A new definition of the fractional Laplacian

Abstract

It is noted that the standard definition of the fractional Laplacian leads to a hyper-singular convolution integral and is also obscure about how to implement the boundary conditions. This purpose of this note is to introduce a new definition of the fractional Laplacian to overcome these major drawbacks.Comment: This study is carred out with the ongoing project of "mathematical and numerical modelling of medical ultasound wave propagation" sponsored by the Simula Research Laborator