The General Solution of Differential Equations with Caputo-Hadamard Fractional Derivatives and Noninstantaneous Impulses

Abstract

Based on some recent works about the general solution of fractional differential equations with instantaneous impulses, a Caputo-Hadamard fractional differential equation with noninstantaneous impulses is studied in this paper. An equivalent integral equation with some undetermined constants is obtained for this fractional order system with noninstantaneous impulses, which means that there is general solution for the impulsive systems. Next, an example is given to illustrate the obtained result.

1. Introduction

Impulsive differential equations are used in modeling of biology and physics and engineering and so forth to describe abrupt changes of the state at certain instants. However, the classical impulsive models in which instantaneous impulses were mainly considered in most of the existing papers can not describe some processes such as evolution processes in pharmacotherapy. Hence, Hernández and O’Regan [1] and Pierri et al. [2] presented and studied a kind of differential equations with noninstantaneous impulses. Next, the fractional differential equations with noninstantaneous impulses were considered in [3, 4].

Recently, Hadamard fractional calculus is getting attention which is an important part of theory of fractional calculus [5]. The works in [6–11] made development in fundamental theorem of Hadamard fractional calculus. A Caputo-type modification of Hadamard fractional derivative which is called Caputo-Hadamard fractional derivative was given in [12], and its fundamental theorems were proved in [13, 14].

Furthermore, some works in [15–21] uncover that there is general solution for several fractional differential equations with instantaneous impulses. Therefore, we will try to consider the general solution for differential equations with Caputo-Hadamard fractional derivatives and noninstantaneous impulses:where , , and and is the left-side Caputo-Hadamard fractional derivative of order . and (here ) are some appropriate functions, and denote noninstantaneous impulses. .

Firstly, we only consider in each interval in (1), and thenSubstituting (2) into (1), we obtainIn fact, satisfies conditions of fractional derivative and noninstantaneous impulses in system (1). However, we will illustrate that is not an equivalent integral equations of system (1).

For system (1), we haveOn the other hand, letting for and all in (3), we getIf is equivalent to system (1), then (5) is equivalent to (6). Therefore, some unfit equations can be obtained asand here , Therefore, is not equivalent with system (1), and will be regarded as an approximate solution of system (1).

Next, considering for () in whole interval , we haveSubstituting (8) into (1), we obtain andHence, substituting (8) and (9) into (1), we getObviously, (10) satisfies the conditions in system (1) andTherefore, (10) is a solution of system (1).

Remark 1. Equation (10) is only a particular solution of system (1) because it does not contain the important part of the approximate solution .

Next, some definitions and conclusions are introduced in Section 2, the equivalent integral equation will be given for differential equation with Caputo-Hadamard fractional derivatives and noninstantaneous impulses in Section 3, and an example will show that there exists the general solution for this fractional differential equation with noninstantaneous impulses in Section 4.

2. Preliminaries

Definition 2 (see [5, p. ]). 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.

The left-sided Caputo-Hadamard fractional derivative is presented in [12] by the following.

Definition 3 (see [12, p. ]). Let and and , . Then exist everywhere on and if ,where differential operator and .

Lemma 4 (see [12, p. ]). Let , , and . If or , then

Lemma 5 (see [12, p. ]). Let or and , then

Lemma 6 (see [15, p ]). Let and is a constant. A function is general solution of the system if and only if satisfies the fraction integral equationprovided that the integral in (17) exists, and here .

3. Main Result

Theorem 7. Let (here ) be some arbitrary constants. System (1) is equivalent toprovided that the integral in (18) exists, and here .

Proof.     
Step  1 (Necessity). We will verify that (18) satisfies all conditions of system (1). For convenience, we divide this section into three steps.
Step  1.1. Equation (18) satisfies the fractional derivative in system (1).
By (18), for (here ), we getSo, (18) satisfies the fractional derivative in system (1).
Step  1.2. We can verify that (18) satisfies noninstantaneous impulses and initial value in system (1).
Step  1.3. Verify that (18) satisfies a hidden condition of system (1).
Letting for and all in (18), we obtainMoreover, it is sure that (20) is the solution of system (4) by Lemma 5. Thus, (18) satisfies all conditions of system (1).
Step  2 (Sufficiency). We will verify that the solution of system (1) satisfies (18). For convenience, we divide this section into three steps.
Step  2.1. Verify that the solution of system (1) satisfies (18) in intervals and .
By Lemma 5, the solution of (1) satisfiesand for .
Step 2.2. Verify that the solution of system (1) satisfies (18) in intervals and .
For , the approximate solution (by the above discussion about ) is given aswith error for . By the particular solution (10), the exact solution of system (1) satisfiesThus, Therefore, supposewhere is an undetermined function with . Thus,On the other hand, letting , we getBy using Lemma 6 to (27), we get , , and here is an arbitrary constant. Thus,And for
Step  2.3. Verify that the solution of system (1) satisfies (18) in intervals and .
The approximate solution as is given bywith error for . Moreover, by the particular solution (10), the exact solution of system (1) satisfiesThus,Therefore, supposewhere is an undetermined function with . Therefore,On the other hand, consider a special case of system (1) as Using Lemma 6 for system (34), we obtain , , and here is an arbitrary constant. ThusBy “Sufficiency” and “Necessity,” system (1) is equivalent to (18). The proof is completed.

4. Examples

In this section, we will give an impulsive fractional system to illustrate that there exists general solution for fractional differential equations with noninstantaneous impulses.

Example 1. Let us consider the following impulsive linear fractional system:By Theorem 7, system (36) has general solutionwhere is a constant. After some elementary computation, (37) can be rewritten byNext, let us verify that (38) satisfies all conditions of system (36). By Definition 3, we haveTherefore, for and in (38), we have Thus, (38) satisfies fractional derivative and noninstantaneous impulses in system (36). Therefore, (38) is general solution of system (36).

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 nos. 21576033, 21636004, 61563023, and 61362038) and the Research Foundation of Education Bureau of Jiangxi Province, China (Grant no. GJJ14738).