# How to solve the given Partial Differential Equation?

I need help on solving the following PDE by Maple or Mathematica:

$$$$D \Big( D - \frac{1}{\alpha} \frac{\partial}{\partial t} \Big) f + \frac{\sin\theta}{\alpha} \bigg( \frac{\partial f}{\partial r} \frac{\partial}{\partial \theta} \Big(\frac{Df}{r^2 \sin\theta} \Big) - \frac{\partial f}{\partial \theta} \frac{\partial}{\partial r} \Big(\frac{Df}{r^2 \sin\theta} \Big) \bigg) = 0$$$$

Where $$f=f(t,r,\theta$$) defined in the spherical coordinate and $$D$$ an operator: $$$$D = \frac{\partial^2}{\partial r^2} + \frac{1}{r^2} \frac{\partial^2}{\partial \theta^2} - \frac{cos\theta}{r^2 sin\theta} \frac{\partial}{\partial \theta}$$$$

Here is the link that provides you the code written in Maple: https://www.dropbox.com/s/iej0mpf6z05k6o4/spherical-outputs.mw?dl=0

With the hint of separation of variables $$f(t,r,\theta)=T(t) \cdot R(r) \cdot \Theta(\theta)$$, Maple couldn't solve the PDE probably because it is not separatable. Can anyone help me to solve this equation with any method in Maple or Mathematica?

Based on the known solution for more simple case, I also formulated the time-dependency $$T(t)=e^{-\lambda^2 \alpha t}$$ and used the change of variable $$f(t,r,\theta)=e^{-\lambda^2 \alpha t} (r \cdot \frac{\partial g(r,\theta)}{\partial r})$$, but still couldn't solve it. Does anyone know a change of variables that could potentially solve this kind of PDE?

• Your PDE doesn't seem to be written correctly: the second D that appears in the formula has no function that it can act upon. Commented Apr 6, 2019 at 23:55
• You're right. I've already fixed it. Function $f(t,r,\theta)$ is multiplied to both $D$ and $\frac{1}{\alpha} \frac{\partial}{\partial t}$. Commented Apr 8, 2019 at 1:19