Page 378 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Unit 22: Solution of Laplace Differential Equation

22.3 Summary Notes

 Laplacian operator is expressed in Cartesian spherical polar co-ordinates and cylindrical
co-ordinates.

 The solution of Laplace equation in these co-ordinate systems is solved.
 Laplace differential equations finds its applications in potential problems, in wave
propagation and diffusion and heat conduction processes.

22.4 Keywords

Method of Separation of Variables helps in finding the solution of Laplace differential equation
in all the three co-ordinate systems.
Partial Differential Equation involve one dependent variable which is a function of more than
one independent variable.

22.5 Review Questions

1. Solve Laplace s equation in cylindrical co-ordinates and independent of Z.
2. Solve

2 u
r 0
r r

subject to the boundary conditions
u ( ) 0 at r a
r
u
and r ( ) u 0 at r 2a
y
x
3. Solve for U ( , ) distribution
2 2
U U 0
x 2 y 2
subject to the conditions

y
l
x
y
U (0, ) U ( , ) 0, U ( ,0) x 2
U
and 0
y
y b
4. Find the potential U(r, ) inside the spherical surface of radius R when its spherical surface
is kept at fixed distribution
U ( , ) U 0 cos
R
Answers: Self Assessment

2(3cos 2 1) r 2
r
1. U ( , ) 3
3r

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