Hall Ticket No Question Paper Code: BCC002
INSTITUTE OF AERONAUTICAL ENGINEERING
(Autonomous)
M.Tech I Semester End Examinations (Supplementary) July, 2018
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. Explain Crank Nicholson explicit method for solving partial differential equations.
Solve by Crank-Nicolson method,
@2u
@t2 subject to the conditions sin
0 x taking h k 1/36.
2. Give examples of parabolic, elliptic, hyperbolic, semilinear, quasi linear partial differential equations.

Derive explicit scheme to solve parabolic equation and using it to solve the equation
@2u
@x2
0 x t 0 subject to the conditions sin 0.
UNIT II
3. Explain conditions for one dimensional heat equation in cylindrical and spherical coordinates.

Discuss about convergence, stability, consistency of implicit methods.
4. Explain five point formula for finite difference by alternative direction implicit formula.
Explain the concepts of local truncation error and global rounding error.
UNIT III
5. Explain the method of characteristics for the hyperbolic partial differential equation.
Solve @2u
@t2 @2u
@x2 0 x t using explicit method given that ut
0 and 100 Compute u for four time steps with h 0.25.
6. Summarize the stability of the finite difference procedure for solving a hyperbolic equation.
Explain an explicit method for solving hyperbolic differential equation.
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UNIT IV
7. Solve the elliptic equation uxx+uyy 0 for the following square mesh boundary values as shown
in the following figure
Figure 1
Solve uxx uyy over the square region bounded by lines x y x y 3
given that u 10 throughout the boundaries taking h 1.
8. Explain analysis of the discretization error of the five point approximation to Poissons equation.

Solve the Poisson equation r2u +y2 over the square mesh with sides x y
x y 3 with u 0 on the boundary and mesh length=1.
UNIT V
9. Explain about the different types of variational methods.
Solve the boundary value problem y00 y x2
0 0 x 1 by Galerkin
method.
10. Explain Stones implicit method with an example.
Solve the boundary value problem y00 1 0 x 1 by weighted residual
method.