On the fractional Laplacian of variable order

TL;DR

This paper introduces a new definition of the variable-order fractional Laplacian on Rn, extending the Riesz potential, and explores its properties, solutions, and Green functions for fractional Poisson's equations.

Contribution

It proposes a novel, generalized definition of the variable-order fractional Laplacian applicable across the entire (0, n/2) range, with analysis of its properties and solutions.

Findings

Defined the variable-order fractional Laplacian using a generalized Riesz potential

Analyzed properties of the fractional Poisson's equation with this operator

Computed Green functions for specific fractional problems

Abstract

We present a novel definition of variable-order fractional Laplacian on Rn based on a natural generalization of the standard Riesz potential. Our definition holds for values of the fractional parameter spanning the entire open set (0, n/2). We then discuss some properties of the fractional Poisson's equation involving this operator and we compute the corresponding Green function, for which we provide some instructive examples for specific problems.