Abstract
Solving fractional relaxation equations requires precisely characterized domains of definition for applications of fractional differential and integral operators. Determining these domains has been a longstanding problem. Applications in physics and engineering typically require extension from domains of functions to domains of distributions. In this work convolution modules are constructed for given sets of distributions that generate distributional convolution algebras. Convolutional inversion of fractional equations leads to a broad class of multinomial Mittag-Leffler type distributions. A comprehensive asymptotic analysis of these is carried out. Combined with the module construction the asymptotic analysis yields domains of distributions, that guarantee existence and uniqueness of solutions to fractional differential equations. The mathematical results are applied to anomalous dielectric relaxation in glasses. An analytic expression for the frequency dependent dielectric susceptibility is applied to broadband spectra of glycerol. This application reveals a temperature independent and universal dynamical scaling exponent.
Highlights
Applications of fractional calculus in physics [3,5,9,21] and mathematics [16,40,49] are enjoying an undiminished surge of attention in recent years
The final Section discusses consequences for fractional calculus and fractional differential equations that result from the previous sections
The restriction to the half-axis is unnecessary, and all definitions and results are readily transported to the operators Dα+ and the space E = E (R) of smooth functions on R vanishing rapidly for t → −∞
Summary
Applications of fractional calculus in physics [3,5,9,21] and mathematics [16,40,49] are enjoying an undiminished surge of attention in recent years. Given the need for concretely characterized extended domains in applications a first step was taken in [18,19] where fractional Weyl integrals have been interpreted as convolutions of Radon measures with continuous functions. Our objective in this work is to combine the benefits of our approach [18,19] with those of Marchaud’s [26], with those of fractional powers [2,12,15,17,20,47], with those of translation invariant distribution spaces [22,41], and with the algebraic benefits of operational calculus [13,30,48]. The main results of the paper are derived from a new method to construct endomorphic domains of distributions for general families of convolution operators that is described in Sect. Dielectric relaxation processes, described by fractional initial value problems on R+, and the response to periodic excitations, described by Fourier multipliers on R, as resulting from a single translation invariant linear differential equation on R
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
