On a connection between the N-dimensional fractional Laplacian and 1-D operators on lattices

Abstract

We show the remarkable fact that the nonlocal property of the discrete N-dimensional fractional Laplacian acting in the second variable of the lattice N×ZN can be exchanged with an equivalent memory corresponding to a power of a one-dimensional operator that acts only on the first variable of the complete lattice Z×ZN. This property allows to reduce the number of calculations and leads to more complete analytical solutions of mathematical models on lattices. The connection is established by showing that a first order equation in the first variable, and of fractional order α>0 in the second, has the same solution as another of order 1/α in the first variable and integer order in the second. As a result, we provide for the first time the fundamental solution for the N-dimensional heat equation discrete in time and space.