Abstract
For a class of nonlinear impulsive fractional differential equations, we first transform them into equivalent integral equations, and then the implicit Euler method is adapted for solving the problem. The convergence analysis of the method shows that the method is convergent of the first order. The numerical results verify the correctness of the theoretical results.
1. Introduction
In recent years, fractional differential equations have become a research hotspot due to their wide application in many fields. We refer the readers to the research papers [1–5] and the monographs by Podlubny [6], Diethelm [7], Kilbas et al. [8], Zhou [9], and the references cited therein. When the fractional differential equations are affected by instantaneous mutation, the impulsive fractional differential equations are obtained. The research of impulsive fractional differential equations can be found in literatures [10–20] and monograph [21]. However, there are few literatures on numerical methods for impulsive fractional differential equations.
In this paper, implicit Euler method is constructed for solving a class of nonlinear impulsive fractional differential equations. It is proved that the method is convergent of the first order. The numerical results also verify the correctness of the theoretical results.
2. Construction of Numerical Scheme
Consider the following impulsive fractional differential equations:where , are constants, and is the -order Caputo derivative of solution defined by (see [6–8]), , , , and represent the left and right limits of at , and and are continuous functions and satisfy the following conditions:where are nonnegative constants with moderate size.
Throughout this paper, let be the Banach space of all continuous functions from into with the norm for . We also define and exists, . The space is a Banach space equipped with the norm .
In addition, due to the need of convergence analysis, for function , there are constants and such that
In order to obtain the numerical scheme for solving problem (1), according to Lemma 3.1 in reference [12], we can express equation (1) as the following equivalent integral equation:
Let , be a given positive integer, the grid points , , , , and express the true solution of equation (1) at . Then, an approximation to the integral equation can be attained by right rectangle formula:
By using the numerical solution instead of the true solution in equation (6), we obtain the implicit Euler method for solving impulsive fractional differential equation (1):
Remark 1. It is well known that there are various forms of definition of fractional calculus in the literature, such as Riemann–Liouville fractional calculus, Grnwald–Letnikov fractional calculus, and Caputo fractional calculus (see [6–8]). The -order Riemann–Liouville derivative of function is defined byWe have the following relationship between the Riemann–Liouville derivative and Caputo derivative:Therefore, the fractional differential equations studied in the present paper only consider Caputo derivative.
3. Convergence Analysis
Let , where and denote the numerical solution and true solution of problem (1) at grid point , respectively. Then, for the error , we havewhere
In order to obtain the convergence result of numerical method (7), we first prove the following lemma.
Lemma 1. Assume the functions and are continuous and satisfy conditions (3) and (4); then, the truncation error of the discrete scheme (7) satisfies
Throughout the paper, will denote a positive constant not necessarily the same at different places, which may depend on , but is independent of and .
Proof. From (11), we haveApplying the integral mean value theorem, we know that there exists such thatUsing conditions (2) and (3) and differential mean value theorem, we know that there exists such thatThis means the proof of Lemma 1 is completed.
Theorem 1. Let and denote the numerical solution and true solution of problem (1) at grid point , respectively. Then, the convergence inequalityholds when is small enough and the conditions of Lemma 1 are satisfied. This means the numerical method (7) is convergent of the first order.
Proof. From (10), we can obtain thatWhen is small enough, i.e.,we havewhereBy using the discrete analogue of Gronwall’s inequality (see Theorem 2 in [22]), it follows thatAccording to Lemma 1, when is small enough, we have , andHence, we obtainwhere we have used inequality (22) and for . Therefore, substituting (22)–(24) into (21) leads towhich means the method is convergent of the first order.
4. Numerical Experiments
In the section, we utilize the following example to verify the theoretical results obtained in the previous section. Here, we will give error estimates and convergence rates for the numerical scheme.
Example 1. Consider the following impulsive fractional differential equation:Because it is difficult to obtain the true solution of the equation, we take the numerical solution as the true solution at with . The numerical scheme (7) is solved by using the Newton iteration method with as an initial value. We iteratively compute untilThe norm of the global error is denoted asThen, the convergence order of the numerical method isWhen takes different values, the error and convergence order of numerical method (7) are shown in Table 1.
Table 1 shows that the numerical method (7) is convergent of the first order, which supports the convergence estimate of Theorem 1.
In Figure 1, we can see that the numerical solution is discontinuous due to the existence of the impulses.
5. Conclusion
In this paper, we focus on the numerical solution of impulsive fractional differential equations. For a class of nonlinear impulsive fractional differential equations, the implicit Euler method is adapted for solving the problem. After careful convergence analysis, we prove that the method is convergent of the first order. For future work, we will study the higher-order methods for solving impulsive fractional differential equations and analyze their convergence.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Acknowledgments
This work was supported by the NSF of China (11571291).