Page 399 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes Boundary Conditions
y
u 0, ,t 0
y
u 1, ,t 0
u x ,0,t 0
u x ,1,t 0
Initial Conditions

u x , ,0 f , x y A sin x sin 2 y
y
u
and 0
t t 0

y
Now u x , ,t A mn cos mn t sinm x sinn y
m 1 n 1
Since C = 1, a = 1, b = 1
and 2 2 m 2 n 2
mn
1 1
where A mn 4 f , x y sinmn .sinn ydxdy
0 0
1 1
4A sin x sinmn x .sin 2 y sinn ydxdy .
0 0
clearly A m 1 A m 3 A m 4 A m 5 ...0

1 1
2
and A m 2 4A sin x sinm x .sin 2 ydxdy .
0 0
1
= 2A sin x sinm xdx .
0
Now A 22 A 32 A 42 ... 0
1
and A 12 2A sin 2 xdx A
0
Hence we have

u x , ,t A 12 cos 12 t sin x 2 y
y
A cos 5 t sin x sin 2 y as all coefficients
,
2 2 2 2
Vanish except 12 1 2 .
or 5
12
Self Assessment

2. Solve one dimensional wave equation

2 2
u 1 u 0
x 2 c 2 t 2

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