Download GTU B.Tech 2020 Summer 3rd Sem 3130107 Partial Differential Equations And Numerical Methods Question Paper

Envlmnt No.

Subject Code: 3130107

GUJARAT TECHNOLOGICAL UNIVERSITY

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BE - SEMESTER- IIl EXAMINATION - SUMMER 2020

Subject Name: Partial Differential Equations and Numerical Methods

Time: 02:30 PM TO 05:00 PM Date:27/10/2020 Total Marks: 70

Instructions:

1. Attempt all questions.

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2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.

Q1

1. Use Iteration method to find the real root of the equation x⁴ + x —1 = 0 correct to six decimal places starting with x₀ =1. [03]
2. Use Bisection method to find the real root of the equation x —cos x = 0 correct upto four decimal places. [04]

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3. Explain the Newton-Raphson method briefly. Also find an iterative formula for √N and hence find √7 correct to three decimal places. [07]

Q2

1. Evaluate ∫₀¹ e^-x2 dx by Simpson’s-1/3 rule with n=10 and estimate the error. [03]
2. Solve the following linear system of equations by Gauss elimination method. [04]
   6x+8y+2z=-7
   

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   3x+5y+2z=8
   6x+2y+8z=26
3. Compute cosh (0.56) using Newton’s forward-difference formula and also estimate the error for the following table. [07]
   

   x	0.5	0.6	0.7	0.8
   f(x)	1.127626	1.185465	1.255169	1.337435

   
   OR

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4. The speed, v meters per second, of a car, t seconds after it starts, is show in the following table. [07]
   

   t	0	12	24	36	48	60	72	84	96	108	120
   v	0	3.60	10.08	18.90	21.60	18.54	10.26	4.50	4.5	5.4	9.0

   
   Using Simpson’s ⅓ rule, find the distance travelled by the car in 2 minutes.

Q3

1. Use Trapezoidal rule to evaluate ∫₀¹ x³ dx using five subintervals. [03]

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2. Check whether the following system is diagonally dominant or not. If not, rearrange the system and solve it using Gauss-Seidel method. [04]
   8x-3y+2z=20
   4x-11y—-z=33
   6x—-3y+12z=35
3. Explain Euler’s method briefly and apply it to the following initial value problem by choosing h =0.2 and hence obtain y(1.0). dy/dx =x+y, y(0)=0. Also determine the error by deriving it analytical solution. [07]
   

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   OR

4. Use Fourth order method to find the approximate value of y(0.2) given that dy/dx =x+y, y(0)=1. [07]

Q4

1. Find the Lagrange Interpolating polynomial from the following data [03]
   

   X	0	1	4	5
   f(x)	1	3	24	39

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2. Derive Secant iterative method from the Newton-Raphson method and use it to find the root of the equation cosx —xe^x = 0 correct to four decimal places. [04]
3. Solve (x —yz)p+(y —xz)q=z² —xy. [07]

Q.5

1. Solve ∂²z / ∂x² +z =0 given that when x=0, z=e^y and ∂z/∂x =1. [03]
2. Obtain the solution of following one-dimensional Wave equation together with following initial and boundary conditions by the method of separation of variables. [04]
   

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   ∂²u / ∂t² = ∂²u / ∂x²
   u(0,t)=u(l,t)=0 ∀t>0
   u(x,0)= f(x) for 0 < x < l
   u_t(x,0)=g(x) for 0 < x < 1
   OR

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3. Solve pr—qz=z² +(x+y). [07]

Q.5

1. Solve ∂²z / ∂x²∂y = x² +y² +1 [03]
2. Obtain the solution of following one-dimensional heat equation with insulated sides by the method of separation of variables. [04]
   ∂u / ∂t = ∂²u / ∂x²
   

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   u(0,t)=u(l,t)=0 ∀t>0
   u(x,0)= f(x) for 0 < x < l
   
3. Solve p²-q² =x-y [07]

Q.5

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1. Solve the given equation ∂u/∂t =2 ∂²u/∂x² +u given that u(x,0) =6e^-3x. [03]
2. Obtain the solution of following one-dimensional heat equation with insulated ends by the method of separation of variables. [04]
   ∂u / ∂t = ∂²u / ∂x²
   u_x(0,t)=u_x(l,t)=0 ∀t>0
   u(x,0)= f(x) for 0 < x < l
   

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   OR

3. Solve (z²(p²+q² + 1) = 1 [07]

Q.5

1. Using method least squares, find the best fit straight line for the following data. [03]
   

   X	1	2	3	4	5
   y	1	3	5	6	5

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2. Obtain the solution following two-dimensional Laplace equation. [04]
   u_xx+u_yy =0
   u(0,y) = u(a,y) = u(x,0)= 0
   u(x,0)= f(x)

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