On evolutionary equations with material laws containing fractional integrals

Rainer Picard, Sascha Trostorff, Marcus Waurick

Research output: Contribution to journalArticlepeer-review

17 Citations (Scopus)


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.
Original languageEnglish
Pages (from-to)3141-3154
Number of pages14
JournalMathematical Methods in the Applied Sciences
Issue number15
Early online date21 Oct 2015
Publication statusPublished - 31 Oct 2015


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


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