# Question on Laplace eq. on a ball.

• November 15th 2013, 04:04 PM
Alan0354
Question on Laplace eq. on a ball.
For $\nabla^2 u(r,\theta, \phi)=0,\;u(r,\theta, \phi)=r^{n}Y_{nm}(\theta,\phi)$

But I have issue with this, for spherical coordinates:

$\nabla^2u=\frac{\partial^2{u}}{\partial {r}^{2}}+\frac{2}{r}\frac{\partial{u}}{\partial {r}}+\frac {1}{r^{2}}\left(\frac{\partial^2{u}}{\partial {\theta}^2}+\cot\theta\frac{\partial{u}}{\partial {\theta}}+\csc\theta\frac{\partial^2{u}}{\partial {\phi}^2}\right)$

Let $u=R(r)Y(\theta,\phi)$ where $Y(\theta,\phi)$ is the spherical harmonics.

$\Rightarrow\; r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+2r\frac{\partial{R}}{\partial {r}}-\mu R=0$

and

$\frac{\partial^2{Y}}{\partial {\theta}^2}+\cot\theta\frac{\partial{Y}}{\partial {\theta}}+\csc^2\theta\frac{\partial^2{Y}}{\partia l {\phi}^2}+\mu Y=0$

For Euler equation: $r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+r\frac{\partial{R}}{\partial {r}}-\mu R=0$ where $\mu=n^2$. and the solution is $R=r^n$.

Here, because of the condition, only ##\mu=n(n+1)## is used for bounded solution.

$r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+2r\frac{\partial{R}}{\partial {r}}-\mu R=r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+2r\frac{\partial{R}}{\partial {r}}-n(n+1) R=r^{2}\frac{\partial^2{R}}{\partial {r}^{2}}+\;r\frac{\partial{R}}{\partial {r}}-(n+1/2)^2R$

Which gives

$R=r^{(n+1/2)}$

What have I done wrong?
• December 2nd 2013, 08:32 AM
Rebesques
Re: Question on Laplace eq. on a ball.
Quote:

Originally Posted by Alan0354
Which gives

$R=r^{(n+1/2)}$

No, it doesn't :)