Abstract

In this paper, we solve the non-local in-time wave and heat equations with damping on an infinite lattice in the linear case. Under suitable assumptions, we derive a representation of the solutions in terms of subordinators and an explicit formula using the discrete Fourier transform, within the framework of Banach algebras. Moreover, we establish sufficient conditions to ensure that the solution is a probability distribution, and highlight the differences with respect to its continuous counterpart. The discrete maximal regularity of the non-homogeneous cases on l^p(Z) also is fully determined.