R05012101-numerical-methods

Set No. 1

Code No: R05012101

I B.Tech Supplimentary Examinations, Aug/Sep 2008 NUMERICAL METHODS (Aeronautical Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Find a real root of the equation x3 − 6x − 4=0 by bisection method. (b) Find a real root of the equation x3 − 2x − 5=0 by Regula falsi method which lies between 2 and 3. [8+8] 2. (a) Find f(8) using Newton’s forward difference formula from the following table x 4 5 7 10 11 13 f (x) 48 100 294 900 1210 2028 (b) Find the unique polynomial P(x) of degree 2 or less such that P(1)=1, P(3)=27, P(4)=64 using Lagrange interpoaltion formula and Newton divided difference formula. [8+8] 3. Evaluate

R4

ex dx taking h = 1 using cubic spline method.

[16]

0

4. (a) Find the Fourier transform of f (x) =



eikx a < x < b 0 b < x, < a

(b) Find the finite Fourier sine and cosine transforms of f (x) = x in (0, l). [8+8] 5. (a) The following indicate the velocity ‘V’ of a body during the time t, specified. Find its acceleration when t=1.1 t V

1.0 1.1 1.2 1.3 1.4 43.1 4.77 52.1 56.4 60.8

(b) A rocket is launched from the ground. Its acceleration measured every 5 seconds is tabulated below. Find the velocity and the position of the rocket at t=40 seconds. Use trapezoidal rule. t 0 5 10 15 20 25 30 35 40 a(t) 40.0 45.25 48.50 51.25 54.35 59.48 61.5 64.3 68.7

[8+8]

6. (a) Find whether the following set of equations are consistent if so, solve them. x + y + z = 9, 2x + 5y + 7z = 52, 2x + y − z = 0. (b) Solve the system of equations by Gauss elimination method x1 + 2x2 + x3 =5, 2x1 -x2 + 2x3 = 6, x1 + 4x2 + 5x3 = 2

[8+8]

dy 7. If dx = x + y, y (0) = 1 by the Runga-kutta 4th order method evaluate y(.5) and y(1) taking h = .5 [16]

1 of 2

Set No. 1

Code No: R05012101 8. Solve the equation

∂2u ∂x2

2

+ ∂∂yu2 = 0 in the domain of the figure 8 by Jacobi’s method [16]

Figure 8 ⋆⋆⋆⋆⋆

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Set No. 2

Code No: R05012101

I B.Tech Supplimentary Examinations, Aug/Sep 2008 NUMERICAL METHODS (Aeronautical Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1 2

h

1. (a) Establish the formula xi+1 = xi + using Newton Raphson method.

N xi

i

and hence compute the value of

(b) Find a real root of x3 + x − 1=0 using bisection method.

√

10

[8+8]

2. (a) Find the missing term in the following table. x: y:

0 1 2 3 4 2 5 7 - 32

(b) Using Gauss Backward interpolation formula find y(8) from the following table. x y

0 5 10 15 20 25 7 11 14 18 24 32

[8+8]

3. (a) Fit a curve of the form y = ax + bx2 x y

1 2 3 4 5 1.8 5.1 8.9 14.1 19.8

(b) Fit a curve of the form y = aebx to the following data x 1 5 7 9 12 y 10 15 12 15 21 4. (a) Find the Fourier sine transform of

[8+8]

1 x(a2 +x2 )

(b) Find the finite cosine transform of f (x) =

π 3

− x+

x2 2π

[8+8]

5. (a) Find f ′ (1.25) and f ′′ (1.25) x 1.0 1.05 1.1 1.15 1.2 1.25 1.3 f(x) 1.0 1.0247 1.04381 1.07238 1.09544 1.11803 1.14017 (b) The velocity V of a particle at distances from a point in its path is given by t 0 10 20 30 40 50 60 v 47 58 64 65 61 52 38 Estimate the time taken to travel 60 ft using simpson’s 13 rd rule 6. Solve the following tridiagonal system by LU decompoisition 3x1 – x2 = 4, 2x1 – x2 + x3 = 6, 2x2 + 3x3 + 2x4 = 11, x3 – 2x4 = -1 1 of 2

[8+8] [16]

Set No. 2

Code No: R05012101

7. Find y(.5) and y(1), y′ = 4–2x, y(0) = 2, with h = 0.5 using Modified Euler method. [16] 8. Solve the equation

∂2u ∂x2

2

+ ∂∂yu2 = 0 in the domain of the figure 8 by Jacobi’s method [16]

Figure 8 ⋆⋆⋆⋆⋆

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Set No. 3

Code No: R05012101

I B.Tech Supplimentary Examinations, Aug/Sep 2008 NUMERICAL METHODS (Aeronautical Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Find a real root of x + logx − 2=0 using Newton Raphson method. (b) Find a real root of x3 − 4x − 9=0 using bisection method.

[8+8]

2. (a) Find f(2.5) using Newtons forward formula from the following table x y

0 1 2 3 4 5 6 0 1 16 81 256 625 1296

(b) Find y(0) given that f(5)=12, f(6)=13, f(9)=14, f(11)=16 using Lagrange’s interpolation formula. [8+8] 3. Fit a parabola of the form y = a + bx + cx2 from the following data. x y w

0 1 2 3 4 1 1.8 1.3 2.5 6.3 .5 1 2 2 3

[16]

4. (a) Find the Fourier transform of f (x) =



eikx a < x < b 0 b < x, < a

(b) Find the finite Fourier sine and cosine transforms of f (x) = x in (0, l). [8+8] 5. (a) Find f ′ (.3) x 0 .1 .2 .3 .4 .5 .6 f(x) 30.13 31.62 32.87 33.64 33.95 33.81 33.24 (b) Evaluate

R1

2

e−x taking h = .2 using

0

i. Simpson’s 31 rd ii. Trapenzoidal rule.

[8+8]

6. (a) Solve the triadiagonal system by LU decomposition 3x1 – x2 = 5, –x1 + 2x2 – 2x3 = 6, 4x2 + 3x3 = 1 (b) Find whether the following equations are consistent, if so solve them. 2x − y − z = 2; x + 2y + z = 2; 4x − 7y − 5z = 2.

[8+8]

7. Using Runga-kutta method with h=0.5 first compute y(0.5), y(1), y(1.5) given that dy = x+y ; y (0) = 2 then compute y(2) using Milne’s predictor corrector method. dx 2 [16] 1 of 2

Set No. 3

Code No: R05012101

8. Solve the equation uxx +uyy =0 in the domain of the figure 8 of Jacobi’s method.[16]

Figure 8 ⋆⋆⋆⋆⋆

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Set No. 4

Code No: R05012101

I B.Tech Supplimentary Examinations, Aug/Sep 2008 NUMERICAL METHODS (Aeronautical Engineering) Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ⋆⋆⋆⋆⋆ 1. (a) Find a real root of x3 − 4x − 9=0 using Regula falsi method. (b) Find a real root of the equation x3 − 9x + 1=0 by bisection method.

[8+8]

2. (a) Find f(2.5) using Newtons forward formula from the following table x y

0 1 2 3 4 5 6 0 1 16 81 256 625 1296

(b) Find y(0) given that f(5)=12, f(6)=13, f(9)=14, f(11)=16 using Lagrange’s interpolation formula. [8+8] 3. (a) Derive normal equations to fit the parabola y = a + bx + cx2 . (b) Fit a curve of the form y=a+bx+cx2 for the following data. x y

0 1 2 3 4 1 1.8 1.3 2.5 6.3

[6+10]

4. (a) Find the Fourier transform of f (x) = Hence evaluate

R∞  x cos x − sin x  0

x2



1 − x2 if |x| < 1 0 if |x| > 1

cos x2 dx.

(b) Find Fourier cosine transform of f (x) =



cos x 0 < x < a 0 x ≥ a

[8+8]

5. Find f ′ (x) at x=.04 x .01 .02 .03 .04 .05 .06 f(x) .1023 .1047 .1071 .1096 .1122 .1148

[16]

6. (a) Solve the system of equations by LU decomposition x1 + x2 – x3 = 5, 2x1 + x2 + 2x3 = 5, 3x + 2x2 + 4x3 = 7 (b) Determine whether the following equations will have a non-trivial solution if so, solve them. 4x + 2y + z + 3ω = 0, 6x + 3y + 4z + 7ω = 0, 2x + y + ω = 0. [8+8] 7. Apply Runge-Kutta 4th order method to find y(.2), y(.4) and y(.6), y ′ = −xy 2 y(0) = 2 using h=0.2. [16]

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Set No. 4

Code No: R05012101 8. Solve the equation

∂2u ∂x2

2

+ ∂∂yu2 = 0 in the domain of the figure 8 by Jacobi’s method [16]

Figure 8 ⋆⋆⋆⋆⋆

2 of 2