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. Copyright © 2014 John Wiley & Sons, Ltd.