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Fractional Calculus and Applied Analysis

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On a fractional Zener elastic wave equation

Sven Näsholm / Sverre Holm
Published Online: 2012-12-27 | DOI: https://doi.org/10.2478/s13540-013-0003-1


This survey concerns a causal elastic wave equation which implies frequency power-law attenuation. The wave equation can be derived from a fractional Zener stress-strain relation plus linearized conservation of mass and momentum. A connection between this four-parameter fractional wave equation and a physically well established multiple relaxation acoustical wave equation is reviewed. The fractional Zener wave equation implies three distinct attenuation power-law regimes and a continuous distribution of compressibility contributions which also has power-law regimes. Furthermore it is underlined that these wave equation considerations are tightly connected to the representation of the fractional Zener stress-strain relation, which includes the spring-pot viscoelastic element, and by a Maxwell-Wiechert model of conventional springs and dashpots. A purpose of the paper is to make available recently published results on fractional calculus modeling in the field of acoustics and elastography, with special focus on medical applications.

MSC: Primary 26A33; Secondary 33E12, 34A08, 34K37, 35L05, 92C50, 92C55, 35R11, 74J10

Keywords: fractional calculus; acoustical wave equations; elastic wave equations; fractional wave equations; fractional viscoelasticity; fractional ordinary and partial differential equations

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About the article

Published Online: 2012-12-27

Published in Print: 2013-03-01

Citation Information: Fractional Calculus and Applied Analysis, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.2478/s13540-013-0003-1.

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