On the General Solution of Impulsive Systems with Hadamard Fractional Derivatives

Xianmin Zhang,Wenbin Ding,Zuohua Liu,Xianzhen Zhang,Hui Cao,Tong Shu

doi:10.1155/2016/2814310

Abstract

This paper is concerned with the solution for impulsive differential equations with Hadamard fractional derivatives. The general solution of this impulsive fractional system is found by considering the limit case in which impulses approach zero. Next, an example is provided to expound the theoretical result.

Highlights

This paper is concerned with the solution for impulsive differential equations with Hadamard fractional derivatives
The general solution of this impulsive fractional system is found by considering the limit case in which impulses approach zero
The theory of fractional calculus has been applied in widespread fields of science and engineering [1–3], and some properties of the solution were researched for fractional differential equations in [4–11]

Summary

Introduction

The theory of fractional calculus has been applied in widespread fields of science and engineering [1–3], and some properties of the solution were researched for fractional differential equations in [4–11]. Impulsive differential equations are often used for description some processes or system with impulsive effects, impulsive (partial) differential equations with Caputo fractional derivative were widely studied in [21–32], and the existence of solutions was considered for impulsive differential equations with Hadamard fractional derivative in [33]. The general solution for some impulsive fractional differential equations was found in [34–39]. We will seek the general solution for the impulsive systems with Hadamard fractional derivatives in present paper: HDqa+ z (t) = g (t, z (t)) , (1a). Using the definition of Hadamard fractional derivative, system (1a)–(1c) is transformed into.

Preliminaries

Main Result

An Example