On representation formulas for solutions of linear differential equations with Caputo fractional derivatives

TL;DR

This paper investigates linear differential equations with Caputo fractional derivatives, analyzing fundamental solutions and deriving new representation formulas for solutions with initial conditions at intermediate points.

Contribution

It introduces new representation formulas for solutions of fractional differential equations with variable coefficients and intermediate initial conditions, expanding existing theoretical frameworks.

Findings

Fundamental solution matrix is Hölder continuous in both variables.

Two new representation formulas for solutions are derived and justified.

Analysis extends to equations with variable coefficients and non-standard initial points.

Abstract

In the paper, a linear differential equation with variable coefficients and a Caputo fractional derivative is considered. For this equation, a Cauchy problem is studied, when an initial condition is given at an intermediate point that does not necessarily coincide with the initial point of the fractional differential operator. A detailed analysis of basic properties of the fundamental solution matrix is carried out. In particular, the H\"{o}lder continuity of this matrix with respect to both variables is proved, and its dual definition is given. Based on this, two representation formulas for the solution of the Cauchy problem are proposed and justified.