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Jacobi's algorithm extended to the diffusion equation in two dimensions, the second derivative

The second derivative with respect to y reads uyyuli,j+12uli,j+uli,j1h2. We use now the so-called backward going Euler formula for the first derivative in time. In its discretized form we have utuli,jul1i,jΔt, resulting in uli,j+4αuli,jα[uli+1,j+uli1,j+uli,j+1+uli,j1]=ul1i,j, where the right hand side is the only known term, since starting with t=t0, the right hand side is entirely determined by the boundary and initial conditions. We have α=Δt/h2.