GUJARAT TECHNOLOGICAL UNIVERSITY
BE - SEMESTER-IV (NEW) EXAMINATION - WINTER 2018
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Subject Code:2140105 Date:22/11/2018Subject 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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Q.1
(a) Name five iterative methods which evaluate the root of equations. 03
(b) Perform five iterations of Bisection method to obtain real root of X’ —x-1=0. 04
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(c) By the method of least squares, find the straight line ¥ =ax+b that best fits the following data: 07
| X | 0 | 5 | 10 | 15 | 20 | 25 |
|---|---|---|---|---|---|---|
| y | 12 | 15 | 17 | 22 | 24 | 30 |
Q.2
(a) Mention atleast two difference between Newton’s forward Interpolation and Newton’s divided difference interpolation. 03
(b) Find second degree polynomial passing through the points (-1, 8), (0, 3), (2, 1) and (3, 12) using Lagrange interpolation. 04
(c) Obtain cubic splines for every subintervals from the following data: 07
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| X | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y | 1 | 2 | 33 | 244 |
OR
(c) Using Newton’s Divided Difference Interpolation find f(x) from the following table: 07
| X | 1 | 2 | 7 | 8 |
|---|---|---|---|---|
| y | 1 | 5 | 5 | 4 |
Q.3
(a) Use Gauss Elimination to solve: 03
x+3y+2z=5
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2x + 4y - 62=-4
x+5y+3z=10
(b) Consider following tabular values: 04
| X | 25 | 25.1 | 25.2 | 25.3 | 25.4 | 25.5 | 25.6 |
|---|---|---|---|---|---|---|---|
| F(x) | 3.205 | 3.217 | 3.232 | 3.245 | 3.256 | 3.268 | 3.280 |
Determine the area bounded by given curve and x-axis between x = 25 to x = 25.6 by the Trapezoidal rule.
(c) Describe Newton-Raphson method and find root of equation xsinx+cosx =0 which is near m correct upto 5 decimal places. 07
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OR
Q.3
(a) Find approximate root of x’ —2x—1=0 starting fromxo=1.5 to X1 = 2 by Secant method correct upto 3 decimal places. 03
b) Evaluate by Simpson’s ? Rule, ? ! dx taking 10 equal parts, hence obtain approximate value of loge5. 04
(c) Solve the following equations using Gauss-Siedel method: 07
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5x+y-z=10
2x +4y+z=14
x+y+8z=20
Q.4
(a) State finite difference quotients for first and second order derivatives. 03
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(b) Solve heat equation ?u/?t = ?2u/?x2 with u(x, 0) =0, u(0, t) =0 and u(l, t)=t with k= l, h= l 04
(c) Solve by Runge-Kutta fourth order dy/dx =2x+y, y(0) = 1, h=0.1 find y (0.1) and y(0.2). 07
OR
Q.4
(a) State Gauss-Seidel method for Laplace equation. 03
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(b) Discuss shooting approach for Boundary Value Problem in brief. 04
(c) Write two differences between finite difference method and finite element method. 07
Q.5
(a) Evaluate (1) (1+?)(1-?)=1 (2) ?=E? 03
(b) Solve by Runge-Kutta second order dy/dx =3x+y, y(1)=1.3, h=0.1 find y (1.2). 04
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(c) Evaluate dy/dx = xvy , y(1)=1 and hence find y(1.5) taking h=0.1 by Euler’s method. 07
OR
Q.5
(a) Describe Galerkin approach in brief. 03
(b) Solve y”- x + y with boundary conditions y(0) =y(1) =0 by finite difference method 04
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(c) Describe Rayleigh Ritz method in brief. 07
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