Page 406 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 23: Wave and Diffusion Equations by Separation of Variable

Notes
1 2 2
we get V x ,t f e x /4c td
2 6 t
x
Putting w , so that dx 2c tdw , we have
2c t

1 w dw
2
V , x t f x 2cw t e ...(xxiii)
which is the required solution.
Case IV: Let there be a bar of length L which is perfectly insulated. Both ends i.e. x = 0 and x = L
are also perfectly insulated and the initial temperature of the bar is
V x ,0 f x

V
The flux of heat across the faces x 0 and x L is proportional t 0 at the end, since these ends
x
are insulated. In this case the boundary conditions are

V 0,t 0 ...(xxiv)
x

V , L t 0 ...(xxv)
x

and the initial condition is
V x ,0 f x 0 x L ...(xxvi)

Proceeding as in Case I, here also we get three solutions. Solution (ii) is inadmissible as in this
V = XT increases indefinitely with time. The solution (iii) by itself is inadequate since in this case
the temperature will tend to zero as t tends to infinity. Therefore general solution will consist of
the solution of (i) and (iii).
Using boundary condition (xxiv) in solution (i), i.e.
X Ax B and T C

or V ' A x B '
we get A ' 0.

Therefore V ' B is one of the solution of (i). Considering solution (iii) i.e.

2 2
c t
X A cos x B sin x , T Ce
2 2
or V , x t C 'cos x D 'sin x e c t
Using boundary condition (xxiv) and (xxv), we get

D ' 0
n
and n 1,2,3,.....
L

LOVELY PROFESSIONAL UNIVERSITY 399