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Any hope for this PDE? Diffusion in population genetics

by Schraiber
Tags: diffusion, genetics, hope, population
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Mar3-10, 11:34 PM
P: 1
I'm working on a problem in multi-allele diffusion in population genetics and I have come to this PDE:

[tex]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}[/tex]

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 :(

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