Page 413 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Let us put the solution as

y
t
x
V ( , , ) A mn e c 2 mnl sin m x sin n y
m 1 n 1 a b
where
2 2 m 2 n 2
mn 2 2
a b
a b
4 x y m x n y
and A mn A sin sin sin sin dx dy
ab a b a b
0 0
a b
4A x m x y n
sin sin sin sin dy dx
ab a a b b
x 0 y 0
1
for n = 2, 3, 4, ... the inner integral vanishes and for n = 1, the value of the integral is , a we have
2
A 11 A

1 1
2
and 11 2 2 .
a b
t
Therefore V ( , , ) Ae c 2 11 t sin x sin y .
x
y
a b
This give the temperature of the plate at any point and time.
Example 3: Find the temperature ( , )u x t of a slab whose ends x = 0 and x = L are kept at
temperature zero and whose initial temperature f(x) is given by

L
f (x) = A when 0 x
2
L
f (x) = 0 when x L
2
Solution: Let L be the length of the slab whose ends are kept at zero temperature and whose
initial temperature is f(x).
The boundary conditions are

u (0, ) 0
t
t
L
u ( , ) 0 for all . …(A )
t
1
The initial conditions are
L
x
x
u ( ,0) f ( ) A when 0 x
2
L
x
f ( ) 0 when x L …(A )
2 2
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