Equations with Three Space Variables in Spherical Coordinates

Abstract

Consider the time-fractional diffusion-wave equation with a source term in spherical coordinates r,θand φ:

$$\frac{\partial^\alpha T}{\partial t^\alpha}\;=\;a\Bigg[\frac{\partial^2 T}{\partial r^2}\;+\frac{2}{r}\;\frac{\partial T}{\partial r}\;+\frac{1}{r^2 \mathrm{sin}\;\theta}\;\frac{\partial}{\partial\theta}\Bigg(\mathrm{sin}\;\theta\frac{\partial T}{\partial\theta}\Bigg)\;+\;\frac{1}{r^2\mathrm{sin}^2\theta}\frac{\partial^2 T}{\partial \varphi^2}\Bigg]+\;\Phi(r,\theta,\varphi,t),\qquad \qquad 0\leq r\leq\infty, 0\leq \theta\leq \pi, 0\leq\varphi\leq 2\pi.$$

I have answered three questions, and that is enough.

Lewis Carroll

“Alice’s Adventures in Wonderland”