# 2D Heat Equation - Exact Solution

I'm trying to solve numerically the heat equation on a rectangle, with homogeneous Dirichlet boundary conditions and a source term.

$u_t = u_{xx} + u_{yy} + Q(x,y,t,u) \qquad \text{in} \quad 0<x<1, \quad 0<y<2, \quad t>0$

with

$u=0 \quad \text{on} \quad x=0,1, \quad y=0,2.$

while

$u (t=0) = u_0(x,y)$

I want to test my code with an exact solution but I can´t find any for this particular case. Do you know any book, website, resource in which example solutions to this equation can be found?

Thank you

• Applied PDEs by Haberman will have it (or something very similar to it). It'll probably be what you need. – NicNic8 Mar 5 '18 at 19:32
• see here: math.psu.edu/wysocki/M412/Notes412_10.pdf the geometry is simple, so you may want to try and separate variables – qbert Mar 5 '18 at 19:38