Phuong Trinh Vi Phan

GIẢI PHƯƠNG TRÌNH VI PHÂN PHƯƠNG PHÁP EULER

y' = 0.25y2 + x2

a= 0

h= 0.1 xi

Ki(h)

y(h) 0 0.1 0.2 0.3 0.4 0.5

ε= 0

-1 -0.98 -0.95 -0.92 -0.89 -0.85

φi(h) 0.25 0.25 0.27 0.3 0.36 0.43

b= 0.5 y(0)= -1

yi+1(h) 0.5 0.51 0.57 0.66 0.79 0.95

-0.98 -0.95 -0.92 -0.89 -0.85 -0.8

y'=y-x2+1 0≤x≤2

y(0)= 0.5 h= 0.2

xi

y(h/2) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

Ki(h/2)

y(h) 0.5 0.81 1.18 1.6 2.05 2.54 3.06 3.58 4.1 4.6 5.06

0.5 0.83 1.21 1.64 2.11 2.62 3.15 3.69 4.24 4.76 5.23

1.5 1.77 2.02 2.24 2.41 2.54 2.62 2.62 2.54 2.36 2.06

Ki(h)

φi(h/2) 1.5 1.79 2.05 2.28 2.47 2.62 2.71 2.73 2.68 2.52 2.23

φi(h) 3.14 3.68 4.16 4.57 4.9 5.13 5.25 5.22 5.01 4.59 3.92

yi+1(h/2) yi+1(h) 3.26 0.81 3.81 1.18 4.3 1.6 4.73 2.05 5.07 2.54 5.32 3.06 5.44 3.58 5.41 4.1 5.21 4.6 4.77 5.06 4.07 5.46

0.83 1.21 1.64 2.11 2.62 3.15 3.69 4.24 4.76 5.23 5.64

GIẢI PHƯƠNG TRÌNH VI PHÂN PHƯƠNG PHÁP RUNGE-KUTTA

y' = 0.25y2 + x2

a= 0

h= 0.1 xi

yi 0 0.1 0.2 0.3 0.4 0.5

ε= 0

K1 -1 -0.98 -0.95 -0.92 -0.89 -0.85

K2 0.25 0.25 0.27 0.3 0.36 0.43

y(0)= -1

K3 0.25 0.25 0.28 0.33 0.39 0.47

b= 0.5

K4 0.25 0.25 0.28 0.33 0.39 0.47

φ(x,y) 0.25 0.27 0.3 0.36 0.43 0.52

0.25 0.26 0.28 0.33 0.39 0.47

y(i+1) -0.98 -0.95 -0.92 -0.89 -0.85 -0.8

y'=y-x2+1 0≤x≤2

y(0)= 0.5 h= 0.2

xi

yi 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

K1 0.5 0.83 1.21 1.65 2.13 2.64 3.18 3.73 4.28 4.82 5.31

K2 1.5 1.79 2.05 2.29 2.49 2.64 2.74 2.77 2.72 2.58 2.31

K3 1.64 1.92 2.17 2.39 2.57 2.69 2.76 2.76 2.67 2.46 2.13

K4 1.65 1.93 2.18 2.4 2.57 2.7 2.77 2.76 2.66 2.45 2.11

φ(x,y) 1.79 2.06 2.29 2.49 2.64 2.74 2.77 2.72 2.58 2.31 1.89

y(i+1) 1.65 1.92 2.17 2.39 2.57 2.7 2.76 2.76 2.66 2.45 2.11

0.83 1.21 1.65 2.13 2.64 3.18 3.73 4.28 4.82 5.31 5.73

GIẢI PHƯƠNG TRÌNH VI PHÂN y' =

3

0≤x≤1

y-

pp Euler xi

x/y

y(0) = 0.4

pp Runge-Kutta Ki(h)

y(h) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

2

h= 0.05

0.4 0.49 0.58 0.69 0.83 0.99 1.19 1.44 1.76 2.16 2.66

φi(h) 1.2 1.26 1.41 1.65 2 2.46 3.07 3.84 4.82 6.05 7.59

yi+1(h) 1.76 1.92 2.21 2.64 3.24 4.03 5.04 6.33 7.96 10 12.54

xi

yi

0.49 0.58 0.69 0.83 0.99 1.19 1.44 1.76 2.16 2.66 3.28

K1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0.4 0.46 0.52 0.58 0.64 0.7 0.78 0.86 0.96 1.07 1.19

K2 1.2 1.16 1.16 1.21 1.29 1.41 1.57 1.78 2.04 2.36 2.75

1.17 1.16 1.18 1.24 1.34 1.48 1.66 1.89 2.18 2.53 2.96

(Thay he so vao cac o tuong ung se co dap an cac bai con lai) 2.75 2.5 2.25 2 1.75 1.5 1.25 1 0.75 0.5 0.25 0

0.05

0.1

0.15

0.2

Column B

0.25

0.3

Column H

0.35

0.4

0.45

0.5

K3

K4 1.17 1.16 1.18 1.24 1.34 1.48 1.67 1.9 2.2 2.55 2.98

cac bai con lai)

φ(x,y) 1.16 1.16 1.21 1.29 1.41 1.57 1.78 2.04 2.36 2.75 3.21

y(i+1) 1.17 1.16 1.18 1.24 1.34 1.48 1.67 1.9 2.19 2.55 2.97

0.46 0.52 0.58 0.64 0.7 0.78 0.86 0.96 1.07 1.19 1.34