Remarks on the Generalized Fractional Laplacian Operator

Chenkuan Li, Changpin Li, Thomas Humphries and Hunter Plowman
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Chenkuan Li: Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada
Changpin Li: Department of Mathematics, Shanghai University, Shanghai 200444, China
Thomas Humphries: Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada
Hunter Plowman: Department of Mathematics and Computer Science, Brandon University, Brandon, MB R7A 6A9, Canada

Mathematics, 2019, vol. 7, issue 4, 1-17

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.

Keywords: distribution; fractional Laplacian; Riesz fractional derivative; delta sequence; convolution (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (3)

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