Convergence Analysis of Implicit Euler Method for a Class of Nonlinear Impulsive Fractional Differential Equations

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.

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. e numerical results also verify the correctness of the theoretical results.

Construction of Numerical Scheme
Consider the following impulsive fractional differential equations: (t, u(t)), t ∈ J ′ ≔ J\ t 1 , t 2 , . . . , t m , J ≔ [0, T], △u t k � I k u t k , △u ′ t k � I k u t k , k � 1, 2, . . . , m, where α ∈ (1, 2), β, c ∈ R are constants, and C 0 D α t u(t) is the α-order Caputo derivative of solution u(t) defined by (see [6][7][8] , and u(t + k ) � lim ε⟶0 + u(t k + ε) represent the left and right limits of u(t) at t � t k , and f: J × R ⟶ R and I k , I k , : R ⟶ R are continuous functions and satisfy the following conditions: where L 1 , L 2 , L 3 are nonnegative constants with moderate size. roughout this paper, let C(J, R) be the Banach space of all continuous functions from J into R with the norm . We also define PC(J, R) ≔ u: J ⟶ R, u ∈ C((t k , t k+1 ], R), k � 0, 1, . . . , m, and u(t + k ) exists, k � 1, 2, . . . m}. e space PC(J, R) is a Banach space equipped with the norm ‖u‖ pc ≔ sup |u(t)|: t ∈ J { }. In addition, due to the need of convergence analysis, for function u(t) ∈ PC(J, R), there are constants L 4 and L 5 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 h k � (t k+1 − t k )/N, N be a given positive integer, the grid points t k, By using the numerical solution u k,i instead of the true solution u(t k,i ) 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, Gru .. nwald-Letnikov fractional calculus, and Caputo fractional calculus (see [6][7][8]). e α-order Riemann-Liouville derivative of func- We have the following relationship between the Riemann-Liouville derivative and Caputo derivative: erefore, the fractional differential equations studied in the present paper only consider Caputo derivative.

Convergence Analysis
where u k,i and u(t k,i ) denote the numerical solution and true solution of problem (1) at grid point t k,i , respectively. en, for the error z k,i , we have where In order to obtain the convergence result of numerical method (7), we first prove the following lemma.  (4); then, the truncation error of the discrete scheme (7) satisfies R k,i ≤ Ch, k � 0, 1, . . . , m, i � 1, 2, . . . , N. (12) roughout the paper, C will denote a positive constant not necessarily the same at different places, which may depend on L j , j � 1, 2, . . . , 5, but is independent of h and N.
Proof. From (11), we have Applying the integral mean value theorem, we know that there exists ξ l,r ∈ (t l,r−1 , t l,r ) such that Using conditions (2) and (3) and differential mean value theorem, we know that there exists η l,r , ζ l,r ∈ (t l,r−1 , t l,r ) such that is means the proof of Lemma 1 is completed.

Theorem 1. Let u k,i and u(t k,i ) denote the numerical solution and true solution of problem (1) at grid point t k,i , respectively. en, the convergence inequality
holds when h is small enough and the conditions of Lemma 1 are satisfied. is means the numerical method (7) is convergent of the first order.
Proof. From (10), we can obtain that When h is small enough, i.e., we have a k,r z k,r , (19) where By using the discrete analogue of Gronwall's inequality (see eorem 2 in [22]), it follows that Mathematical Problems in Engineering According to Lemma 1, when h is small enough, we have 0 < w k ≤ C, k � 0, 1, . . . , m, and Hence, we obtain where we have used inequality (22) and (1 + x) ≤ e x for x ≥ − 1. erefore, substituting (22)-(24) into (21) leads to which means the method is convergent of the first order. □

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 u k,i as the true solution u(t k,i ) at t � t k,i with N � 6400. e numerical scheme (7) is solved by using the Newton iteration method with u 0 k,i � 0 (k � 0, 1, . . . , m, i � 1, 2, . . . , N) as an initial value. We iteratively compute u j k,i until max 0≤k≤m, 1≤i≤N e L ∞ norm of the global error is denoted as en, the convergence order of the numerical method is When α 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 eorem 1.
In Figure 1, we can see that the numerical solution is discontinuous due to the existence of the impulses.

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.