PDE Cylindrical and Spherical Symmetry

[walter9459]

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)])

Homework Equations

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]

Start with the general soltion to Laplace's equation for cylindrical symmetry...what's that?

Now, what are your boundary conditions for this problem?

 

[walter9459]

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]

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?

 

[walter9459]

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

 

[gabbagabbahey]

Well then, try applying it...show an attempt

 

[walter9459]

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]

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...