Page 408 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 23: Wave and Diffusion Equations by Separation of Variable

show that the function Notes
1 x 2
U , x t exp
t 4xt

is also the solution of heat equation.

23.2.2 Heat Flow in Two Dimensional Rectangular System

To illustrate the solution of the two dimensional diffusion equation, let us consider the following
problem.
Figure 23.5

Y
t = 0

b
O X
a
A thin rectangular plate whose surface is impervious to heat flow has at t = 0 an arbitrary
distribution of temperature. Its four edges are kept at zero temperature. It is required to determine
the subsequent temperature of the plate as t increases.
Let the plate extend from x = 0 to x = a and from y = 0 to y = b. Expressing the problem
Mathematically, we must solve the equation

2 V 2 V V
c 2 . ...(i)
x 2 y 2 t
Subject to the boundary conditions

y
V 0, ,t 0
y
V a , ,t 0
for all t. ...(ii)
V x ,0,t 0
b
V x , ,t 0
The initial conditions are
V x , ,0 F , x y for 0 x a ,0 y b
y
V x , , 0 ...(iii)
y
To solve equation (i) assume a solution of the form

y
V x , ,t e t X x Y y e t XY say . ...(iv)
where X is a function of x only and Y is function of y only. Substituting (iv) in (i) we get

2
2
1 d X 1 d Y
X dx 2 Y dy 2 c 2
2
2
1 d X 1 1 d Y 2
or 2 2 2 . ...(v)
X dx c Y dy

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