Page 414 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 23: Wave and Diffusion Equations by Separation of Variable

Let the solution of the heat equation Notes
u 2 2 u
c , …(1)
t dx 2
is of the form

t
x
u ( , ) X ( ) ( ) …(2)
T
t
x
where X is a function of x only and T is that of t only.
Substituting in (1), we get
2
2
1 d X 1 d T
…(3)
2
X dx 2 c T dt 2
Since L.H.S. is a function of x only and R.H.S. is a function of t only, both sides will be equal if
they are constant i.e. equal to 2
2
1 d X 1 dT 2
2
X dx 2 c T dt
Thus

2
d X 2 X 0
dx 2
and
dT 2 2
c T 0 ...(4)
dt
The solutions of equations (4) are

2 2
;
X A cos x B sin x T Ce e t …(5)
using boundary conditions (A ), the solution (5) gives
1
X (0) 0 A and ( ) 0 B sin L 0 …(6)
X
L
Now B 0 hence
sin L 0
or L n , for n = 1, 2, 3, … …(7)

e
i . . n /L
Hence for each value of n

n c n t /L 2
2 2 2
t
x
u n ( , ) B n sin x e …(8)
L
are solution of equation (i) satisfying the given boundary conditions (A ). So the general solution
1
is
2 2 2
c n t
n x 2
t
t
x
u ( , ) u n ( , ) B n sin e L …(9)
x
n 1 n 1 L
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