Page 415 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes The coefficients B are given by
n
2 L n x
x
B n f ( )sin dx
L 0 L
L /2
n x
cos
2A L /2 n x 2A L
sin dx
L 0 L L (n / )
L
0
2A n
cos
n 2
or

2A n
B n 1 cos
n 2L

2A 2 n
2sin
n 4L
4A 2 n
sin …(10)
n 4L
Thus the solution (9) becomes

2 n 2 2 2
sin n c t
4A 4L n x 2
u ( , ) sin e L ...(11)
t
x
n L
n 1
So the solution of equation (i) subject to the conditions (A ) and (A ) is given by equation (11).
1 2
Self Assessment
4. Find the solution of heat equation.

2 2
V V V
dx 2 dy 2 dt

Subject to the boundary conditions

V 0 when t , when x 0 or x  and when y 0 or .
Also initially
y
V ( , ,0) f ( , )
x
x
y
23.3 Summary
 Wave equation is written in Cartesian co-ordinates, cylindrical co-ordinates and spherical
polar co-ordinates.

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