53124-mt----numerical Methods For Partial Differential Equation

NR

Code No: 53124/MT M.Tech. – I Semester Supplementary Examinations, September, 2008

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATION (CAD/CAM) Time: 3hours

Max. Marks:60 Answer any FIVE questions All questions carry equal marks ---

1.

Deduce five point standard formula and five point diagonal formula to solve a two dimensional Laplace equation ∂ 2u ∂ 2u + = 0. ∂x 2 ∂y 2

2.

Explain Galerkin’s method of weighted residuals and using it obtain an approximate solution of the differential equation d2y − 10 x 2 = 5, 0 ≤ x ≤ 1 2 dx with boundary conditions y(0) = y(1) = 0.

3.

Using Crank Nichloson method solve the ODHE ft = t xx for 0 ≤ t ≤ 10τ subject to the conditions f (0, t ) = 10 , f (1, t ) = t in steps of 1 h = with the initial distribution f ( x,0) = 0 . 4

4.

Given that u ( x, y ) satisfies the equation ∆ 2u = 0 and the boundary 1 condition u (0, y ) = 0, u (4, y ) = 8 + 2 y, u ( x, 0) = x 2 and u ( x, 4) = x 2 . Find 2 the values of u (i, j ), i = 1, 2,3 j = 1, 2,3 by the method of relaxation.

5.

Apply Liebmann iteration process to solve ∆ 2u = − 10( x 2 + y 2 + 10) over the square mesh with sides x = 0, y = 0, x = 3, y = 3 with u = 0 on the boundary and mesh length 1 unit. 6. Derive the complete solution of the one dimensional heat equation in thin uniform rod of length L laterally insulated and the ends are kept at 00C while the initial temperature in the rod is f ( x) .

Contd…2.,

Code No: 53124/MT 7.

8.

::2::

∂ 2u ∂ 2u + = 0 subject to the ∂x 2 ∂y 2 boundary conditions u (0, y ) = u ( x,0) = u (a, y ) = 00 C and u ( x, b) = u0 .

Solve the initial boundary value problem

The temperature u ( x, y ) satisfies Laplace equation at all the points within and on the square as indicated in the following figure. Compute the solution correct up to two decimal places by using Gauss Seidal Method.

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