On the General Solution of Impulsive Systems with Hadamard Fractional Derivatives

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

doi:https://doi.org/10.1155/2016/2814310

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Acknowledgments References Copyright Related Articles

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Theory and Applications of Fractional Order Systems 2016

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Volume 2016 | Article ID 2814310 | https://doi.org/10.1155/2016/2814310

On the General Solution of Impulsive Systems with Hadamard Fractional Derivatives

Xianmin Zhang,¹Xianzhen Zhang,²Zuohua Liu,³Wenbin Ding,¹Hui Cao,¹and Tong Shu¹

Academic Editor: Josè A. Tenereiro Machado

Received20 Jan 2016

Accepted08 May 2016

Published15 Jun 2016

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 [1–3], and some properties of the solution were researched for fractional differential equations in [4–11]. Moreover, the Hadamard-type fractional calculus which is a branch of fractional calculus was developed in [12–17]. 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 [21–32], 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 [34–39]. 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) bywhere is a constant. Next, it will be verified that (40) satisfies all conditions of system (39).(i)For in (40), we have(ii)For in (40), we getSo (40) satisfies Hadamard fractional derivative of system (39).

By the definition of Hadamard fractional integral, we haveTherefore,That is, (40) satisfies impulsive condition in system (39).

Finally, for system (39), there exists an implicit condition

By (40), we havewhich shows that (40) satisfies condition (3). So (40) is the general solution of system (39).

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The work described in this paper is financially supported by the National Natural Science Foundation of China (Grant no. 21576033) and the Natural Science Foundation of Jiangxi Province (Grant no. 20151BAB207013).

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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.

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