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Mathematics 2019, 7(4), 320; https://doi.org/10.3390/math7040320

Remarks on the Generalized Fractional Laplacian Operator

1
Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada
2
Department of Mathematics, Shanghai University, Shanghai 200444, China
*
Author to whom correspondence should be addressed.
Received: 25 February 2019 / Revised: 21 March 2019 / Accepted: 25 March 2019 / Published: 29 March 2019
(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2019)
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PDF [277 KB, uploaded 29 March 2019]

Abstract

The fractional Laplacian, also known as the Riesz fractional derivative operator, describes an unusual diffusion process due to random displacements executed by jumpers that are able to walk to neighbouring or nearby sites, as well as perform excursions to remote sites by way of Lévy flights. The fractional Laplacian has many applications in the boundary behaviours of solutions to differential equations. The goal of this paper is to investigate the half-order Laplacian operator ( Δ ) 1 2 in the distributional sense, based on the generalized convolution and Temple’s delta sequence. Several interesting examples related to the fractional Laplacian operator of order 1 / 2 are presented with applications to differential equations, some of which cannot be obtained in the classical sense by the standard definition of the fractional Laplacian via Fourier transform. View Full-Text
Keywords: distribution; fractional Laplacian; Riesz fractional derivative; delta sequence; convolution distribution; fractional Laplacian; Riesz fractional derivative; delta sequence; convolution
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Li, C.; Li, C.; Humphries, T.; Plowman, H. Remarks on the Generalized Fractional Laplacian Operator. Mathematics 2019, 7, 320.

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