On evolutionary equations with material laws containing fractional integrals. (21st October 2014)

• Record Type: Journal Article
• Authors: Picard, Rainer; Trostorff, Sascha; Waurick, Marcus
• Is Part Of: Mathematical methods in the applied sciences. Volume 38:Number 15(2015:Oct. 15)
• Page Start: 3141
• Page End: 3154
• Publication Date: 2014-10-21
• Subjects: fractional derivatives -- fractional integrals -- evolutionary equations -- visco‐elasticity -- Kelvin–Voigt model -- fractional Fokker–Planck equation -- sub‐diffusion -- super‐diffusion -- causality; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.3286
• Languages: English
• ISSNs: 0170-4214

Abstract

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 L 2 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.