# PDE Cylindrical and Spherical Symmetry

## Homework Statement

Show that the solution u(r,theta) of Laplace's equation (nabla^2)*u=0 in the semi-circular region r<a, 0<theta<pi, which vanishes on theta=0 and takes the constant value A on theta=pi and on the curved boundary r=a, is
u(r,theta)=(A/pi)[theta + 2*summation ((r/a)^n*((sin n*theta)/n)])

## The Attempt at a Solution

I didn't now where to even begin. I am struggling and am requesting help in explaining what I do not understand. Thanks!

gabbagabbahey
Homework Helper
Gold Member
Start with the general soltion to Laplace's equation for cylindrical symmetry...what's that?

Now, what are your boundary conditions for this problem?

This was all the data I was given. Our book is very brief and assumes you are very proficient with ODE's. I am struggling and need to understand these concepts. Thanks!

gabbagabbahey
Homework Helper
Gold Member
Well, I'm not here to teach an entire course on PDE's...you must at least know the method of Separation of variables right?

Yes I do understand and know the method of Separation of variables.

gabbagabbahey
Homework Helper
Gold Member
Well then, try applying it....show an attempt

okay I have been studying my book all weekend and this is what I had.
Laplace Equation (1/r) d/dr(rdu/dr) + (1/r^2)d^2u/ds = 0 (s represents theta)
u(r,s)=R(r)S(s)
(1/Rr)d/dr(rdR/dr) + (1/(r^2S))(d^2S/ds^2) = 0
(1/S)d^2S/ds^2 = -w^2 and (r/R)d/dr(rdR/dr)= w^2
Therefore S = A cos ws + B sin ws
r(d/dr(rdR/dr)) - w^2R = 0
w = n n=integer
r^2(d^2R/dr^2) + r(dR/dr) - n^2R = 0
R(r) = (C/r^n) + Dr^n
u(r,s) = ((C/r^n) + Dr^n)(A cos ns + B sin ns)

r=a therefore un(r,s) = r^n(An cos ns + Bn sin ns)

This is a interior Dirichlet problem therefore the solution is
u(r,s) = A0/2 + Summation (r/a)^n(An cos ns + Bn sin ns)

This is where I get lost, I think I need to do the following:
Bn = 1/(2*pi) integral 0 to pi m sin nm dm.

gabbagabbahey
Homework Helper
Gold Member
okay I have been studying my book all weekend and this is what I had.
Laplace Equation (1/r) d/dr(rdu/dr) + (1/r^2)d^2u/ds = 0 (s represents theta)
u(r,s)=R(r)S(s)
(1/Rr)d/dr(rdR/dr) + (1/(r^2S))(d^2S/ds^2) = 0
(1/S)d^2S/ds^2 = -w^2 and (r/R)d/dr(rdR/dr)= w^2

So far so good, but you will need to analyze 3 different cases:

(1)w^2>0
(2)w^2=0
(3)w^2<0

You may find non-trivial solutions in more than one of these cases for this problem ....

Therefore S = A cos ws + B sin ws
This is correct for the first case , but don't forget to analyze the other two cases as-well!

r(d/dr(rdR/dr)) - w^2R = 0
w = n n=integer

How are you deducing that w is an integer? This can only come from applying your boundary conditions....