From: "Dr. Richard L. Hall" <richlh@ix.netcom.com>
Subject: Re: Differential equation.
Date: Tue, 09 Mar 1999 12:22:33 -0500
Newsgroups: sci.math
To: Marco Haschka <s_mhasch@mathematik.uni-ulm.de>
Keywords: Elementary example of separation of variables

Do a separation of variables.  Let f(x,y,t) = G(x,y)T(t).  Substituting
and dividing by f(x,y,t), you can write

1/T(t) d/dt(T(t)) = g(x) + (1/G(x,y))[{(d/dx)^2 + (d/dy)^2}G(x,y)].  The
left hand side depends only on t and the right hand side depends only on x
and y.  Therefore they each must be constant.  Let the constant be K.  So

d/dt(Tt)) = K T(t)  or T(t) = A exp (Kt) and

[{(d/dx)^2 + (d/dy)^2}G(x,y)]  -(K - g(x))G(x,y) = 0.  Now you should be
able to separate this out again.  Let G(x,y) = X(x) Y(y).  Substitute and
divide as before and get two equations.  One for x and one for y.  by
inspection, I would say they should be like

1/X(x) {(d/dx)^2} X(x) - (K - g(x)) = - 1/Y(y) {(d/dy)^2}Y(y) = L where L
is a constant as before since the right hand side is a function only of y
and the left hand side is a function only of x.

Hope this helps.

   Richard

Marco Haschka wrote:

> Hello,
>
> I am a student of mathematics and physics, and
> we try to solve the following partial differential equation.
>
>  d                                ( d   d             d  d           )
>  -- f (x,y,t) = g(x)*f(x,y,t) + y*| --  -- f(x,y,t) + -- -- f(x,y,t) |
>  dt                               ( dx  dx            dy dy          )
>
> with a given g(x) and a given f(x,y,t=0).
>
> We should show, that
> int(f*y,x=-inf..inf,y=0..inf)/int(f,x=-inf..inf,y=0..inf) --> 0
> for t--> inf.
>
> This Equation seems to be very simple, but we have not been
> able to solve this problem for two weeks  by using Maple
> and Derive.
>
> It would be very helpfull, if you could give some hints,
> about the type of PDE we have, and how we could solve it.
>
> Marco