Hall Ticket No Question Paper Code: BCC002
INSTITUTE OF AERONAUTICAL ENGINEERING
(Autonomous)
M.Tech I Semester End Examinations (Regular) February, 2017
Regulation: IARE-R16
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(CAD/CAM)
Time: 3 Hours Max Marks: 70
Answer ONE Question from each Unit
All Questions Carry Equal Marks
All parts of the question must be answered in one place only
UNIT I
1. Solve the heat conduction equation
@2u
@x2 subject to the initial and boundary conditions
u sinx; 0 x 1 and u u 0 using Crank-Nicolson method for h 1
3 and k 1
36 Integrate upto two time levels.
Explain Stability and Convergence analysis of difference schemes
2. Discuss the classification of second order partial differential equations .
Show that the Crank-Nicolson method is unconditionally stable.
UNIT II
3. Explain five point formula for finite difference
Discuss the stability of heat equation using Von Neumann method.
4. Explain conditions for one dimensional diffusion equation in cylindrical and spherical coordinates.

Explain Alternating Direction Implicit methods.
UNIT III
5. Find the solution of the initial-boundary value utt uxx 0 x 1 subject to the initial and
boundary conditions u sinx; 0 x 1 and u u t 0 using explicit
scheme for h 1
4 and r 3
4 . Integrate upto two time levels.
Prove that the Wenderoff's scheme is unconditionally stable.
6. Find the solution of the differential equation



u2
2

0 subject to conditions u
u 0 x 1 Using the Lax-Wenderoff's formula h=0.2 ,r=0.5 and integrate for one
time step.
Derive Wenderoff's formula.
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UNIT IV
7. Solve the mixed boundary value problem r2u 0 y u=2x
x ux u 2 u 2 x x 0 y using five point formula
with
Solve the boundary value problem (ux uy) 􀀀5e2xcosy (cosy siny),0 y 1
using second order method with h 1
2
8. Solve the boundary value problem
urr 1
rur uzz r z 1
on the boundary.
Using five point formula with
Explain Weighted Residual Methods
UNIT V
9. Discuss Variation methods, least square method and Galerkin method .
Obtain a one parameter approximate solution of the boundary value problem
r2u 0,jxj jyj 1
u=0 ,on the boundary using Galerkin method.
10. Discuss Finite element method.
Find a one parameter Galerkin solution of the boundary value problem
r2u 0,jxj jyj 1
2 u=0 On the boundary.
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