Existence and uniqueness of solutions to discrete diffusion equations

AbstractMotivated 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 Green's function for the appropriate partial difference operator, and almost immediately we establish an existence and uniqueness result for the discrete nonhomogenous problem