This download link is referred from the post: GTU BE 2020 Summer Question Papers || Gujarat Technological University
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:
- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
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Q1
- 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]
- Use Bisection method to find the real root of the equation x —cos x = 0 correct upto four decimal places. [04]
- Explain the Newton-Raphson method briefly. Also find an iterative formula for √N and hence find √7 correct to three decimal places. [07]
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Q2
- Evaluate ∫01 e-x2 dx by Simpson’s-1/3 rule with n=10 and estimate the error. [03]
- Solve the following linear system of equations by Gauss elimination method. [04]
6x+8y+2z=-7--- Content provided by FirstRanker.com ---
3x+5y+2z=8
6x+2y+8z=26 - 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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- 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
- Use Trapezoidal rule to evaluate ∫01 x3 dx using five subintervals. [03]
- 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 - 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
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- Use Fourth order method to find the approximate value of y(0.2) given that dy/dx =x+y, y(0)=1. [07]
Q4
- Find the Lagrange Interpolating polynomial from the following data [03]
X 0 1 4 5 f(x) 1 3 24 39 - Derive Secant iterative method from the Newton-Raphson method and use it to find the root of the equation cosx —xex = 0 correct to four decimal places. [04]
- Solve (x —yz)p+(y —xz)q=z2 —xy. [07]
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Q.5
- Solve ∂2z / ∂x2 +z =0 given that when x=0, z=ey and ∂z/∂x =1. [03]
- 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
ut(x,0)=g(x) for 0 < x < 1
OR
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- Solve pr—qz=z2 +(x+y). [07]
Q.5
- Solve ∂2z / ∂x2∂y = x2 +y2 +1 [03]
- Obtain the solution of following one-dimensional heat equation with insulated sides by the method of separation of variables. [04]
∂u / ∂t = ∂2u / ∂x2--- Content provided by FirstRanker.com ---
u(0,t)=u(l,t)=0 ∀t>0
u(x,0)= f(x) for 0 < x < l
- Solve p2-q2 =x-y [07]
Q.5
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- Solve the given equation ∂u/∂t =2 ∂2u/∂x2 +u given that u(x,0) =6e-3x. [03]
- Obtain the solution of following one-dimensional heat equation with insulated ends by the method of separation of variables. [04]
∂u / ∂t = ∂2u / ∂x2
ux(0,t)=ux(l,t)=0 ∀t>0
u(x,0)= f(x) for 0 < x < l--- Content provided by FirstRanker.com ---
OR
- Solve (z2(p2+q2 + 1) = 1 [07]
Q.5
- 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 - Obtain the solution following two-dimensional Laplace equation. [04]
uxx+uyy =0
u(0,y) = u(a,y) = u(x,0)= 0
u(x,0)= f(x)
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This download link is referred from the post: GTU BE 2020 Summer Question Papers || Gujarat Technological University
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