Cauchy Problem for a Linear System of Ordinary Differential Equations of the Fractional Order

TL;DR

This paper studies the initial value problem for linear systems of fractional differential equations using the Dzhrbashyan -- Nersesyan operator, proving existence and uniqueness, and explicitly constructing solutions with Mittag-Leffler functions, generalizing prior fractional derivatives.

Contribution

It introduces the Dzhrbashyan -- Nersesyan fractional operator for systems, proving fundamental theorems, and providing explicit solutions that unify and extend existing fractional differential equation results.

Findings

Existence and uniqueness of solutions established.

Explicit solutions expressed via Mittag-Leffler functions.

Generalization of classical fractional derivatives achieved.

Abstract

The paper investigates the initial problem for a linear system of ordinary differential equations with the fractional differentiation operator Dzhrbashyan -- Nersesyan with constant coefficients. The existence and uniqueness theorems of the solution of the boundary value problem under study are proved. The solution is constructed explicitly in terms of the Mittag-Leffler function of the matrix argument. The Dzhrbashyan -- Nersesyan operator is a generalization of the Riemann -- Liouville, Caputo and Miller-Ross fractional differentiation operators. The obtained results as special cases contain results related to the study of initial problems for systems of ordinary differential equations with Riemann -- Liouville, Caputo and Miller -- Ross derivativess, and the investigated initial problem generalizes them