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Fractional Calculus and Applied Analysis

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Volume 22, Issue 6

Issues

Dispersion analysis for wave equations with fractional Laplacian loss operators

Sverre Holm
Published Online: 2019-12-31 | DOI: https://doi.org/10.1515/fca-2019-0082

Abstract

Several wave equations for power-law attenuation have a spatial fractional derivative in the loss term. Both one-sided and two-sided spatial fractional derivatives can give causal solutions and a phase velocity dispersion which satisfies the Kramers–Kronig relation. The Chen–Holm and the Treeby–Cox equations both have the two-sided fractional Laplacian derivative, but only the latter satisfies this relation. There also exists several seemingly different expressions for the phase velocity for these equations and it is shown here that they are approximately equivalent. Causality of the Chen–Holm equation has also been a topic of some discussion and it is found that despite the lack of agreement with the Kramers–Kronig relation, it is still causal.

MSC 2010: Primary 26A33; Secondary 35R11

Key Words and Phrases: fractional partial differential equations; fractional Laplacian; wave equations; dispersion

This paper is dedicated to the memory of the late Professor Wen Chen

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About the article

Received: 2019-07-05

Published Online: 2019-12-31

Published in Print: 2019-12-18


Citation Information: Fractional Calculus and Applied Analysis, Volume 22, Issue 6, Pages 1596–1606, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.1515/fca-2019-0082.

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