Page 410 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 23: Wave and Diffusion Equations by Separation of Variable

and integrating w.r.t. x and y from x = 0 and y = 0 to y = b, because of the orthogonality properties Notes
of the sin all the terms in the summation vanish except the term for which m = r and n = s and
we obtain the result.

a b
4 r x y
B r F , x y sin sin dxdy
ab a b ...(xi)
x 0 y 0
This determines the arbitrary constants of the general solution (ix)

Three Dimensional Heat Flow

The heat equation in three dimensions is given by

V 2 2 2 2 V 2 V 2 V
c V c 2 2 2 ...(i)
t x y z

k
2
where c
cp
Consider now a slab of dimensions a, b, c, the boundary conditions are

z
V 0, , ,t 0,
y
y
V a , , ,t 0,
z
z
V x ,0, ,t 0,
and ...(ii)
z
b
V x , , ,t 0,
y
V x , ,0,t 0,
y
c
V x , , ,t 0,
for all t.
z
y
V x , , ,0 F x , ,z for 0 x a , 0 y b and 0 z . c ...(iii)
y
To solve equation (i) we assume as usually a solution of the form
z
y
V x , , ,t e t X x Y y Z z ...(iv)
and then find the solutions similar to the case of two dimensions.
23.2.3 Temperature Inside a Circular Plate

Consider a thin circular plate whose faces are impervious to heat flow and whose circular edge
is kept at zero temperature. At t = 0 the initial temperature of the plate is a function f(r) of the
distance r from the center of the plate only. It is required to find the temperature u (r, t). Let the
radius of the plate be a.
The equation of heat conduction is

u 2 2
h u ...(i)
t

LOVELY PROFESSIONAL UNIVERSITY 403