I have a heat equation to be solved for in the form of $$u_{t}= u_{xx}+u_{yy} $$ for t = [0,+inf) and $0<x<1 ,0<y<1 $ and $u=u(x,y,t)$ and following B.C: $$ u(0,0,t)=0 $$ $$ u(1,1,t)=1 $$ I work out the solution roughly using the following procedure:
$u(x,y,t)=f(\xi,\eta)$, with $\xi = \frac{x}{\sqrt{t}}$ and $\eta = \frac{y}{\sqrt{t}}$
Obtain PDE as: $$ -\frac{1}{2}(\xi f_\xi+\eta f_\eta)=f_{\xi\xi}+f_{\eta\eta} $$ So, my question is
What is the appropriate way to solve this PDE ?
Thank you all in advance
This problem isn't well posed. Two non-identical solutions can be identified as: $$u_1(t,x,y) = x \ \ \ \ \ \ u_2(t,x,y) = y$$
Notice that each solution satisfies defining PDE and the boundary conditions.