Solving PDE with interesting initial condition

Question

I have the PDE (Fokker-Planck) which reads

$$ \frac{\partial p}{\partial L}(L,x) = \bigg((x-1)^2\frac{\partial^2p}{\partial x^2}(L,x)+4x(x-1)\frac{\partial p}{\partial x}(L,x)+2xp(L,x)\bigg),$$ with initial condition $p(L=0,x)=\delta(x-1),$ where $\delta$ is a dirac delta function.

I'm looking for advice on the best way to solve this. Am I correct that the RHS of the PDE can be written as a Sturm-Louiville problem, and then some sort of transform can be used? Thanks in advance.

Answer 1

Yes the right hand side can be written as a Sturm-Liouville equation, check out the WolframAlpha solution! If you are interested in solving it yourself, then read below:

Particularly for the equilibrium distribution you may set the LHS to zero, then you have two options for solution: 1. because $x=1$ is a regular singular point of the differential equation, around which you may find a series expansion using Frobenius method, i.e. $p(x)=\sum_{n=0}^{\infty}c_n x^{n+\sigma}$ where $\sigma$ is a constant to be found.

2. Also, you will be able to convert the "RHS=0" into a hypergeometric differential equation for which analytical solutions exist, i.e. with the caveat that solutions are hypergeometric functions, yet they are very well known functions so that is no problem!