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.
- evolutionary equations
- fractional derivatives
Picard, R., Trostorff, S., & Waurick, M. (2015). On evolutionary equations with material laws containing fractional integrals. Mathematical Methods in the Applied Sciences , 38(15), 3141-3154. https://doi.org/10.1002/mma.3286