### Abstract

Language | English |
---|---|

Pages | 3141-3154 |

Number of pages | 14 |

Journal | Mathematical Methods in the Applied Sciences |

Volume | 38 |

Issue number | 15 |

Early online date | 21 Oct 2015 |

DOIs | |

Publication status | Published - 31 Oct 2015 |

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### Keywords

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

### Cite this

*Mathematical Methods in the Applied Sciences*,

*38*(15), 3141-3154. https://doi.org/10.1002/mma.3286

}

*Mathematical Methods in the Applied Sciences*, vol. 38, no. 15, pp. 3141-3154. https://doi.org/10.1002/mma.3286

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

Research output: Contribution to journal › Article

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 -