Page 360 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 22: Solution of Laplace Differential Equation

Notes
2
2
2
r d R r dR 2 2 = 1 d µ 2 ...(12)
r
R dr 2 R dr d 2
which gives
2
2 d R dR 2 2 2
r 2 r ( r µ )R = 0 ...(13)
dr dr
and
d 2 2
µ = 0 ...(14)
d 2

x
In equation (13) if we use the substitution r , it reduces to

2
d R 1 dR 1 µ 2 R = 0 ...(15)
dx 2 x dx x 2

Equation (15) is Bessel s differential equation and so the solution is given by
R = A J µ ( ) B J –µ ( )
x
x
or R = A J µ ( r ) B J –µ ( r ) ...(16)

where µ is not an integer and

R = A J µ ( r ) B Yy ( r ) ...(17)
1
1
when µ is an integer. The solutions of equations (10), (14) are given by
Z = A e .z B e z ...(18)
2
2
and = A 3 cos(µ ) B 3 sin (µ ) ...(19)
Hence the total solution is

V = R Z A J µ ( r ) B J –µ ( r ) A e z B e z A 3 cos(µ ) B 3 sin(µ ) ...(20)
2
2
where µ is a fraction and  = 1, 2, 3... and

V = R Z A J µ ( r ) BY ( r ) A 3 cos(µ ) B 3 sin( µ ) A e z B e z ...(21)
1
2
2
When µ is an integer and = 1, 2,...
The solutions (20) and (21) depend upon the parameters µ, . If we see a solution that is finite at
r = 0 and also be single valued in then µ be a positive integer and taking all values from 0 to .
Thus for a fixed ,

)
V = A J ( r A 3 cos µ A 4 sin µ A e z A e z ...(22)
2
1 µ
2
µ 0
Thus the above solution is known as cylindrical Harmonics and will be useful for certain physical
problems.
The solution (22) V for a single value of µ is called general cylindrical Harmonics.

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