From: "Dr. Richard L. Hall" Subject: Re: Differential equation. Date: Tue, 09 Mar 1999 12:22:33 -0500 Newsgroups: sci.math To: Marco Haschka 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