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Mathematical Problems in Engineering
Volume 2016, Article ID 2814310, 12 pages
http://dx.doi.org/10.1155/2016/2814310
Research Article

On the General Solution of Impulsive Systems with Hadamard Fractional Derivatives

1School of Electronic Engineering, Jiujiang University, Jiujiang, Jiangxi 332005, China
2School of Chemical and Environmental Engineering, Jiujiang University, Jiujiang, Jiangxi 332005, China
3School of Chemistry and Chemical Engineering, Chongqing University, Chongqing 400044, China

Received 20 January 2016; Accepted 8 May 2016

Academic Editor: Josè A. Tenereiro Machado

Copyright © 2016 Xianmin Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

1. Introduction

The theory of fractional calculus has been applied in widespread fields of science and engineering [13], and some properties of the solution were researched for fractional differential equations in [411]. Moreover, the Hadamard-type fractional calculus which is a branch of fractional calculus was developed in [1217]. Next, the Caputo-Hadamard fractional derivative which is a Caputo-type modification of the Hadamard fractional derivative is suggested in [18], some fundamental theorems of this fractional derivative were proved in [19], and Cauchy problems of a differential equation with a left Caputo-Hadamard fractional derivative were studied in spaces of continuously differentiable functions in [20].

On the other hand, 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 [2132], and the existence of solutions was considered for impulsive differential equations with Hadamard fractional derivative in [33].

Furthermore, the general solution for some impulsive fractional differential equations was found in [3439]. Therefore, we will seek the general solution for the impulsive systems with Hadamard fractional derivatives in present paper:where and , denotes left-sided Hadamard fractional derivative of order and , and denotes left-sided Hadamard fractional integral of order . and () are some appropriate functions and impulsive points . and represent the right and left limits of at , respectively.

Considering limited case in system (1a)–(1c), we haveTherefore,

Next, using the definition of Hadamard fractional derivative, system (1a)–(1c) is transformed intoAccording to each interval (here ) in (4), define a functionwith .

By Definition 2, we haveTherefore, satisfies condition (1a), and also satisfies conditions (1b) and (1c). But does not satisfy condition (3). Thus, is not a solution of system (1a)–(1c) and will be considered as an approximate solution to seek the exact solution of system (1a)–(1c).

Next, we will give some definitions and conclusions in Section 2 and prove the formula of general solution for impulsive differential equations with Hadamard fractional derivative in Section 3. Finally, we will provide an example to expound the main result in Section 4.

2. Preliminaries

Definition 1 (see [2]). Let be finite or infinite interval of the half-axis . The left-sided Hadamard fractional integral of order () of function is defined bywhere is the Gamma function.

Definition 2 (see [2]). The left-sided Hadamard fractional derivative of order with on is defined bywhere and differential operator with .

Lemma 3 (see [2]). Let and , . The fractional Cauchy problemis equivalent to the following nonlinear integral equation:

3. Main Result

Theorem 4. Let , , and , and let be an arbitrary constant. System (1a)–(1c) is equivalent to the integral equation:provided that the integral in (11) exists.

Proof.
Necessity. Letting () in (11), we haveAccording to Lemma 3, (12) is equivalent to (2). Therefore, (11) satisfies condition (3).
Next, using Definition 2 for (11), we have So (11) satisfies (1a).
By (11), we getTherefore, (11) satisfies (1b). Thus, (11) satisfies all conditions of system (1a)–(1c).
Sufficiency (by Mathematical Induction). By Lemma 3, the solution of system (1a)–(1c) satisfiesUsing (15), we obtain Therefore, the approximate solution is provided by Let for . Moreover, by (15), the exact solution of system (1a)–(1c) satisfiesTherefore,By (19), letwhere is an undetermined function with . Thus,Using (21), we get Then the approximate solution is given byLet for . By (21), the exact solution of (1a)–(1c) satisfiesTherefore, Then, by (25), we obtainThus,In addition,By using (21) and (27) for systems (29) and (28), we obtainSo for , where is an arbitrary constant. Thus, (21) and (27) are rewritten intoFor , letUsing (32), we haveTherefore, the approximate solution is provided asLet for . On the other hand, by (32), the exact solution of (1a)–(1c) satisfiesTherefore,By (36), we obtainThenThus, the solution of system (1a)–(1c) satisfies (11).
So system (1a)–(1c) is equivalent to (11). The proof is now completed.

4. An Example

In this section, an example will be provided to illustrate the usefulness of Theorem 4, which will show that there is general solution for impulsive systems with Hadamard fractional derivatives.

Example. The analytical solution of (1a)–(1c) is difficultly obtained when is a nonlinear function. Therefore, let us consider an impulsive fractional linear system:Using Theorem 4, we give general solution of system (39) by