6.3.1 Spherical coordinates

   6.3.1.1 [416] No I.C. no B.C.

6.3.1.1 [416] No I.C. no B.C.

problem number 416

Added Jan 10, 2019.

Solve for \(u(r,\theta ,\phi ,t)\) the wave PDE in 3D \[ u_{tt} = c^2 \nabla ^2 u \] Using the Physics convention for Spherical coordinates system.

pict
Figure 6.109:PDE specification

Mathematica


\[\left \{\left \{u(r,\theta ,\phi ,t)\to \begin {array}{cc} \{ & \begin {array}{cc} \frac {\sqrt {2} e^{-\frac {1}{2} \sqrt {c_{10}} (2 \phi +\pi )-t \sqrt {c_{11}}} \left (J_{\frac {1}{2} \sqrt {\frac {4 c_9}{c^2}+1}}\left (\frac {r \sqrt {-c_{11}}}{\sqrt {c^2}}\right ) c_1+Y_{\frac {1}{2} \sqrt {\frac {4 c_9}{c^2}+1}}\left (\frac {r \sqrt {-c_{11}}}{\sqrt {c^2}}\right ) c_2\right ) \left (e^{2 \phi \sqrt {c_{10}}} c_5+c_6\right ) \left (e^{2 t \sqrt {c_{11}}} c_7+c_8\right ) \left (c_4 \, _2F_1\left (\frac {1}{4} \left (-\frac {\sqrt {c^2+4 c_9}}{\sqrt {c^2}}+2 \sqrt {-c_{10}}+1\right ),\frac {1}{4} \left (\frac {\sqrt {c^2+4 c_9}}{\sqrt {c^2}}+2 \sqrt {-c_{10}}+1\right );\frac {1}{2};\cos ^2(\theta )\right )+c_3 \cos (\theta ) \, _2F_1\left (\frac {1}{4} \left (-\frac {\sqrt {c^2+4 c_9}}{\sqrt {c^2}}+2 \sqrt {-c_{10}}+3\right ),\frac {1}{4} \left (\frac {\sqrt {c^2+4 c_9}}{\sqrt {c^2}}+2 \sqrt {-c_{10}}+3\right );\frac {3}{2};\cos ^2(\theta )\right )\right ) \sin ^{i \sqrt {c_{10}}}(\theta )}{\sqrt {r}} & c\neq 0 \\ \text {Indeterminate} & \text {True} \\\end {array} \\\end {array}\right \}\right \}\]

Maple


\[u \left (r , \theta , \phi , t\right ) = \frac {\left (c_{7} {\mathrm e}^{2 t \sqrt {\textit {\_c}_{4}}}+c_{8}\right ) \left (c_{5} {\mathrm e}^{2 \phi \sqrt {\textit {\_c}_{3}}}+c_{6}\right ) \left (c_{1} \BesselJ \left (\frac {\sqrt {\frac {c^{2}-4 \textit {\_c}_{1}}{c^{2}}}}{2}, \frac {\sqrt {-\textit {\_c}_{4}}\, r}{c}\right )+c_{2} \BesselY \left (\frac {\sqrt {\frac {c^{2}-4 \textit {\_c}_{1}}{c^{2}}}}{2}, \frac {\sqrt {-\textit {\_c}_{4}}\, r}{c}\right )\right ) \left (c_{4} \sqrt {2}\, \hypergeom \left (\left [-\frac {-2 \sqrt {-\textit {\_c}_{3}}\, c -3 c +\sqrt {c^{2}-4 \textit {\_c}_{1}}}{4 c}, \frac {2 \sqrt {-\textit {\_c}_{3}}\, c +3 c +\sqrt {c^{2}-4 \textit {\_c}_{1}}}{4 c}\right ], \left [\frac {3}{2}\right ], \frac {\cos \left (2 \theta \right )}{2}+\frac {1}{2}\right ) {| \cos \left (\theta \right )|}+c_{3} \hypergeom \left (\left [\frac {2 \sqrt {-\textit {\_c}_{3}}\, c +c +\sqrt {c^{2}-4 \textit {\_c}_{1}}}{4 c}, -\frac {-2 \sqrt {-\textit {\_c}_{3}}\, c -c +\sqrt {c^{2}-4 \textit {\_c}_{1}}}{4 c}\right ], \left [\frac {1}{2}\right ], \frac {\cos \left (2 \theta \right )}{2}+\frac {1}{2}\right )\right ) \left (-1\right )^{\frac {i \sqrt {\textit {\_c}_{3}}}{2}} \left (\sin ^{i \sqrt {\textit {\_c}_{3}}}\left (\theta \right )\right ) {\mathrm e}^{-\phi \sqrt {\textit {\_c}_{3}}-t \sqrt {\textit {\_c}_{4}}}}{\sqrt {r}}\]

________________________________________________________________________________________