GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER-IV(NEW) — EXAMINATION - SUMMER 2019
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Subject Code:2140105 Date:09/05/2019Subject Name: Numerical Methods
Time:02:30 PM TO 05:00 PM Total Marks: 70
Instructions:
- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
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MARKS
Q.1 (a) Name two interpolation methods used for unequal intervals. Also state their formulas. 03
(b) Perform four iterations to find a root of the equation X —4x-9=0 04 using Bisection method.
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(c) Using fourth order Runge Kutta method, find »(0.1) for differential 07 equation % =2x+y,y(0) =1 by taking h= 0.1
Q.2 (a) Solve the following system by Gauss elimination method. 03
x+3y+2z=5,2x+4y—-6z=—4, x+5y+3z=10
(b) Find a real root of the equation 3x =cosx+1, correct up to four 04 decimal places using Newton Raphson method.
(c) Fit a second degree polynomial using least square method to the 07 following data:
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X 1 2 3 4 5Y 5 12 26 60 97
Also estimate y at X=06.
OR
(c) Fit a curve of the form y=ae™ to the following data: 07
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X 1 3 5 7 9Y 115 105 95 85 80
Q.3 (a) Using Newton’s forward interpolation formula, find the value of 03 £(1.6) .
X 1 1.4 1.8 2.2
f(x) 3.49 4.82 5.96 6.5
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(b) Use trapezoidal rule to evaluate ?dx, dividing the interval into four equal parts. 04
(c) Use Gauss-Siedel method to solve the following system: 07
6x+y+z=105, 4x+8y+3z=155, 5x+4y—-10z =65
Q.3 (a) Evaluate f(9) by using Lagrange’s interpolation method from the following data: 03
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x 5 7 11 13 17F(x) 150 392 1452 2366 5202
(b) Evaluate ?dx with n=6 by using Simpson’s 3/8 rule. 04
(c) Compute y(1.5) & y'(1) from the following data using Cubic 07 Spline.
X 1 2 3
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Y -8 -1 18Q.4 (a) Use Taylor’s series method to find y at x=0.03 given that 03 Yy 1, y0)=1.
(b) Find the root of xlog,x—1.9=0, correct up to three decimal 04 places with x, =3 and x, =4 using Secant method.
(c) Using Shooting method , Solve the boundary value problem: 07 V"=, ¥(0)=0and y(1)=1.17
OR
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Q.4 (a) Solve the following system by Gauss Jordan method: 03
x=2y=—4,-5y+z=-9,4x-3z=-10
(b) Solve the equation )"=x+y with the boundary conditions 04 1(0)=y(1)=0 by finite difference method.
(c) Using Picard’s method of successive approximation, obtain a solution 07 up to fifth approximation of the equation % =x+y,y(0)=1.
Q.5 (a) Explain Initial value problem'and boundary value problem with 03 example.
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(b) Solve = a—L; in 0<x<5,t=0 given that 04 u(x,0) =20, u(0,7) =0, u(5,) =100 . Compute u(x,t) with h=1 by Crank-Nicholson method.
(c) Solve the boundary value problem 07 V'=x=0,y(0)=0and y'(1)= —% by the Rayleigh-Ritz method.
OR
Q.5 (a) State the difference between finite difference method and finite 03 element method.
(b) Discuss the concept of Laplace equation 8_L2t + a—L; =0 04
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(c) Solve the boundary value problem " +y=-x, ¥(0)=0, y(1)=0 07 by the Galerkin method.
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