Existence and uniqueness of solutions to discrete diffusion equations

Motivated by the classical one-dimensional diffusion equation, utkuxx = f(x,t), xεR, t≥0, u(x,0)=ε(x), xεR, we find the unique solution of the partial difference equation δtu(x,t)-kδ2x,xu(x-1,t)=f(x,t), xεZ, t≥0, u(x,0)=ε(x), xεZ, where k ε (0, 1). Our approach is to construct the Greenʹs function for the appropriate partial difference operator, and almost immediately we establish an existence and uniqueness result for the discrete nonhomogenous problem.