Jacobi's algorithm extended to the diffusion equation in two dimensions
We assume that we have a square lattice of length \( L \) with equally
many mesh points in the \( x \) and \( y \) directions. Setting the diffusion
constant \( D=1 \) and using the shorthand notation
\( u_{xx}={\partial^2 u}/{\partial x^2} \) etc for the second
derivatives and \( u_t={\partial u}/{\partial t} \) for the time
derivative, we have, with a given set of boundary and initial
conditions,
$$
\begin{equation*}
\begin{array}{cc}u_t= u_{xx}+u_{yy}& x, y\in(0,L), t>0 \\
u(x,y,0) = g(x,y)& x, y\in (0,L) \\
u(0,y,t)=u(L,y,t)=u(x,0,t)=u(x,L,t)0 & t > 0\\
\end{array}
\end{equation*}
$$