Any hope for this PDE? Diffusion in population genetics

[Schraiber]

I'm working on a problem in multi-allele diffusion in population genetics and I have come to this PDE:

$$0 = (tp_1(1-p_1)-sp_2)\frac{\partial u}{\partial p_1}+(sp_2(1-p_2)-tp_1)\frac{\partial u}{\partial p_2} + \frac{p_1(1-p_1)}{2}\frac{\partial^2 u}{\partial p_1^2} + \frac{p_2(1-p_2)}{2}\frac{\partial^2 u}{\partial p_2^2} - p_1p_2 \frac{\partial^2 u}{\partial p_1 \partial p_2}$$

The boundary conditions are u(0,p2) = 0 (for all p2), u(1,0) = 1

I'm not entirely sure it's possible to solve, though it's highly symmetric so I thought I'd give it a shot and see if anyone has any ideas. I tried separating which obviously doesn't work (unless I'm a failure) and I've spent some time searching for clever transformations that might work, but it seems to be of no use :(

Thanks!

 

[jvicens]

Thank you for sharing your problem with us. The PDE you have presented is indeed a challenging one, but it is not impossible to solve. There are a few approaches you can take to tackle this problem.

One approach is to use numerical methods to approximate the solution. This involves discretizing the domain and solving the PDE using techniques such as finite difference or finite element methods. This approach may give you a good approximation of the solution, but it may not provide an analytical expression for the solution.

Another approach is to use analytical methods, such as separation of variables or the method of characteristics, to try and find a solution. While it may not be possible to find an exact solution, these methods can help you find approximate solutions or provide insight into the behavior of the solution.

It may also be helpful to simplify the PDE by making some assumptions or transformations. For example, you could assume that p1 and p2 are small and use Taylor series expansions to approximate the terms in the PDE. Or, you could transform the PDE into a more manageable form by making a change of variables.

Lastly, it may be beneficial to seek help from a colleague or a mentor who has experience in solving PDEs in population genetics. They may be able to provide valuable insights and guidance on how to approach this problem.

I wish you the best of luck in solving this challenging PDE. Keep exploring different methods and don't be discouraged if it takes some time to find a solution. With persistence and determination, I'm sure you will make progress in solving this problem.