Hall Ticket No Question Paper Code: BCC002
INSTITUTE OF AERONAUTICAL ENGINEERING
(Autonomous)
M.Tech I Semester End Examinations (Regular) January/February, 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. Summarize the discrete grid point method for finite difference approximation.
Solve by Crank-Nicolson method, @2u
@x2
0 x t 100(x
0. Compute u for one time step with h
4
2. Define finite difference method. Summarize the advantages and disadvantages of finite difference
method.
Compute u for one step by Crank-Nicolson method if
@2u
@x2 0 x 5
and
UNIT II
3. Explain the meanings of the concepts of consistency, stability, and convergence of numerical
methods.
"Prove that sum of consistency analysis and stability analysis is consistency convergence analysis".
Justify your answer.

4. Explain five point formula for finite difference by alternate direction implicit method.
Explain the stability analysis of implicit methods and describe the types of errors.
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
0 and 100 Compute u for four time steps with h 0.25.
6. Prove that there is no explicit, unconditionally stable system for hyperbolic partial differential
equation
Explain the Lax-Wendroff for the partial differential equation with an example in multiple dimensional.

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UNIT IV
7. Given the values of on the boundary of the square in the figure evaluate the function
satisfying the Laplace equation uxx uyy 0 at the pivotal points.
Figure 1
Solve uxx uyy y)2 over the square region bounded by lines x y x
y 3 given that u 0 throughout the boundaries taking h 1.
8. Explain solution of Laplace's 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 the convergence of iteration methods to solve large linear systems.
Using Galerkin's method to solve the boundary value problem y00-y+x 0 x
0.
10. Explain weighted residual method with an example
Using finite element method to solve y"+1 0 x 0.