Page 392 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 23: Wave and Diffusion Equations by Separation of Variable

L.H.S. is purely a function of t and R.H.S. is a function of x and y. Hence both sides will be equal Notes
1 2 Y
only when both reduce to some constant value. Again in R.H.S. the sum of two terms 2 and
X x
1 2 X
2 cannot be equal to a constant unless each of these is constant.
X y
Thus we have following three possibilities

1 2 T 1 2 X 1 2 Y
(a) 0, 0, 0,z
2
c T t 2 X x 2 Y y 2
1 2 T 1 2 X 1 2 Y
(b) 2 , 2 1 , 2 2 ,
2
c T t 2 X x 2 Y y 2
where 2 2 2 and
1 2
1 2 T 2 1 2 X 2 1 2 Y 2
(c) , 1 , 2 ,
2
c T t 2 X x 2 Y y 2
where again 2 2 1 2 2

The general solution in above three cases are
X A x B 1 , Y A y B 2 , T A t B 3 , ...(iv)
3
2
1
X A e 1x B e 1x , Y A 2 2e 2y B 2 2e 2y and T A e ct B e ct ...(v)
3
1
1
3
X A cos x B sin x
1 1 1 1
Y A cos x B sin x
2 2 2 2
T A cos C t B sin C t ...(vi)
3 3
From the boundary conditions (i) it is clear that (iv) and (v) are not the solution of the wave
equation. Therefore (vi) must be required solution which is periodic in time. Hence we have

u x , ,t A cos x B sin x A cos y B sin y A cosc t B sinc t ...(vii)
y
1 1 1 1 2 2 2 2 3 3
Using the boundary condition (i), we get
t
y
u (0, , ) A A 2 cos 2 y B 2 sin 2 y A 3 cosc t B 3 sinc t 0
1
A 1 0
u a , ,t B 1 sin 1 a A 2 cos 2 y B 2 sin 2 y A 3 cosc t B 3 sinc t 0;
y

Sin a 0
1
or 1 a m

m
1 m 1,2,3,...
a

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