Find soln of PDE for given boundary conditions: (1/r)*∂/∂r(r*∂Φ/∂r) + ∂^2(Φ)/∂z^2...

[Amit Hasan]

Find soln of PDE for given boundary conditions: (1/r)*∂/∂r(r*∂Φ/∂r) + ∂^2(Φ)/∂z^2...

I need to solve the following equation

(1/r)*∂/∂r(r * ∂Φ/∂r) + ∂^2(Φ)/∂z^2 + B^2*Φ = 0

for the situation where these boundary values are active

Ф(r>=R,z)=0, Ф(r<=a,z<=H/2)=0, Ф(r<=a, z>=-H/2)=0

My question is that which method can I use to find a solution?

I tried separation of variables but turns out that it only works when the boundary values can be reduced so that one BV only depends on one variable.

If I write Ф=X(r)Z(z), the BV Ф(r>R,z)=0 can be reduced to X(R)=0 because it does not depend on z.

But in this case the BVs Ф(r<a,z<H/2)=0 and Ф(r<a, z>-H/2)=0 cannot be reduced in such a way. These BVs depend on both r and z.

How can I solve this problem? Which method to use? I don't want a solution, just a point in the right direction. Suggest what I can study to gain knowledge for this please.

Thanks in Advance

 

[HallsofIvy]

Wait, you have a homogeneous equation and \(\displaystyle \phi\) is 0 on each boundary? Then \(\displaystyle \phi\) equal to 0 for all r and z is an obvious solution!