Integral Representation of the Solutions for Neutral Linear Fractional System with Distributed Delays

Hristo Kiskinov,Ekaterina Madamlieva,Magdalena Veselinova,Andrey Zahariev

doi:10.3390/fractalfract5040222

Hristo Kiskinov, Ekaterina Madamlieva + Show 2 more

Open Access

https://doi.org/10.3390/fractalfract5040222

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Journal: Fractal and Fractional	Publication Date: Nov 15, 2021
Citations: 8	License type: CC BY 4.0

Affiliation: Plovdiv University

Abstract

In the present paper, first we obtain sufficient conditions for the existence and uniqueness of the solution of the Cauchy problem for an inhomogeneous neutral linear fractional differential system with distributed delays (even in the neutral part) and Caputo type derivatives, in the case of initial functions with first kind discontinuities. This result allows to prove that the corresponding homogeneous system possesses a fundamental matrix C(t,s) continuous in t,t∈[a,∞),a∈R. As an application, integral representations of the solutions of the Cauchy problem for the considered inhomogeneous systems are obtained.

Highlights

It is well known that fractional calculus and fractional differential equations are an efficient tool for investigations in various fields of science
The book of Diethelm [4] is devoted to an application-oriented exposition, and some generalizations concerning distributed order fractional differential equations are considered in Jiao et al [5]
In comparison with the integer-order case, the main advantage of the delayed fractional differential equations is the possibility to describe the impact of history on the evolution of the processes, taking information from two sources

Summary

Introduction

It is well known that fractional calculus and fractional differential equations are an efficient tool for investigations in various fields of science. From the works devoted to the problem of establishing an integral representation for fractional differential equations and/or systems with delay, we point out [15,16,17] for the case of singular systems Both problems (the existence of a fundamental matrix and integral representation of the solutions) for fractional systems with a single-order Caputo-type derivative of retarded and/or neutral type with distributed delays are studied in [18,19,20,21,22,23,24,25,26]. It is known that in the general case, the problem of the solvability of an initial problem for this system with discontinuous initial function is the basic result from which, as a corollary, we can prove the existence of a fundamental matrix for a homogeneous delayed (or neutral) fractional differential system.

Preliminaries and Problem Statement

Main Results

Applications

Comments and Conclusions