Fractional Differential Equations with Sonine Kernels

Abstract

The method of Mikusinski operational calculus relies on an abstract algebra ´ understanding of integral and derivative operators, which can be used for solving differential equations. This has been used for fractional operators of many types, including with Sonine kernels. Here, we use the general Sonine kernel to construct a different algebraic setting in which the general fractional derivatives and general fractional integrals can be understood in a different way. By considering algebraic properties, we construct embeddings and isomorphisms to connect the different rings and fields involved in the fractional Mikusinski calculus, and we demonstrate the ´ advantages and limitations of the more general algebraic setting. In this work, we also study linear fractional ODEs with continuous variable coefficients and general fractional derivative (GFD) operators of Riemann–Liouville and Caputo types defined using Sonine kernels. Using the Banach fixed point theorem, we prove existence and uniqueness of continuous solution functions, and construct solutions explicitly by means of uniformly convergent infinite series involving Sonine kernels. This work is done both for the classical Luchko-type GFDs with Sonine kernels, and for the m-fold versions of these operators, akin to sequential fractional derivatives. Keywords: Mikusinski Operational Calculus, Sonine Kernels, Fractional Calculus, Fractional Differential Equations, Fixed Point Theory.