Existence and uniqueness of solutions to discrete diffusion equations (Q1827166)

The authors establish a formula for the unique solution of the nonhomogeneous discrete diffusion equation \[ \begin{aligned} \Delta_t u(x,t)-k\Delta^2_{x,x}u(x-1,t)&=f(x,t),\quad x\in\mathbb Z,\quad t\geq0,\\ u(x,0)&=\psi(x),\qquad x\in\mathbb Z, \end{aligned} \] where \(k\in(0,1)\). The key of the proof is the construction of the Green's function for the associated partial difference operator.