# Question:PDE separation of variables

## Question:PDE separation of variables

Maple

Hi everyone,

I am having some trouble separting terms in a pretty gnarly PDE.  I have a fairly complicated 2+1 PDE with auxillary conditions that I am trying to solve with a formal separation of variables approach.  I have a PDE that looks like this:  expr:= 2((fxyt + u1xfxy + u1fxxy ) + fxxxy )f 2
+2(−fxfyu1x − (u1fxx + fxt + fxxx )fy
− (fxxy +2u1fxy + fyt )fx − fxyfxx − ft fxy )f
+4f f (u fx + fxx + ft ) = 0.

where the t,x, and y's following f are partial derivatives.  I let f be separated in the following manner:

f(x,y,t) = 1 + m*p(x,t) + n*q(y,t) + A*p(x,t)*y(y,t)

In maple after this is done I correctly get a super complicated expression in p and q with all the correct derivates.  I want to collect derivates of p and derivatives of q.  The answer is this:

[2(m + n3q) − (a0 + mp + nq + Apq)px^{-1}
∂x ](pt + pxx + u1px )
+[2(n + Ap) − (a0 + mp + nq + Apq)qy^{-1}∂y ]qt = 0.

It's kind of a mess but it's separable and allows for a slick solutions.  My problem is I can't get to the last expression with collect, factor, or any other combination I have tried so far.  In other words I get to the second to last step but cannot separate p and q.  Here are a few basic questions:

1)  Collect works nicely for things like collect(expr,diff(p,t)); but keeps returning an error if I try to collect terms that are sums, such as collect(expr,diff(p,t) + diff(p,x,x)), which I know apriori is possible.  How can I do this?

2) In general, once I get to the expression with p and q, how can I get maple to separate the expression in terms of p and it's derivatives and q and its derivatives?

3) This example is from the paper the method comes from, but in general there may be some more complicated algabraic manipulations depending on the PDE after the substitution for f.  Is there some algorithmic way to look for such a complicated separation?

Thanks for the help,

Ryan   ﻿