25)
Units: **undefined**
124
40 40 60 400 600 600
2
6
1 56
-40 -40 0 -400 -600 -600
1
1
1
4
4
12
2
4
5
3
56
2
5
5 2
6
3
4
456
345 1
3
4
6
5
12
2
2
2 1
1 6
4
56
6
3
5
12
3
3 4 3
3
4
3
0
1
2
3
4
5 6 Time (Second)
7
8
9
10
1 1 1 1 1 1 1 1 1 x : Current 2 2 2 2 2 2 2 2 2 y : Current 3 3 3 3 3 3 3 3 3 z : Current 4 4 4 4 4 4 4 4 "dx/dt" : Current 4 5 5 5 5 5 5 5 5 5 "dy/dt" : Current 6 6 6 6 6 6 6 6 "dz/dt" : Current
1
Current x 40 20
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
0
1
1
1
1
1
1 1
1
1
1
1
1
1
-20 -40 "dx/dt" 400 200 1
0
1
1
1
1
1
1
1
1
1
1
-200 -400
0
2.5
5 Time (Second)
125
7.5
10
1
Current y 40 20 0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1
1 1
1
-20 -40 "dy/dt" 600 300 1
0
1
1
1
1
1
1
1
1
1
1 1
-300 -600
0
2.5
5 Time (Second)
126
7.5
10
1
Current z 60
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
45 1
1
30 1
15
1 1
1
1
1
1
1
1
1
1
1
1
1
0 "dz/dt" 600 300 1
0
1
1
1
1
1
1
1
1
1
1
1
1
-300 -600
0
2.5
5 Time (Second)
7.5
10
xy-plane 40 1
1
1
1
20
1
1 1 1
0
-20
-40 -19 y : Current
-9 1
1
2 x 1
1
1
1
12 1
1
1
127
1
22 1
1
1
1
xz-plane 60
45 1 1
30 1
15
1 1 1
1
1
0 -19
-9
z : Current
1
1
2 x 1
1
1
12
1
1
1
1
1
22 1
1
1
1
yz-plane 60
45
30 1
15
1 1
1
1
1
1
1
0 -24
-20
-16
-12
-8
-4
0
4
8
12
16
20
24
28
y z : Current
1
1
1
1
1
1
1
1
128
1
1
1
1
1
1
# -8ذج ا#س وا: Prey and Predator Model % G ه6ا 2ذج Oم 2/*8ن 2رد و * ] Tوا6ي " # d;= 8ه6ا ا 2رد .ا 2ذج ه6ا * 8ض إ *Gاف ا 2ارد وYfKة " #ا Q *Jا 3 `:ا6ي 2/8 L ^ = .T # 8ن ا 2رد Q:Lا0را Rوا * Q:L ] Tا^ Y/*K ،Rا0را Rدورة ز 9ى ل # Q R *8ده = 3اورة ا ; 9O Q 9أن ا 2ارد ا *=2ة راg KS R fن 8أآ^ 500أر*;8 .Rت " #ه 56ا0را Q *J Rا^ Rوا6ي 2 8وG8داد آ آن هك #د آف آ ا0را .Rرس Oم ا دSت ا`و 9Lا* lK 3ه6ا اOم ا د 9اآ 9راR r rk +1 = rk + g 1 − k rk − 0.001 rk f k 500
ا د 9اآR^ 9 f k +1 = f k + 0.001 rk f k − 0.02 f k
ا د 9اآ 9را K Rأن ا0راGK Rداد ل # Q R *8ده " #ا# 8G8Sده 500 #أر Rو; L*Kار # 0.001د ا^ .Rا 9اآ K R^ 9ان ا^G8 Rداد #ده * # 0.001 Qد ا0را Rو; L*Kار # 0.02ده آN دورة. %#0ا #$ذج 8&fام :Vensim ا aا*)* + 3ام l8 Vensimآ 9اOم
Rabbits change changeRate Foxes foxesChange
)R(t)=R(t-1)+G[1-R(t-1)/500]R(t-1)-0.0001R(t-1)F(t-1 )F(t)=F(t-1)+0.0001R(t-1)F(t-1)-0.02F(t-1
129
0 200 400
3
1
3
3
1
3
3
3
3
1
3
3
1
3
1
3
3
1
3
1
-1 100 300
3 3
1 1 1
-2 0 200
2
0
2
10
change : Current Foxes : Current Rabbits : Current
2
2
2
2
2
2
2
2
1
2
2
2 1
1 1
20
30
1
1 2
40 1
2 3
50 60 Time (Day)
1 2
3
1 2
3
1 2
3
1 2
3
70 1
2 3
1 2
3
change
3
changeRate
(Rabbits)
Rabbits foxesChange
Loop Number 1 of length 1 Rabbits change Loop Number 2 of length 3 Rabbits foxesChange Foxes Change
130
1 2
3
Foxes
1
90
1 2
Rabbits
change
1
80
Foxes (Rabbits)
2
2
2
1 2
3
100 1 2
3
3
(01)
change = changeRate*(1-Rabbits/500)*Rabbits-0.0001*Rabbits*Foxes
(02)
changeRate = 0.01
(03)
FINAL TIME = 100 Units: Day
(04)
Foxes = INTEG ( foxesChange, 20)
(05)
foxesChange = 0.001*Rabbits*Foxes-0.02*Foxes
(06)
INITIAL TIME = 0 Units: Day
(07)
Rabbits = INTEG ( change, 400)
(08)
SAVEPER = TIME STEP Units: Day
The final time for the simulation.
The initial time for the simulation.
The frequency with which output is stored.
(09)
TIME STEP = 1 Units: Day
Current 1 Rabbits 400 1
1
1
1
1
1
1
The time step for the simulation.
1
1
1
1 1
350
1 1
300 250 200 change 0
1 1
-0.5
1 1
-1
1
-1.5
1 1
-2
0
25
50 75 Time (Day)
100
131
Graph for Rabbits 1
400
1
1
1
1
1
1
1
1
1 1
1
350
1 1 1 1
300
250
200 0
10
Rabbits : Current
20 1
30 1
40 1
50 60 Time (Day)
1
1
1
1
70 1
80 1
1
90 1
1
100 1
: %99 ت اFaIأ
(Foxes) foxesChange
Foxes
Rabbits
change
Rabbits
Foxes foxesChange
(Foxes) : %2ورات ا8ا
Loop Number 1 of length 1 Foxes foxesChange Loop Number 2 of length 3 Foxes change Rabbits FoxesChange
132
1
Current Foxes 200
1
1
1
1
1
1
1
150 1
100
1 1 1
50 1
1
1
1
0 foxesChange 2 1.5 1
1
1
1
1
1 1
0.5 0
1
1
0
25
50 75 Time (Day)
100
Graph for Foxes 200
150
100
50 1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0 0
10
Foxes : Current
20 1
30 1
40 1
1
50 60 Time (Day) 1
133
1
1
70 1
80 1
1
90 1
1
100 1
: 1
ا 9 0وا HAت *رات ا ;9
134
ا ا ا ث ا ت2 ء وA اL V 203 ء ا ذج: ا دة 9 ` ل ا# ولS *ر اSا هـ1422/1421 3^ اN`ا *# : Gا :9* ا9 0 اQ H 3 # RHأ :ال اولr ا :T N/ S^ 3:# وأ9* ت ا: ف ا# ب ا)رجA )د( أInflow N ب ااA )ج( أFlow بA )ب( أStock )أ( ا *ع 9` 986ZK )ز( دورةSink (9#2ر )ا2Z )و( اSource )هـ( ا رOutflow System مO ا9H6 ( )طAuxiliary Variable # اZ* )ح( اFeedback Loop Dynamic Mathematical Model (3/8 *ك )د3U8ذج ر2 ( )يModeling
( =;ةN/ 9# l _ ت# 9 ) :ال اr ا د ا زل# 3 # * !ة8 تH^ ا$ Q8 ي6 أن ا ل ا9G تHY زع2 S إذا.تHY ] K 3*د ا زل ا# د8زد+ L*8 ا ا ل6ت وأن هHY T H28S 3*ا وT " إ*آ# ي2K / و9HY ] K S 3* زل ا3 %0 اد اN^ 8 H أنU*=إ .9HY ] K 3*د ا زل ا# x G وة ا6 ) م اO ا9H6 Difference Equation 9L ا`و9)أ( أن ا د :3م( ه2ا xt = xt −1 + k ( H − xt −1 )
(l* و#) وH = 20000 زل3 %0ذج د ا2 ا ا6 هN ، t = 0 # x = 0 )ب( إ=*ض أن (l* و#) t = 0,1, 2,⋯,10 ; ، k = 0.05 R K X Y
135
)ج( #ا *ع و ا Aب وا *Zات ا #ة وا ر ودورة ا* 986Zا) `9#) 9 واة( )د( أر !* 3::)K N/ع وأ Aب ، Stock and Flow Diagramو`)* + 56ام Vensimو ) .]J@* NJأر # 9ت(
136
ا HAت ا 2ذ:9H )– 1أ( ا *ع Stockأي ! Q Z*8ا) Gد* ،3/8ك( G8داد و ;8و " 8اi8 *2ى Levelاو * State Variable 9 Zو N^ 8آ*:3
Stock Source/Sink Flow
Information Link
AuxiliaryVariable
أ^ :9آ 9ا G 3= 5ان# ،د ا T 0ا* ^* T/ * 8 3ا )-1ب( ا 0ب Flowه 2ل 8G8 Rateأو * ;8ع .أ^ :9آ 9ا 5ا G) 986Zان او ا # ،$ 9 2د ا T 0ا *Pاة أو ا 9# )-1ج( أ Aب اا Nه 2ل 8G8ا *ع .أ^ :9آ 9ا 5ا G) 986Zان# ،د اT 0 ا *Pاة )-1د( أ Aب ا)رج ه 2ل ;8ا *ع .أ^ :9آ 9ا 5ا 9 2ا)Gان# ،د ا T 0ا 9# )-1هـ( ا ر ه 2ا Qاو ا N%0ا6ي 6Z8ي ا *ع .أ^ G :9ان 9ا ة ،اT 0 ا :و* 9اول )-1و( ا2Zر )ا (9#2ه 2ا;ع او ا = او ا Jري ا6ي ;8ا *ع .أ^ :9ا Jري او اف ا 3ة ا)ر 9Hا)Gان ،ا T 0ا :و* 9دل )2/8 L :9Oن ا ر وا2Zر ` cا Pآ = 3ا ^ل ا( 0 )-1ز( دورة 9` 986ZKه 3ا* T= YB8 3ا *ع " #ا 0ب ا 6Zي .$أ^ :9ا)Gان ا 6Zى 5ل ا2ب ،9=D $ /*K 5آ Q`KKا ة = 3ا)Gان $ Q`KK ا 9=:و N;Kآ 9ا 5اا 9 )-1ح( ا * Zا #ه 3= N 8 Z* 2و l%و N #اOم و 3= YB8ا Aب وYf*8 L *ع .أ^ :9ا 9ا;2ى )Gان ،رأس ا ل ا *ح ^* Aر 137
lK 9U8ت رL# N^ .مO اN # l8 3U8ذج ر2 QU و،مO ا9H6 (ط-1) T 0 " ا# R : اض وا9 lK 3 L *دLت إS د،5 انG)ل إ*ء ا ن2/*8 وG اQ Z*K تS دl%28 ي6 ا2( ه3/8 *ك )د3U8ذج ر2 (ي-1) 9 !*;قA " ا# لK Gق ا2= 9:; )اxɺ = kx N^ .9L او =و9 U`K تS د 3 ا2 9 3= مO N^ Kو" وS ا9H ار9 U`K 9 د3( وهG (أ-2) x ( t ) − x ( t − dt ) ∝ H − x ( t − dt ) dt =k H − x ( t − dt ) x ( t ) − x ( t − dt ) = k H − x ( t − dt ) dt x ( t ) = x ( t − dt ) + k H − x ( t − dt ) dt = x ( t − 1) + k H − x ( t − 1)
،9L =و9 إ" دT2 Y (92T ) 9 U`K 9 أ ء د2 Ni=0!*;ق اA ا3= xt = xt −1 + k ( H − xt −1 )
(ب-2) xt = xt −1 + 0.05 ( 20000 − xt −1 ) , x0 = 0, t = 0,1, 2,⋯ ,10 x1 = x0 + 0.05 ( 20000 − 0 ) = 1000
x2 = x1 + 0.05 ( 20000 − x1 ) = 1000 + 0.05 ( 20000 − 1000 ) = 1950 x3 = 1950 + 0.05 ( 20000 − 1950 ) = 2852.5 x4 = 3709.875
⋮ x 9HY ] K 3*د ا زل ا# 2ج( ا *ع ه-2) xt − xt −1 تH^ ا$ عK ي6 ا ل ا2 ب هAا H Tى إ*آ2K / و9HY ] KS 3* زل ا3 %0 اد ا# اZ* ا
N ب ااA واxt − xt −1 ;8 3* وH − x ;8 x دادG8 آ9` ) ا986Z*دورة ا .ا6/ وهx 8G8 ا6T و9 9 ن آ2/8 ان/ 8S (د-2)
138
x dx/dt
k
H
(x) H
dx/dt
x
k
x
dx/dt
(x)
Loop Number 1 of length 1 x dx/dt
139
1
Current x 20,000 15,000
1
1
1
1
1
1
1
1
1
1
1
10,000 5,000 1
0 "dx/dt" 1,000 750
1
500 250 0
1 1
0
50 Time (Day)
100
Graph for x 20,000
1
1
1
1
1
1
1
1
1
1
1 1
15,000 1 1
10,000 1
5,000
0
1
0
10
x : Current
x
1
0
20 1
30 1
1
1000
40 1
50 60 Time (Day) 1
1
1950
140
1
70 1
1
80 1
2852.5
90 1
1
100 1
3709.88
4524.38
5298.
ا ا ا
9Hا ] 2د آ 9ا 2م LاAء و 2ث ا ت ا* Aر ا^ #0 3ل ا` Nا^ 1422/1421 3هـ دة ) V 203ء ا ذج ( ا_ 4& 2 :6 أ Q H " # RHا 9 0ا*:9 ا Bال ا0ول: Nا دSت ا`و 9Lا* ; 9ا0و 9ا :ة /Kارt = 1, 2,...,15 ; (Iterate) 8 x0 = 5, x1 = −1 x0 = 3, x1 = −15
xt + xt −1 − 2 xt −2 = 0, xt + 6 xt −1 + 9 xt −2 = 0,
و= 3آ 9 Nأر . t = 1, 2,...,15 Q xt ا Bال ا^:3 هك ا ^/ا2Oاه ا N^ 9:ا2Oاه ا 982وا 9H2وا 9# *HAوا*LAد 98ا ) 2ت اآ^ إLأ 2/K ( Fibonacci Numbers #ن = Tآ O 9م د # 3/8اG ! *K tة 3 #آ 9اOم #ا0ز t − 1 9و += . t − 2ذا آN^ K T V xt , t ∈ T X =*ة ز ;K ،9 9أن ا د 9ا`و 9Lار 9Hا^ 9وا* lK 3د 9/8اOم #أي 9Oز 9ه:3 xt − xt −1 − xt −2 = 0, ∀t ∈ T
)أ( xt = λ t , λ ≠ 0 QU2أن ا د 9ا #ة " # R*/Kا λ 2 − λ − 1 = 0 N/PوT ا Nا*:3
141
1+ 5 ≈ 1.6180339 2 1− 5 λ2 = ≈ 0.6180339 2
λ1 =
N/P ا3 # R*/K xt )ب( أن t
t
1+ 5 1− 5 xt = A + B , ∀t ∈ T 2 2
:N/P ا3 # R*/K xt ن+= 3* وB = xt =
(
) (
−1 1 وA= ن+= x1 = 1 وx0 = 0 9و; أو 5 5
)
t t 1 1 + 5 − 1 − 5 , t = 0,1, 2,... 2 5
(T = {0,1, 2,...})
:QU2 ] وذt = 1, 2,..., 25 ; xt H أوExcel *)ام+ ()ج A1=1 A2=1 A3=A1+A2 . . . A(n)=A(n-1)+A(n-2) A4-A25 8) إ *)م ا وا ا:9O
; انK B1=A2/A1 QU2 B(n) → λ1 ≈ 1.6180339, As n → ∞
; انK C1=A1/A2 QU2 و C(n) → λ2 ≈ 0.6180339, As n → ∞
n=30 6 :9O
:V^ال اB ا
142
Markovian 9=2 ا آ9%) اT 9 98*دL*د أن هة إLAل أ اء ا2;8 ()أ
؟38 = ذا9^ ا9H ارProperty 9 ا دR 9^ ا9H ار9=2 ا آ9%) اQ*8 9 !آT `لLA)ب( ا 9Lا`و xt − axt −1 − bxt −2 = ε t , ∀t ∈ T , ε t ~ N ( 0, 4 )
(Second Order one State
وا9 Z* 9Y 9H در9 ا د56ل ه2
]( وذFirst Order two State Variables) 9 ىZ* " أو9H إ" درVariable) . yt −1 = xt −2 QU2 ; (Steady State) 5م وأ * إ *;ارOا ا6T ذج2 ن2 آVensim *)ام+ ()ج 1) a = 1.2, b = −0.7 2) a = −1.2, b = −0.7
9* إ *)م ا; ا:9O RANDOM NORMAL(-3.99,3.99 , mean ,standard deviation ,seed=Prime Number ) INITIAL TIME = 0 FINAL TIME = 2 SAVEPER = TIME STEP TIME STEP = 0.01
143
ا ا ا V 203 *رA 9/ ل ا2 أ ا :ول0ال اB $ Hإ xt = − xt −1 + 2 xt − 2 , x0 = 5, x1 = −1 5,-1,11,-13,35,-61,131,-253,515,-1021,2051,-4093,8195,-16381,32771
30000
C2
20000 10000 0 -10000 -20000 Index
5
10
15
xt = −6 xt −1 − 9 xt −2 , x0 = 3, x1 = −15 3,-15,-39,-57,-21,-93,51,-237,339,-813,1491,-3117,6099,-12333,24531
20000
C3
10000
0
-10000 Index
5
10
15
:3^ال اB 9 Hإ
144
أي9f ق ا2: R xt −2 وxt −1 3 # *K xt 8 xt = xt −1 + xt −2 xt − xt −1 − xt −2 = 0, ∀t ∈ T
J xt = λ t , λ ≠ 0 QU2 ()أ λ t − λ t −1 − λ t −2 = 0
λ t −2 ( λ 2 − λ − 1) = 0
∵λ ≠ 0 ∴λ 2 − λ − 1 = 0
λ1,2 = λ1 =
if
1± 1+ 4 2
ax 2 + bx + c = 0, then x1,2 =
−b ± b 2 − 4ac 2a
1 5 1 5 + ≈ 1.6180339, λ2 = − ≈ 0.6180339 2 2 2 2
2 ه9L ا`و9 ام دNن ا+= 9` *) و9;; ة# ا9ور ا دGH )ب( ان xt = Aλ1t + Bλ2t , ∀t ∈ T t
t
1+ 5 1− 5 xt = A + B , ∀t ∈ T 2 2 ∵ x0 = 0, x1 = 1 ∴ x0 = A + B = 0 ⇒ A = − B 1+ 5 1− 5 and x1 = A + B =1 2 2 1+ 5 1− 5 ∴− B + B =1 2 2 −1 1 ∴B = ⇒ A= 5 5 t t 1 ∴ xt = 1 + 5 + 1 − 5 , t = 0,1, 2,... 2 5
(
) (
)
()ج A
B
C
1
1
1
1
1
1
2
2
0.5
3
1.5
0.666666667
5
1.666666667
0.6
145
8
1.6
0.625
13
1.625
0.615384615
21
1.615384615
0.619047619
34
1.619047619
0.617647059
55
1.617647059
0.618181818
89
1.618181818
0.617977528
144
1.617977528
0.618055556
233
1.618055556
0.618025751
377
1.618025751
0.618037135
610
1.618037135
0.618032787
987
1.618032787
0.618034448
1597
1.618034448
0.618033813
2584
1.618033813
0.618034056
4181
1.618034056
0.618033963
6765
1.618033963
0.618033999
10946
1.618033999
0.618033985
17711
1.618033985
0.61803399
28657
1.61803399
0.618033988
46368
1.618033988
0.618033989
75025
1.618033989
0.618033989
121393
1.618033989
0.618033989
196418
1.618033989
0.618033989
317811
1.618033989
0.618033989
514229
1.618033989
0.618033989
832040
1.618033989
0.618033989
أن B ( n ) → λ1 ≈ 1.6180339, As n → ∞
C ( n ) → λ2 ≈ 0.6180339, As n → ∞
:V^ال اB 9 Hإ Markovian Property of order k 9H ار9=2 ا رآ9%))أ( ا
9ز0 ا# T L " # *K t G ا# 9@ا2Pهة اO ا9 L " ان# ل2;K ن+= { yt , −∞ < t < ∞} X = ^ إذا آ.d;= t − 1, t − 2,..., t − k
146
(
)
P yt < s | yt −1 , yt −2 ,..., yt −k , yt −( k +1) ,... = P ( yt < s | yt −1 , yt −2 ,..., yt −k )
()ب xt − axt −1 − bxt −2 = ε t , ∀t ∈ T , ε t ~ N ( 0, 4 )
∵ yt −1 = xt −2 ⇒ yt = xt −1 ∴ xt = axt −1 + byt −1 + ε t yt = xt −1
( )ج
x dx
a
eps b y dy
(01) a = 1.2
(02) b = -0.7
(03) dx = a*x+b*y+eps
(04) dy = x
(05) eps = RANDOM NORMAL(-3.99,3.99 , 0 ,2 ,19 )
(06) FINAL TIME = 2
147
(07) INITIAL TIME = 0
(08) SAVEPER = TIME STEP
(09) TIME STEP = 0.01
(10) x = INTEG ( dx, 0)
(11) y= INTEG ( dy, 0)
Current x 0.2 0.1 0 -0.1 -0.2 dx 4 2 0 -2 -4
0
0.50
1 Time (Day)
1.50
2
;* \ مOا
(01) a = -1.2 148
(02) b = -0.7
(03) dx = a*x+b*y+eps
(04) dy = x
(05) eps = RANDOM NORMAL(-3.99,3.99 , 0 ,2 ,19 )
(06) FINAL TIME = 2
(07) INITIAL TIME = 0
(08) SAVEPER = TIME STEP
(09) TIME STEP = 0.01
(10) x = INTEG ( dx, 0)
(11) y = INTEG ( dy, 0)
149
Current x 0.2 0.1 0 -0.1 -0.2 dx 4 2 0 -2 -4
0
0.50
1 Time (Day)
1.50
2
;* مOا
150
ا ا ا
ﻗﺴﻢ ﺍﻹﺣﺼﺎﺀ ﻭﺑﺤﻮﺙ ﺍﻟﻌﻤﻠﻴﺎﺕ ﻛﻠﻴﺔ ﺍﻟﻌﻠﻮﻡ ﺟﺎﻣﻌﺔ ﺍﻟﻤﻠﻚ ﺳﻌﻮﺩ
هـ1422/1421 3^ اN` 3@T *ر اAا ( ) ء ا ذجV 203 دة ت4& 3 6 _ا :9* ا9 0 اQ H " # RHأ :ول0ال اB ا :9* اLinear Differential Equations 9:) ا9 U`*ت اS ا د9#2 J xɺ = −0.5 x + ay ,
a < 0.4
yɺ = x − 0.5 y
ةG ور اGJ اH وأوxɺ = Ax State Space 9ء اi= N/! " # T*)أ( أآ .9 U`*ت اS *;ا د3:K ة: اa 9 L و أنA 9=2` Eigenvalues 9#2 J 9=/ اLinear Difference Equations 9:) ا9Lت ا`وS ا دR*)ب( أآ طP اN ه. x i +1 = Axi , i = 0,1, 2,… State Space 9ء اi= N/! " # T* وأآ9; ا .i8 *;ا ه ا3:8 a 9 L " # ; وx0 = 1, y0 = 1 9و0 ; ا9Lت ا`وS ا د9#2 J N Excel *)ام+ ()ج . a = 0.3 وi = 1, 2,… , 20 :3^ال اB ا Drilling رة2` ر اb )اN ب دا+ دادG8 *ولN; 3= x (Wells رbد ا#) d
(Drilling Fraction ` ا9 ) 3 # ر وbد ا# 3 # *K 3* واxɺ ( Wells
YfK) وn (Normal Drilling Fraction 98 ا` اد9 ) بU N% 2ي ه6وا ا6 وهe ( Effict of Reserves on Drilling Fraction ` ا9 " # 3D*Aا 3;* اr
(Petroleum Reserves ا*ول3D* )إ9H2 9L# 3:8 0ا 151
3D*A 3ى ا2* ا9 # رة# 2 ا*ول ه3D* إ3;* ء اGJ ا.` ا9 و ; واi
(Initial Petroleum Reserves ا*ول3D*A 9و0 ا9 ;ا*ول ا" )ا
/ ا*ول ا9 آX L ا*ول آ3D* إNL ;ار أ3;K ان آ38 9 ا56 ه3= .;K ` ا9 ن+= ا6/ وهTH إ ا3= 98*دLA ا9 ; اX L وآTHإ *)ا 3 ه3* وا− xɺ ( Closing Wells 9; Z ر اb )ا2 ب رج ه+ L*8 رbد ا# *)اجA اYfK) وc (Normal Closing Fraction 98\ق ادA ا9 ) بU N% *)اجA اYfK . f (Effect of Extraction on Closing Fraction \قA ا9 3 # (Extraction per Well N/ *)اجA )ا3 # Yf* اK 9 دا2\ق هA ا9 3 # m (Maximum Extraction per Well N/ O#0 *)اج اA )اQ 9 ;ر
p
3K N/ O#0 *)اج اA إ" اN/ *)اجA ا9 3= ;f= . \قA ا9 3 # دادGK ا6T و98*دLA ا5واH ا مN`L Ni=0 ا$ وا9 2% ^ أآg% *)اج اAأن ا YfK) وN/ O#0 *)اج اAب اU N% 2 هN/ *)اجA ا.\قA ا9 ( Effect of Reserves on Extraction per Well N/ *)اجA " ا# 3D*Aا 9 XLK /= .3D*A ا*ول ا *)ج وا9H2 9L# ي8 0ا ا6 وهh .L*8 N/ *)اجAن ا+= 3* و9 2% ^ اآg8 *)اجAن ا+= ا*ول3D*إ .N إ ب دا$ c ا*ول3D*ى إ2* ن+= دJ* \ ر2ن ا*ول ه0أ ا و g ( Extraction *)اجA )ا9 / ا9 / ا2 ا*ول ه3D* إrɺ ب ا)رجAا
. رbد ا# وN/ *)اجAب اU N% 3 ه3* وا982 ا 9Lت ا`وS ا دR*م وأآOا ا6T (9 تL# ) ى وإ ب2* d:) )أ( أر .$= /*K 3* اDifference Equations Vensim *)ام+ *)اج ا*ولA ذج2 ن2)ب( آ
:V^ال اB ا . Jاء اGH اQ H " إ9@ اJ آS 3Z ام اN;K 9)i ن آA اR L *ر# إ/ 8 *دد8 مO آR ; ا. Oscillator ا( آ *ددH 38;K N/P ) نA اR L 9H6 / 8 آ . ;K 9 أيSystole )ء و إ;ضK إر9 أيDiastole إ ط:م# N/P * Electro-Chemical
3@ وآT آ6` R ;ت اi# إ;ض وإ ط3= R *8
9 آv وR ; ا3= Muscle Fiber 9 i# 9` ل2D 2 هx أنU*=ذا إ+= . Stimulus
152
R ; ا9 i# 9` ل2D 3= ;ضA ط واA أن ل ا9 رب اJ* اH =; و،6` ا 9` اJ V Y ا`قQ ( µ > 0 R K X ^ ) * ;8 و6` ا9 آQ دادG8 .9` ل ا2D Q L*K 6` ا9 أن ل آH آ و.T2D( و98 د * و0* ا#)أ 9 i# 9` 9ذج آ2 Difference Equations 9 U`*ت اSن ا د2 آ:Sأو .9; ت ا: " ا# * اR ;ا :a@* اL و9* ; اVensim ذج2 ن2 آ:Y µ = 2 cm / sec, x ( 0 ) = 2 cm
v ( 0 ) = 1 microgrm, t = 0 ( 0.1) 100 sec
:Q ال ااB ا Logistic
9* H2 ا9 اT* آl%2K 3* ا9:اه ا2O^ ا/هك ا
3 ه ا2 g8 Y ( c f* ا9 ) : 2 أK هةlK 9 اا56 ه.Function ه2 lL2*8 Y (ai ا9 ) ه2 BD*8 Y (زدهرA ا9 ) `*ةExponential " إGK n0 ن+= n ( t ) G t G ا# هةO ا9 ; Gذا ر+= .( *;ارA ا9 ) 2 )وهM 2هة هO ا56T 2 اl; وإذا آنInitial Value 9@ او ا9و0 ا9 ;ا (98TS " إGول اB8 # $ K 9;; ا3= – هةO ا9 L $ اNK ان/ 8 "Lأ :3 ه9ز0 اN آ# هةO اlK 3* اDynamic Model 9 اآ9ن ا د+= n (t ) =
M , t≥0 1 + ( M − n0 ) n0 e − ct
.هةO ا56T Vensim d:) ن2)أ( آ :9*)ب( ات ا Year
Volume
Year
Volume
1984
0.0
1992
15.0
1985
1.5
1993
18.5
1986
2.0
1994
20.0
1987
2.5
1995
22.5
1988
4.0
1996
23.5
1989
5.0
1997
23.0
153
1990
8.5
1991
11.5
1998 1999
27.0 27.5
. c وn0 وM ر اL أ *)مy =
a N/P " ا# 9 ا د6 وExcel أوCurve Expert أ *)م:9O 1 + be − cx
.9 هJK ا ىL أو أيa0 = 30, b0 = 30, c0 = 0.5 9آ; او .a@* اLم وO ي آةH ا`;ة )أ( وأ3= d:) ا; ا ;رة آ6 ()ج . ب
dn ( t ) M − n (t ) = c× × n ( t ) 9L أ *)م ا:9O dt M
154
ا ا ا هـ1422/1421 3^ اN` 3@T *ر ا9 * تHإ V 203 ا دة :ول0ال اB 9 Hإ ( )أ a x xɺ −0.5 yɺ = 1 −0.5 y a −0.5 A= −0.5 1 −0.5 − λ det ( A − Iλ ) = det 1
( −0.5 − λ )
2
=0 −0.5 − λ a
−a =0
λ1 = − a − 0.5 λ2 = a − 0.5
. *;اNن ا2/8 !وط أن56 وهλ2 < 1 وλ1 < 1 أن آJ −0.4 < a < 0.4 ; ()ب xi +1 = −0.5 xi + ayi ,
a < 0.4, i = 0,1, 2,...
yi +1 = xi − 0.5 yi a xi xi +1 −0.5 , i = 0,1, 2,... y = 1 −0.5 yi i +1
.i8 ه ا:K 9; ا`;ة ا3= 9 HAن ا+= $ # وT ` 3 هState Matrix 9 ا9=2` ()ج x
y
1
1
=-0.5*A2+0.3*B2
=A2-0.5*B2
=-0.5*A3+0.3*B3
=A3-0.5*B3
=-0.5*A4+0.3*B4
=A4-0.5*B4
=-0.5*A5+0.3*B5
=A5-0.5*B5
=-0.5*A6+0.3*B6
=A6-0.5*B6
=-0.5*A7+0.3*B7
=A7-0.5*B7
=-0.5*A8+0.3*B8
=A8-0.5*B8
=-0.5*A9+0.3*B9
=A9-0.5*B9
155
=-0.5*A10+0.3*B10
=A10-0.5*B10
=-0.5*A11+0.3*B11
=A11-0.5*B11
=-0.5*A12+0.3*B12
=A12-0.5*B12
=-0.5*A13+0.3*B13
=A13-0.5*B13
=-0.5*A14+0.3*B14
=A14-0.5*B14
=-0.5*A15+0.3*B15
=A15-0.5*B15
=-0.5*A16+0.3*B16
=A16-0.5*B16
=-0.5*A17+0.3*B17
=A17-0.5*B17
=-0.5*A18+0.3*B18
=A18-0.5*B18
=-0.5*A19+0.3*B19
=A19-0.5*B19
x
y
1
1
-0.2
0.5
0.25
-0.45
-0.26
0.475
0.2725
-0.4975
-0.2855
0.52125
0.299125
-0.546125
-0.3134
0.5721875
0.32835625
-0.59949375
-0.34402625 0.628103125 0.360444063 -0.65807781 -0.377645375 0.689482969 0.395667578 -0.72238686 -0.414549847 0.756861008 0.434333226 -0.79298035 -0.455060718 0.830823401 0.476777379 -0.87047242 -0.499530415 0.912013589 0.523369284 -0.95553721
:3^ال اB 9 Hإ
156
x ( t ) = x ( t − dt ) + (u − v )dt , u = xɺ , v =< − xɺ > y ( t ) = y ( t − dt ) − wdt , w = yɺ
xɺ = bx,
yɺ = px, < − xɺ >= gx, g = cf
b = ne,
p = mh,
f = p m, e = y a, h = y a
()ب
c
n dx/dt
<-dx/dt> x
m
u
v f
g
p
b dy/dt y w
h
e a
b u (x) x g v (x)
157
x
w
y
u
(x)
v
(x)
p w
y
e
b
h
p
x
y
a h y
p m
c g m
f
158
:V^ال اB 9 Hإ ;8 و6` ا9 آQ دادG8 R ; ا9 i# 9` ل2D 3= ;ضA ط واA)أ( أن ل ا إذاT2D و9` اJ V Y ا`قQ ( µ > 0 R K X ^ ) * dx ( t ) = v ( t ) − µ x 3 ( t ) 3 − x ( t ) dt
إذا9` ل ا2D Q L*K 6` ا9 و أن ل آ dv ( t ) = − x (t ) dt
d N/P T*/و dx = v − µ ( x3 3 − x ) dt
dv = −x dt
Vensim a N^ ذج2 )ب( ا x dx
mu
v dv
(01) dv= -x Units: microgrm/sec
(02) dx= v-mu*(((x^3)/3)-x) Units: cm/sec
(03) FINAL TIME = 100
159
Units: Second The final time for the simulation.
(04) INITIAL TIME = 0 Units: Second The initial time for the simulation.
(05) mu= 2 Units: cm/sec
(06) SAVEPER = TIME STEP Units: Second The frequency with which output is stored.
(07) TIME STEP = 0.1 Units: Second The time step for the simulation.
(08) v= INTEG ( dv, 1) Units: microgrm
(09) x= INTEG ( dx, 2) Units: cm
160
Current 1 x 4 2 1 0 -2
1
1
1
1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
1
1
1
1
1
1 1
1
1
1
1
-4 dx 1
4 2 0 -2 -4
1
1 1
1
1
1
1
1
1
1
1
1
1
1 1
0
25
50 Time (Second)
75
100
(1) N/! 1
Current v 4 2 0 -2
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
-4 dv 4 2 0 -2 -4
1
1
0
1
1
1
25
1
1
1
1
50 Time (Second)
1
1
1
1
75
100
(2) N/!
161
Heart Rate v Fiber Length 4
2
0
-2
-4 -3
-2
dx : Current
-1
1
1
1
0 x 1
1
1
1 1
1
2
1
1
1
3 cm/sec
1
(3) N/! Heart Rate v Stimulus 4
2 1 1
1
1
1
0
-2
-4 -3 dx : Current
-2 1
-1 1
1
0 v 1
1
1
1 1
1
1
2 1
1
1
3 cm/sec
(4) N/!
162
Stimulus v Fiber Length 4
2
0
-2
-4 -3 v : Current
-2
-1
1
1
1
0 x 1
1
1
1 1
1
2
1
1
1
3 microgrm
1
(5) N/!
Stimulus v Heart Rate 4 1
1
1
1
1
1
1
2 1
0
-2
1
-4 -4 v : Current
-3
-2 1
1
-1 1
1
0 dx 1
1
1 1
1
2 1
1
3 1
1
4
microgrm
(6) N/! :a@* ا9PL Systole ;ضA( إ" ا9 82D )أفDiastole طA*;ل اA( أن ا1) N/! 8
ي6 وا3` إ ب3= R *8S 3* ) ول0 ا3= ءd ;ضA اT= ثK (ةL )أف 163
" ام إQ= Q8 N/P j;K ف0ن ا+= 6` ا9# 9 آ# /( وR ;ذي اB8 (3) ل/!0 ا.5ZK ول6` ي ا2* 3= Oscilation ا*دد8 (2) N/! .ا)رج :ت82* =" ا98 ا*ددات اور9D gU2K (6) "إ (3) N/! 9 iل ا2D U R ; ا9 i# تi ل-1 (4) N/! 6` ا9 آU R ; ا9 i# تi ل-2 (5) N/! 9 iل ا2D U 6` ا9 آ-3 (6) N/! R ; ا9 i# تi لU 6` ا9 آ-4 :Q ال ااB 9 Hإ ( )أ
M-n(t)
n(t) dn(t)/dt M
n(0) c
ا8;K ()ب Excel *)ام+ Sاو
A
B
C
D
c=
0.473488084598268
M=
27.218780487999
n0=
0.691567114138888
E
Year
Delta Year
Value
Logistic
Sq Error
1980
=A5-$A$5
0
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B5))
=(D5-C5)^2
1981
=A6-$A$5
1.5
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B6))
=(D6-C6)^2
1982
=A7-$A$5
2
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B7))
=(D7-C7)^2
1983
=A8-$A$5
2.5
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B8))
=(D8-C8)^2
1984
=A9-$A$5
4
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B9))
=(D9-C9)^2
164
1985
=A10-$A$5
5
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B10))
=(D10-C10)^2
1986
=A11-$A$5
8.5
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B11))
=(D11-C11)^2
1987
=A12-$A$5
11.5
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B12))
=(D12-C12)^2
1988
=A13-$A$5
15
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B13))
=(D13-C13)^2
1989
=A14-$A$5
18.5
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B14))
=(D14-C14)^2
1990
=A15-$A$5
20
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B15))
=(D15-C15)^2
1991
=A16-$A$5
22.5
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B16))
=(D16-C16)^2
1992
=A17-$A$5
23.5
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B17))
=(D17-C17)^2
1993
=A18-$A$5
23
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B18))
=(D18-C18)^2
1994
=A19-$A$5
27
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B19))
=(D19-C19)^2
1995
=A20-$A$5
27.5
=$D$2/(1+(($D$2-$D$3)/$D$3)*EXP(-$D$1*B20))
=(D20-C20)^2
Sum=
=SUM(E5:E20)
A
B
C
D
c=
0.473488085
M=
27.21878049
n0=
0.691567114
E
Year
Delta Year
Value
Logistic
Sq Error
1980
0
0
0.691567114
0.478265073
1981
1
1.5
1.093543839
0.165206611
1982
2
2
1.714073963
0.081753699
1983
3
2.5
2.650992755
0.022798812
1984
4
4
4.019328309
0.000373584
1985
5
5
5.923649251
0.85312794
1986
6
8.5
8.403403286
0.009330925
1987
7
11.5
11.36711781
0.017657676
1988
8
15
14.56684923
0.187619589
1989
9
18.5
17.66362481
0.699523461
1990
10
20
20.35934383
0.129127985
1991
11
22.5
22.49780667
4.81069E-06
1992
12
23.5
24.07261592
0.327888994
1993
13
23
25.16994472
4.708660072
1994
14
27
25.90542485
1.198094761
1995
15
27.5
26.38562558
1.241830348
Sum=
10.12126434
Curve Expert *)ام+ Y 165
166
167
3 ا; ا;رة هJو User-Defined Model: y=a/(1+((a-b)/b)*exp(-c*x)) 168
Coefficient Data: a=
27.218854
b=
0.69156932
c=
0.47348677 M = 27.218854, n0 = 0.69156932, c = 0.47348677
أي
()ج
M-n(t)
n(t) dn(t)/dt M
n(0) c
(01) c = 0.47
(02) dn(t)/dt = c*(M-n(t)/M)*n(t)
(03) FINAL TIME = 15
(04) INITIAL TIME = 0
(05) M = 27.22
(06) M-n(t) = INTEG ( -dn(t)/dt, 30)
169
(07) n(0) = 0.69
(08) n(t) = INTEG ( dn(t)/dt, 1)
(09) SAVEPER = TIME STEP
(10) TIME STEP = 1 Current "n(t)" 40 30 20 10 0 "dn(t)/dt" 6 4.5 3 1.5 0
0
3.8
7.5 11.3 Time (Month)
15
Current "M-n(t)" 40 30 20 10 0 "dn(t)/dt" 6 4.5 3 1.5 0
0
7.5 Time (Month)
15
170
M-n(t) (n(t)) dn(t)/dt
n(t)
c M
(M-n(t)) n(t) dn(t)/dt
M-n(t)
c M
M-n(t) n(t)
dn(t)/dt (n(t))
(M-n(t)) M-n(t)
dn(t)/dt n(t) :a@* ا9PL
أن ا *عV الB ا3= ف2%2 ا3* H2 ا*ف ا9JK ا9ت ا2 ا dn ( t ) dt Flow بA ا.lL2*8 Y 3 ا2 " إ52 ل2*8 Y ءd 2 8 n ( t ) Stock
L* اfD*8 وQ8 LKدة و8G ا3= lL2K Y 98 دة8 ا" ز9:دة ا8G اZ*8 .lL2*8 "* ا*طء3= * 8و
171
ا ا ا
ﻗﺴﻢ ﺍﻹﺣﺼﺎﺀ ﻭﺑﺤﻮﺙ ﺍﻟﻌﻤﻠﻴﺎﺕ ﺍﻟﻤﺎﺩﺓ :ﺑﻨﺎﺀ ﺍﻟﻨﻤﺎﺫﺝ 203ﺑﺤﺚ ا* Sر اSول #ل ا` 9 ا` Nا^ 1423/1422 3هـ ا*# : G أ Q H 3 # RHا 9 0ا*:9 ا rال اول: أ( #ف ا*:3 اOم ، Systemا ، Entity @/ا` ، Attribute 9اPط 9 ، ActivityاOم System State
ب( أذآ أ2اع ا.9 O0 ج( آ N^ 8 lاOم = 3ا*` /ا.3O ا rال ا: )* +ام N Excelا دSت ا`و 9Lا* 9وار ر ا ; Nا :ة: 1) xn = 0.3 xn −1 + 10, x0 = 0, n = 1,...,50
2) xn = xn −1 (1 + xn −1 ) , x0 = 10, n = 1,...,50
3) xn = xn2−1 + 0.7 xn −1 + 0.2, x0 = 1, n = 1,...,10
ا rال ا: Dا 2ذج " # y = a + bx cات ا*Excel Solver )* 9 9
8
7
6
5
4
3
2
1
0
x
690
509
312
275
190
132
92
65
47
32
y
ار ا; ا :ة وا; ا .9;:
172
ﺑﺴﻢ ﺍﷲ ﺍﻟﺮﺣﻤﻦ ﺍﻟﺮﺣﻴﻢ ﺟﺎﻣﻌﺔ ﺍﻟﻤﻠﻚ ﺳﻌﻮﺩ ﻗﺴﻢ ﺍﻹﺣﺼﺎﺀ ﻭﺑﺤﻮﺙ ﺍﻟﻌﻤﻠﻴﺎﺕ
( ذج#$ء ا$) 203 دة ﻫـ1423/1422 ﺍﻹﺨﺘﺒﺎﺭ ﺍﻝﻨﻬﺎﺌﻲ ﻝﻠﻔﺼل ﺍﻝﺜﺎﻨﻲ ﺴﺎﻋﺎﺕ3 ﺍﻝﺯﻤﻥ : % ا2[& ا%#! 324 k!أ :ال اولr ا gPة ا#L ( أ او2) Occam's Razor مTس اوآ2
(1) : *ر+ ف و# Parsimony Principle
(5) Endogenous Activities 9 اا9:P0( ا4) System State مO ا9 (3) Stochastic
9@ا2P ا9:P0( ا6)
Exogenous Activities 9H ا)ر9:P0ا
System مO ا9H6 (8) Discrete Systems 9 ` ا9 O0( ا7) Activities Flow ب0( ا10) Stock ( ا *ع9) Modeling
(12) Auxiliary Variables ة# ات اZ* أو اConverters تS2 ( ا11) Information Link ت2 اd أو رواConnectors ت%2 ا
:ال اr ا ا98*دLA وا9# *HA وا9H2 وا982اه ا2O اN^ 9:اه ا2O^ ا/هك ا # Z* ع آ ت ا2 J ويK t G ا# 3/8م دO 3= Z* 9 آT= ن2/K =*ةN^ K T V t G ا# Z* ا9 آN^ K xt , t ∈ T Xذا آ+= . t − 2 وt − 1 9ز0ا .9 9ز 9 ز9O أي# مO ا9/8 دlK 3* وا9^ ا9H ار9L ا`و9( إ!* ا د1 . t ∈T
173
9 ءi= N/! 3= 9; ا9L اQU (2State Space Form N/P ا3 # xt = Axt −1 , t = 0,1,..., T
V 1 1 A= 0 1
. A 9=2` Eigenvalues ةG ور ا6J اH( أو3 T # X 3* ا9L ا`و9 ا دN EXCEL *)ام+ وx1 = 1, x2 = 1 9 أوL 6 f (4 .( A1, A2,..., A30 8) ا3= N اQU) . t = 1, 2,...,30 ; (1) 3= ;رن اL C1 = A2 A1,..., C 29 = A30 A29 وB1 = A1 A2,..., B 29 = A29 A30 QU2 (5 . A 9=2` ةG ور ا6J اQ 9JKا :ال اr ا Markovian 9=2 ا آ9%) اT 9 98*دL*د أن هة إLAل أ اء ا2;8 ()أ
؟38 = ذا9^ ا9H ارProperty R 9^ ا9H ار9=2 ا آ9%) اQ*8 9 !آT 32`ل اLA)ب( ا 9L ا`و9ا د xt − axt −1 − bxt −2 = ε t , ∀t ∈ T , ε t ~ N ( 0, 4 )
(Second Order one State
وا9 Z* 9Y 9H در9 ا د56ل ه2
]( وذFirst Order two State Variables) 9 ىZ* " أو9H إ" درVariable) . yt −1 = xt −2 QU2 ; (Steady State) 5م وأ * إ *;ارOا ا6T ذج2 ن2 آVensim *)ام+ ()ج وy0 = 1 وx0 = 1 1) a = 1.2, b = −0.7 2) a = −1.2, b = −0.7
9* إ *)م ا; ا:9O RANDOM NORMAL(-3.99,3.99 , 0 ,2 ,seed=19 ) INITIAL TIME = 0
FINAL TIME = 2
SAVEPER = TIME STEP
TIME STEP = 0.01
174
ﺑﺴﻢ ﺍﷲ ﺍﻟﺮﺣﻤﻦ ﺍﻟﺮﺣﻴﻢ هـ1423/1422 ا2 R;$ر ا9\t 2# إ!ت ( ذج#$ء ا$ ) 203 دة :ال اولr 2 !إ !ء0 اd * :ل2;8 3 # 3` = أ2 وهOccam’s Razor مTس اوآ2 (1) N%`*ه اf أ8 إNL0 " ا# أو98ورi \ اN%`* اN( إ * آNآP )ا (9 /P )اPا ا6د هK 3*ا إذا: ل2;8 ي6 وا3 اV ا3= Parsimony Principle gPة ا#L ( أ او2) NLي ا28 ي6ذج ا2 = * مa@* اc` 3:8 ذج وا2 ^آن هك أآ .Ni=0ذج ا2 ا2ات وا هZ* د ا# TK`%@ت و/ اN آlK اتZ* 3 وهSystem State
مO ا9 (3)
3= اتZ* اQ** مOر ا2:K رس8 و.9 9O # مO ا3= 9:P0وا .$* .مO اN دا9:P0 اlK وEndogenous Activities 9 اا9:P0( ا4) مO ا9 3= 9:P0 اlK وExogenous Activities 9H ا)ر9:P0( ا5) Z مO 3 8 9H ر9:Pf Yf*8S ي6م اO ا.مOا ا6 ه3 # YBK 3*وا .ح2*` مO $f l%28 ي6 وا9H ا)ر9:P0 Yf*8 ي6م اO اc/ N/P هYfK Z*8 3* ا3 وهStochastic Activities 9@ا2P ا9:P0( ا6) ي6 اXL2 = ^ ا3 * إQ8ز2* l%2K ت *دة/ إTJ@* ن2/K و3@ا2P# Z*8 9& ل:# إG آ ان ا3 * إQ8ز2* l%28 Q J* 9& $LZ* K .3@ا2P# N/P Q:;* N/P مO ا9 T= Z*K 3* واDiscrete Systems 9 ` ا9 O0( ا7) 3 # R D ل2% و،Q:;* N/P ث8 Q ا3= 9#i = ^ إآ لG اQ . اQ:;* N/P ث8 Q@iا
175
) 9H6 (8اOم System Modelingرا O 9م RJ8ان 2/ن او 2 3ذج l%2 Modelه6ا اOم Zض إHاء ا*Jرب 3 # 9 Hأ 9وإ=*اUت / 8SإHا@ 3 # TاOم !ة وذ] *:i8S 3ب اOم ا 3 %0و8ث إرKك =B8 $ # 3دي ا ZK 3اOم و=;ا2) $ا $%ا 9 %0آ ان درا 9 ا 2ذج SاOم # R8JK / Kة 2ارات 2ل اOم وذ] #+دة ا 2ذج ا 3ا 9ا # 9 %0إHاء آ2 Nار c/اOم ا 3 %0ا6ي إذا / 8S ZKإ#د 9Kة ا ى * 9ا ^ = 9 %0را O 9م إ*Lدي *Z ت اض واB8 L R :دي ا .T /# / 8S a@* 3آ أن ا 2ذج / 8 ان 8رس = 3أز 9إ=*ا / 8 ^ = 9UإHاء آة Oم )* +ام ا 2ذج و==K j 9ت اOم `*ات #ة ا! Tاو = 3د .9 L @Lوآ]6 8D # / 8ا 2ذج درا 9اOم NLإ $@Pوو2Hد 5ا 8 ^ = %ء Qو# 8ة رات ء = * 8اي ر ا=2/ Niن 2ذج N/ر وآK 3ف ا XK Qه 56ا)رات. ) (9ا *ع Stockأي ! Q Z*8ا) Gد* ،3/8ك( G8داد و ;8و" 8 ا2* i8ى Levelاو * ، State Variable 9 Zا *ع R#2* 8ا!0ء `*ة ز 9وه 56ا!0ء ا * TOKS 9#2أو O 3`*)Kإذ اiK Tف أو (N;K) R Kل =*ة ز.9 ) (10ا Aب Flowه 2ل 9 Z8 Rateا *ع 8Gُ8 2T= ،أو ِ;ُ8ا *ع. ا Aب ا6ي 8G8ا *ع " 8أ Aب اا Nأو ا ر ،Sourceوا6ي ;8ا *ع " 8 أ Aب ا)رج أو ا2Zر . Sink ) (11ا S2ت Convertersأو ا *Zات ا #ة 2K Auxiliary Variablesى أرLم2;K ،م ت 9أو 98Hأو 9;:و *Kا * 3= Controllers 9 /ا 2ذج. ) (12ا %2ت Connectorsأو روا dا 2ت Information Linkوه 3ا*3 N Kأو d K ا 2ت GHء إ" & = 3ا 2ذج ،ا Tأن ان ا %2ت N;KS آ ت دN^) 98 176
ءGJ N;*K ذج2 ء اGH 9 ;= d;= ت2 N;K 3 هN (د2; او ا9L: او ا5 ا . & :ال اr 2 !إ xt −1 3 هt − 1 G ا# وxt 3 هt G ا# Z* ا9 ال آB ت ا: (1
إذاxt −2 3 هt − 2 G ا#و xt = xt −1 + xt − 2 , t ∈ T
:9L =و9 دN/! " # أو xt − xt −1 − xt − 2 = 0, t ∈ T
:N/P ا3 # 9L ا`و9 ا دQi (2 xt = xt −1 + xt − 2 , t ∈ T
J yt −1 = xt −2 QU2 و xt = xt −1 + yt −1 yt = xt −1
N/P " ا# 3وه xt 1 1 xt −1 y = t 1 0 yt −1
.ب2 : اN/P ا2وه 9L *)ج اK ةG ور ا6J( ا3 A − λI = 0 1 1 1 0 −λ =0 1 0 0 1 1− λ
1
1
−λ
=0
(1 − λ )( −λ ) − 1 = 0 λ 2 − λ −1 = 0 1+ 5 1− 5 , λ2 = 2 2 λ1 = 1.618034
∴ λ1 =
λ2 = 0.618034
177
(4 1
1
1
2
0.5
1
1.5
0.666667
2
1.666667
0.6
3
1.6
0.625
5
1.625
0.615385
8
1.615385
0.619048
13
1.619048
0.617647
21
1.617647
0.618182
34
1.618182
0.617978
55
1.617978
0.618056
89
1.618056
0.618026
144
1.618026
0.618037
233
1.618037
0.618033
377
1.618033
0.618034
610
1.618034
0.618034
987
1.618034
0.618034
1597
1.618034
0.618034
2584
1.618034
0.618034
4181
1.618034
0.618034
6765
1.618034
0.618034
10946
1.618034
0.618034
17711
1.618034
0.618034
28657
1.618034
0.618034
46368
1.618034
0.618034
75025
1.618034
0.618034
121393
1.618034
0.618034
196418
1.618034
0.618034
317811
1.618034
0.618034
514229 832040
أن ا; BK Bi 3= 9ول إ" λ1 = 1.618034و ا; BK Ci 3= 9ول إ" λ2 = 0.618034
إ! r 2ال ا:
178
" ان# ل2;K Markovian Property of order
k 9H ار9=2 ا رآ9%) ا: ()ا
.d;= t − 1, t − 2,..., t − k 9ز0 ا# T L " # *K t G ا# 9@ا2Pهة اO ا9 L ن+= { yt , −∞ < t < ∞} X= ^ إذا آ
(
)
P yt < s | yt −1 , yt −2 ,..., yt −k , yt −( k +1) ,... = P ( yt < s | yt −1 , yt −2 ,..., yt −k )
(Second Order one State وا9 Z* 9^ ا9H ار9L ا`و9)ب( ا د Variable) xt − axt −1 − bxt −2 = ε t , ∀t ∈ T , ε t ~ N ( 0, 4 )
(First Order two State Variables) 9 ىZ* " أو9H درN/! " # QU2K xt = axt −1 + byt − 2 + ε t , ε t ~ N ( 0, 4 ) yt = xt −1
Vensim *)ام+ مO ذج2 )ج( ا a
epsil on
x(t) dx(t) b y(t) dy(t) a = 1.2, b = −0.7 ; -1
(01)
a= 1.2
Units: **undefined**
(02)
b= -0.7
Units: **undefined**
179
(03)
"dx(t)"= a*"x(t)"+b*"y(t)"+epsilon
Units: **undefined**
(04)
"dy(t)"= "x(t)"
Units: **undefined**
(05)
epsilon= RANDOM NORMAL(-3.99, 3.99 , 0 , 2 , 19 )
Units: **undefined**
(06)
FINAL TIME = 2 Units: Day
The final time for the simulation.
(07)
INITIAL TIME = 0
Units: Day
The initial time for the simulation.
(08)
SAVEPER = TIME STEP
Units: Day The frequency with which output is stored.
(09)
TIME STEP = 0.01
Units: Day
The time step for the simulation.
(10)
"x(t)"= INTEG ("dx(t)", 1)
Units: **undefined**
(11)
"y(t)"= INTEG ("dy(t)", 1)
Units: **undefined**
180
Graph for epsilon 4
2
0
-2
-4 0
0.50
1 Time (Day)
1.50
epsilon : Current
Current "x(t)" 2 1.7 1.4 1.1 0.8 "dx(t)" 6 3 0 -3 -6
0
0.50
1 Time (Day)
181
1.50
2
2
Current "y(t)" 4 3 2 1 0 "dy(t)" 2 1.7 1.4 1.1 0.8
0
0.50
1 Time (Day)
1.50
2
6
3
0
-3
-6 0.80
0.95
1.10 "x(t)"
1.25
"dx(t)" : Current
a = 1.2, b = −0.7 ; ;* ذج2 ان ا0 اN/P اgUوا a = −1.2, b = −0.7 ; -2
(01)
a= -1.2
Units: **undefined**
(02)
b= -0.7
Units: **undefined** 182
1.40
(03)
"dx(t)"= a*"x(t)"+b*"y(t)"+epsilon
Units: **undefined**
(04)
"dy(t)"= "x(t)"
Units: **undefined**
(05)
epsilon= RANDOM NORMAL(-3.99, 3.99 , 0 , 2 , 19 )
Units: **undefined**
(06)
FINAL TIME = 2
Units: Day
The final time for the simulation.
(07)
INITIAL TIME = 0
Units: Day
The initial time for the simulation.
(08)
SAVEPER =
TIME STEP
Units: Day
The frequency with which output is stored.
(09)
TIME STEP = 0.01
Units: Day
The time step for the simulation.
(10)
"x(t)"= INTEG ("dx(t)", 1)
Units: **undefined**
(11)
"y(t)"= INTEG ("dy(t)", 1)
Units: **undefined**
183
Current "x(t)" 1 0.5 0 -0.5 -1 "dx(t)" 6 3 0 -3 -6
0
0.50
1 Time (Day)
0.50
1 Time (Day)
1.50
2
Current "y(t)" 2 1.7 1.4 1.1 0.8 "dy(t)" 1 0.5 0 -0.5 -1
0
184
1.50
2
2
1.7
1.4
1.1
0.8 -1
-0.50
0 "x(t)"
0.50
1
-0.50
0 "x(t)"
0.50
1
"y(t)" : Current
6
3
0
-3
-6 -1 "dx(t)" : Current
a = −1.2, b = −0.7 ; ;* \ ذج2 ان ا0 اN/P اgUوا
185
:QHا ا 1) A Course in Mathematical Modeling By: Douglas D. Mooney and Randall J. Swift Published and Distributed by: The Mathematical Association of America.
2) Mathematics for Dynamic Modeling, 2nd ed. By: Edwaed Beltrami Published by: Academic Press
3) An Introduction to Mathematical Modeling By: Edward A. Bender Published by: Dover Publications, inc.
4) Mathematical Modelling Techniques By: Rutherford Aris Published by: Dover Publications, inc.
5) Introduction to Difference Equations By: Samuel Goldberg Published by: Dover Publications, inc.
6) Matrix Computations By: Gene H. Golub & Charles F. Van Loan Published by: North Oxford Academic
7) The State Space Equations and Their Time Domain Solution Lecture Notes By: Dr. J. R. White, UMass-Lowell 186
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