Page 416 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 23: Wave and Diffusion Equations by Separation of Variable

 It is shown that depending upon the nature of the process the suitable wave equation can Notes
be set up and solved.

 One dimensional wave equation suits in most problems. So the solution of wave equation
in one dimension is solved.
 Two dimensional wave equation depending upon the symmetry of the problem is solved
both in rectangular and circular cases. Also heat conduction is studied.

23.4 Keywords

Heat Conduction: It is an other process that occurs in so many processes. Diffusion process is
very very similar to conduction process.
Wave Motion: It can be obtained in mechanical vibrations, electrical vibrations and other
processes.
23.5 Review Questions

1. Show that the solution of the wave equation

1 r 2 u 1 2 v
r 2 r r a 2 t 2

can be of the form

1
r
t
u ( , ) ( t r at ) F (r at )
r
where f and F are arbitrary functions.
2. Solve the one dimensional wave equation
2 2
u 1 u
0
2
x 2 c dt 2
with the boundary conditions
u (0, ) 0
t
for all t
l
u ( , ) 0
t
and
u(x, 0) = A sin 2x

u
0
t t 0
3. Solve the heat equation in one dimension:

u 2 V
k 0
t x 2
subject to the conditions
u(0, t) = u(, t) =0
and V(x, 0) = sin 3x

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