GUJARAT TECHNOLOGICAL UNIVERSITY
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BE - SEMESTER- III (New) EXAMINATION — WINTER 2019
Subject Code: 3130107 Date: 26/11/2019
Subject Name: Partial Differential Equations and Numerical Methods
Time: 02:30 PM TO 05:00 PM Total Marks: 70
Instructions:
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- Attempt all questions.
- Make suitable assumptions wherever necessary.
- Figures to the right indicate full marks.
| Marks | |
|---|---|
| Q.1 (a) Discuss in brief least square method for straight line | 03 |
| (b) Fit a straight line to | 04 |
| X | 1 | 2 | 3 | 4 | 5 |
| y | 14 | 27 | 40 | 55 | 68 |
| (c) State the method of false position and solve | 07 |
f(x) = 2x - 5 = 0
| Q.2 (a) Apply Newton Raphson to solve | 03 |
xsinx+cosx=0, x0=3.1416
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| (b) Solve using Gauss elimination method | 04 |
5x-2y+3z=18
x+7y-3z=-22
2x—y+6z=22
| (c) Find the value of y when x=390 using Newtons formula from the table | 07 |
| x | 100 | 150 | 200 | 250 | 300 | 350 | 400 |
| y | 10.63 | 13.03 | 15.04 | 16.81 | 18.42 | 19.9 | 21.47 |
OR
| (c) Use Lagranges formula to obtain y for x=2 using, | 07 |
| X | 0 | 1 | 3 | 4 |
| y | -12 | 0 | 6 | 12 |
| Q.3 (a) Apply Trapezoidal rule to evaluate | 03 |
?15 log10 x dx with h=0.5
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| (b) State the formula for Secant method | 04 |
| (c) Apply Runge kutta 4th order method to solve | 07 |
dy/dx = y/x , y(0)=1. Compute value for y(0.4)
OR
| Q.3 (a) Apply the Simpson 3/8 rule | 03 |
?01 1/(1+x) dx with h=0.25
| (b) State the formula for Cubic spline interpolation | 04 |
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| (c) State the Picard’s method to solve | 07 |
dy/dx = x + y2 subject to condition y=1,x=0.
| Q.4 (a) State the classification of partial differential equation. | 03 |
| (b) Solve pq=k | 04 |
| (c) Solve 1) 2r+5s+2t=0 and 2) y2p+x2q=x2y2z2 | 07 |
OR
| Q.4 (a) State the parabolic equation with initial conditions | 03 |
| (b) Solve p2+q2=m | 04 |
| (c) Solve one dimensional heat equation | 07 |
?u/?t = ?2u/?x2 with u(0,t)=0
u(L,t)=0 and u(x,0)=f(x)
| Q.5 (a) State the Taylors series formula to solve initial value problem. | 03 |
| (b) Solve ?2z/?x2 - 3 ?2z/?x?y + 2 ?2z/?y2 = 0 | 04 |
| (c) State the methods to solve linear partial differential equations and solve (mz —ny)p +(nx— lz)q =ly—mx | 07 |
OR
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| Q.5 (a) Give the formula for Newton’s backward interpolation formula | 03 |
| (b) Solve px(x+y)=gy(x+y)—(x—y)(2x+2y+z) | 04 |
| (c) Using Lagrange interpolation formula form a polynomial for, | 07 |
| X | 2 | 2.5 | 3 |
| y | 0.69315 | 0.91629 | 1.09861 |
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