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.

This study is carred out with the ongoing project of "mathematical and numerical modelling of medical ultasound wave propagation" sponsored by the Simula Research Laboratory

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