Generalised Operational Calculus Approach for Fractional Differential Equations

Mikusinski’s operational calculus is a method for interpreting and solving fractional ´ differential equations, formally similar to Laplace transforms but more rigorously justified. This formalism was established for Riemann–Liouville and Caputo fractional calculi in the 1990s, and more recently for other types of fractional calculus. In this thesis, we consider the operators of Riemann–Liouville and Caputo fractional differentiation of a function with respect to another function, and discover that the approach of Luchko can be followed, with small modifications, in the more general settings too. We establish all the function spaces, formalisms, and identities required to build the versions of Mikusinski’s operational calculus which cover ´ Riemann–Liouville and Caputo derivatives with respect to functions. In the process, we gain a deeper understanding of some of the structures involved in applying Mikusinski’s operational calculus to fractional calculus, such as the existence of a ´ group isomorphic to R. The mathematical structure established here is used to solve fractional differential equations using Riemann–Liouville and Caputo derivatives with respect to functions, the solutions being written using multivariate Mittag-Leffler functions, in agreement with the results found in other recent work. It is useful to understand how the various operators of fractional calculus relate to each other, especially relations between newly defined operators and classical wellstudied ones. In this work, we also focus on an important type of such relationship, namely conjugation relations, also called transmutation relations. We define a general abstract setting in which such relations are relevant, and indicate how they can be used to prove many results easily in general settings such as fractional calculus with respect to functions and weighted fractional calculus.