I am trying to solve the 2D PDE $$\frac{\partial p(\mathbf{x},t)}{\partial t}=-{f}(\mathbf{x})\frac{\partial }{\partial x_1}p(\mathbf{x},t) -{g}(\mathbf{x})\frac{\partial }{\partial x_2}p(\mathbf{x},t)+a\frac{\partial ^2}{\partial x_1^2}p(\mathbf{x},t)+b\frac{\partial ^2}{\partial x_2^2}p(\mathbf{x},t)$$ so that I can test the accuracy of a numerical solver that I wrote. Here, $a$ and $b$ are real valued, nonzero, scalars.
I learned how to solve this when $f$ and $g$ are constants, but I am not sure if it is possible to analytically solve this when they are not constant (Or if one is nonconstant and the other is constant). I have been searching in the literature but am having trouble finding solutions.
Are there any known solutions to this that I can use to test my code in this situation that either $f$ or $g$ or both are nonconstant? Even for a specific case would be helpful.
Thank you!
