On evolutionary equations with material laws containing fractional integrals

A well‐posedness result for a time‐shift invariant class of evolutionary operator equations involving material laws with fractional time‐integrals of order α ϵ ]0, 1[ is considered. The fractional derivatives are defined via a function calculus for the (time‐)derivative established as a normal opera...

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Bibliographic Details
Published in:Mathematical methods in the applied sciences Vol. 38; no. 15; pp. 3141 - 3154
Main Authors: Picard, Rainer, Trostorff, Sascha, Waurick, Marcus
Format: Journal Article
Language:English
Published: Freiburg Blackwell Publishing Ltd 01.10.2015
Wiley Subscription Services, Inc
Subjects:
Boundary value problems
Calculus
causality
Derivatives
Evolutionary
evolutionary equations
fractional derivatives
fractional Fokker-Planck equation
fractional integrals
Integrals
Invariants
Kelvin-Voigt model
Mathematical analysis
Mathematical models
Operators (mathematics)
sub-diffusion
super-diffusion
visco-elasticity
ISSN:0170-4214, 1099-1476
Online Access:Get full text
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Summary:A well‐posedness result for a time‐shift invariant class of evolutionary operator equations involving material laws with fractional time‐integrals of order α ϵ ]0, 1[ is considered. The fractional derivatives are defined via a function calculus for the (time‐)derivative established as a normal operator in a suitable L2 type space. Employing causality, we show that the fractional derivatives thus obtained coincide with the Riemann‐Liouville fractional derivative. We exemplify our results by applications to a fractional Fokker‐Planck equation, equations describing super‐diffusion and sub‐diffusion processes, and a Kelvin‐Voigt type model in fractional visco‐elasticity. Moreover, we elaborate a suitable perspective to deal with initial boundary value problems. Copyright © 2014 John Wiley & Sons, Ltd.
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ISSN:0170-4214
1099-1476
DOI:10.1002/mma.3286