On evolutionary equations with material laws containing fractional integrals

Rainer Picard, Sascha Trostorff, Marcus Waurick

Research output: Contribution to journalArticle

9 Citations (Scopus)

Abstract

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.
LanguageEnglish
Pages3141-3154
Number of pages14
JournalMathematical Methods in the Applied Sciences
Volume38
Issue number15
Early online date21 Oct 2015
DOIs
Publication statusPublished - 31 Oct 2015

Fingerprint

Fractional Integral
Fractional
Fractional Derivative
Derivatives
Riemann-Liouville Fractional Derivative
Superdiffusion
Subdiffusion
Normal Operator
Viscoelasticity
Kelvin
Operator Equation
Fokker-Planck Equation
Causality
Fokker Planck equation
Well-posedness
Diffusion Process
Initial-boundary-value Problem
Calculus
Boundary value problems
Derivative

Keywords

  • well-posedness
  • evolutionary equations
  • fractional derivatives
  • causality

Cite this

@article{f3afb9a33cf7441c8bc514289ea58e73,
title = "On evolutionary equations with material laws containing fractional integrals",
abstract = "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.",
keywords = "well-posedness, evolutionary equations, fractional derivatives, causality",
author = "Rainer Picard and Sascha Trostorff and Marcus Waurick",
year = "2015",
month = "10",
day = "31",
doi = "10.1002/mma.3286",
language = "English",
volume = "38",
pages = "3141--3154",
journal = "Mathematical Methods in the Applied Sciences",
issn = "0170-4214",
number = "15",

}

On evolutionary equations with material laws containing fractional integrals. / Picard, Rainer; Trostorff, Sascha; Waurick, Marcus.

In: Mathematical Methods in the Applied Sciences , Vol. 38, No. 15, 31.10.2015, p. 3141-3154.

Research output: Contribution to journalArticle

TY - JOUR

T1 - On evolutionary equations with material laws containing fractional integrals

AU - Picard, Rainer

AU - Trostorff, Sascha

AU - Waurick, Marcus

PY - 2015/10/31

Y1 - 2015/10/31

N2 - 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.

AB - 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.

KW - well-posedness

KW - evolutionary equations

KW - fractional derivatives

KW - causality

UR - http://onlinelibrary.wiley.com/journal/10.1002/(ISSN)1099-1476

U2 - 10.1002/mma.3286

DO - 10.1002/mma.3286

M3 - Article

VL - 38

SP - 3141

EP - 3154

JO - Mathematical Methods in the Applied Sciences

T2 - Mathematical Methods in the Applied Sciences

JF - Mathematical Methods in the Applied Sciences

SN - 0170-4214

IS - 15

ER -