Representations of solutions of time-fractional multi-order systems of differential-operator equations

TL;DR

This paper develops a general theory for solutions of time-fractional multi-order differential-operator systems, introducing new representation formulas and properties of vector-indexed Mittag-Leffler functions.

Contribution

It provides the first known representation formulas for arbitrary fractional multi-order systems and introduces vector-indexed Mittag-Leffler functions.

Findings

Derived representation formulas for solutions.

Proved existence and uniqueness theorems.

Introduced and analyzed properties of vector-indexed Mittag-Leffler functions.

Abstract

This paper is devoted to the general theory of systems of time-fractional differential-operator equations. The representation formulas for solutions of systems of ordinary differential equations with single (commensurate) fractional order is known through the matrix valued Mittag-Leffler function. Multi-order (incommensurate) systems with rational components can be reduced to single order systems, and hence, representation formulas are also known. However, for arbitrary fractional multi-order (not necessarily with rational components) systems of differential equations this question remains open even in the case of ordinary differential equations. In this paper we obtain representation formulas for solutions of arbitrary fractional multi-order systems of differential-operator equations along with proving the existence and uniqueness theorems in appropriate topological-vector spaces. Moreover, we introduce vector-indexed Mittag-Leffler functions and prove some of their properties.