Page 404 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 23: Wave and Diffusion Equations by Separation of Variable

In this result by making use of (xiii), we get Notes
2 2
V 0,t ' A e c t 0
t
A ' 0
Then making use of (xiv) in (xvi), we get

V t , L t B 'sin L 0

sin L 0
or L n

n
or (n = 1, 2, 3, ...)
L
Therefore a solution for V x ,t is
t
n x 2 2 2
c t L
B sin . x e / 2 ( 1,2,3,...)
n
n
L
Now adding the solutions for different n the general solution may be written as
n x 2 2 t L
V t , x t B n sin . x e / 2 ...(xvii)
n 1 L
In this result if we use (xv), we get

t n
V x ,0 t i 0 x B n sin x
t
L L
n 1
L
2 t n
which gives B n t i 0 x sin xdx
L L L
0
Integrating by parts, we get

2
B t 1 n t t
n i i 0
n
Therefore

t 2 n x 2 2 2 / t L n x
c
V t , x t 0 x t i 1 t i t e sin ...(xviii)
0
L L
n 1
Here if the initial temperature of the wire is zero then, we get
t 2 n 2 2 2 n x
V t , x t 0 x 1 e x c / t L sin . ...(xix)
L L
n 1
Case III: Let there is a bar of infinite length (i.e. extending up to infinity on both sides) which is
insulated laterally. Then we have to find out the solution of heat equation (1) if the initial
temperature of the bar is f(x).

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