Problems and Solutions for Partial Di erential Equations

Problems and Solutions for Partial Differential Equations

by Willi-Hans Steeb International School for Scientific Computing at University of Johannesburg, South Africa

Yorick Hardy Department of Mathematical Sciences at University of South Africa, South Africa

Contents

1 Linear Partial Differential Equations

1

2 Nonlinear Partial Differential Equations

26

3 Lie Symmetry Methods

109

Bibliography

133

Index

134

vi

Chapter 1

Linear Partial Differential Equations

Problem 1. equation

is given by

Show that the fundamental solution of the drift diffusion

u 2u u

t

=

x2

+2 x

u(x, t) = exp - 1 (x - x0 + 2t)2 .

4t

4t

Solution 1.

Problem 2. (i) Show that

Dxm(f

?

1)

=

mf xm .

(1)

(ii) Show that

Dxm(f ? g) = (-1)mDxm(g ? f ).

(2)

(iii) Show that

Dxm(f ? f ) = 0, for m odd

(3)

Solution 2. 1

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download