13 Laplace PDE in Spherical coordinates

 13.1 Laplace in a sphere

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13.1 Laplace in a sphere

problem number 108

Taken from Maple pdsolve help pages

Solve for \(u\left ( r,\theta ,\phi \right ) \)

\begin{align*} \frac{\partial }{\partial r} \left (r^2 \frac{\partial u}{\partial r} \right ) + \frac{1}{\sin \theta } \frac{\partial }{\partial \theta } \left (\sin \theta \frac{\partial u}{\partial \theta } \right ) + \frac{1}{\sin ^2\theta } \frac{\partial ^2 u}{\partial \phi ^2}=0 \end{align*}

Mathematica

\[ \text{DSolve}\left [\frac{\frac{f^{(0,2,0)}(r,\theta ,\phi )}{r}+f^{(1,0,0)}(r,\theta ,\phi )}{r}+f^{(2,0,0)}(r,\theta ,\phi )+\frac{\csc (\theta ) \left (\sin (\theta ) f^{(1,0,0)}(r,\theta ,\phi )+\frac{\cos (\theta ) f^{(0,1,0)}(r,\theta ,\phi )}{r}+\frac{\csc (\theta ) f^{(0,0,2)}(r,\theta ,\phi )}{r}\right )}{r}=0,f(r,\theta ,\phi ),\{r,\theta ,\phi \},\text{Assumptions}\to 0\leq \theta \leq \pi \right ] \]

Maple

\[ F \left ( r,\theta ,\phi \right ) ={\frac{ \left ( -1 \right ) ^{1/2\,\sqrt{{\it \_c}_{{2}}}}\sqrt{2} \left ( \sin \left ( \theta \right ) \right ) ^{\sqrt{{\it \_c}_{{2}}}} \left ({\it \_C5}\,\sin \left ( \sqrt{{\it \_c}_{{2}}}\phi \right ) +{\it \_C6}\,\cos \left ( \sqrt{{\it \_c}_{{2}}}\phi \right ) \right ) \left ({\it \_C1}\,{r}^{1/2\,\sqrt{1+4\,{\it \_c}_{{1}}}}+{\it \_C2}\,{r}^{-1/2\,\sqrt{1+4\,{\it \_c}_{{1}}}} \right ) \left ( \cos \left ( \theta \right ){\mbox{$_2$F$_1$}(1/2\,\sqrt{{\it \_c}_{{2}}}+1/4\,\sqrt{1+4\,{\it \_c}_{{1}}}+3/4,1/2\,\sqrt{{\it \_c}_{{2}}}-1/4\,\sqrt{1+4\,{\it \_c}_{{1}}}+3/4;\,3/2;\,1/2\,\cos \left ( 2\,\theta \right ) +1/2)}{\it \_C3}+{\it \_C4}\,{\mbox{$_2$F$_1$}(1/2\,\sqrt{{\it \_c}_{{2}}}+1/4\,\sqrt{1+4\,{\it \_c}_{{1}}}+1/4,1/2\,\sqrt{{\it \_c}_{{2}}}-1/4\,\sqrt{1+4\,{\it \_c}_{{1}}}+1/4;\,1/2;\,1/2\,\cos \left ( 2\,\theta \right ) +1/2)} \right ) }{\sqrt{r}}} \]