- #1
Fgard
- 15
- 1
Homework Statement
Solve the Laplace equation inside a sphere, with the boundary condition:
\begin{equation}
u(3,\theta,\phi) = \sin(\theta) \cos(\theta)^2 \sin(\phi)
\end{equation}
Homework Equations
\begin{equation}
\sum^{\infty}_{l=0} \sum^{m}_{m=0} (A_lr^l + B_lr^{-l -1})P_l^m(\cos \theta) [S_m\sin(m\phi) + C_m\cos(m\phi)]
\end{equation}
The Attempt at a Solution
I have derived the general solution for the Laplace equation in spherical problems and that went okej. The problem is when I try to match it with the boundary conditions. If I use the method of identification it is imminently obvious that:
\begin{equation}
C_m=0, m=1 \text{ and that for} \quad l\neq1 \rightarrow A=0
\end{equation}
, m has to equal one so the arguments of sin is correct and A=0 so the solution does not blow up, when r approaches 0. Which means I have two different solutions: one where l=1 and another solution for all the rest of l´s. When l equals one the Legendre polynomials dose not match the boundary conditions so that solution can be discarded.
Which means that I have an equation that looks like:
\begin{equation}
\sum^{\infty}_{l=0} ( B_l3^{-l -1})P_l^1(\cos \theta) S_1\sin(\phi) = \sin(\theta) \cos(\theta)^2 \sin(\phi)
\end{equation}
When I look at table's for Legendre polynomials I do not find any that looks like it can solve my equation. I would really appreciate some guidance in the right direction.