Abstract
This paper derives a computationally efficient and fast-running solver for the approximate solution of fractional differential equations with impulsive effects. In this connection, for approximating the fractional-order integral operator, a B-spline version of interpolation by corresponding equal mesh points is adopted. An illustrative example illustrates the accuracy of the new solver results as compared with those of the previous study. The proposed solver’s performance is evaluated by the fractional Rössler and susceptible-exposed-infectious impulsive systems. Moreover, the effect of impulsive behaviors is shown for various values of impulsive.
1. Introduction
The impulsive differential equations (IDEs) are mostly investigated systems together with short-time perturbations [1–4]. Impulsive control systems have been studied in many fields such as economics [5], chemostat [6], population ecology [7, 8], engineering [9], and neural networks [10, 11]. Many theoretical and numerical researchers have investigated IDEs in many studies. In [12–15], the existence and uniqueness theorems on IDEs have been analyzed. In addition, analytical and numerical solutions of this kind of equation have been investigated in [16–21] and etc.
Nowadays, one of the most famous branches of mathematical science is the fractional calculus with arbitrary fractional order [22]. The fractional calculus is applied to the model of many phenomena including control [23], mechanics [24], physics [25–27], stock market [28], electronics [29], biology [30], and epidemiology [31, 32]. Recently, fractional impulsive differential equations (FIDEs) are considered in simulations of many systems including chaotic and hyperchaotic systems [33–35], control [36], and neural networks [37]. The existence of the solutions of FIDEs is studied in [38] by using the fixed point method. The existence of solutions for FIDEs with the integral jump and antiperiodic conditions is investigated in [39]. Furthermore, the existence of solutions of these equations is analyzed through a global bifurcation approach in [40]. The existence and stability results are presented in [41].
To the best of the author’s knowledge, developing a fast-running solver requires FIDE up to date. This motivates our interest to designate an accurate computational technique for solving the following FIDE: where , , , where every satisfies , and plus is jointly continuous function. Moreover, , and , where denotes the impulsive interval. Furthermore, and indicate the left and right limits of at , respectively.
Throughout this paper, we do choose the Riemann-Liouville fractional integral [42] and fractional derivative in the Caputo sense [43, 44] which are formulated as where , and . In addition, the unknown function, , is continuously differentiable times.
The rest of the paper is arranged as follows. Section 2 suggests an implicit numerical technique, by using base spline interpolation for discretizing the FIDE. Section 3 investigates the performance and accuracy of the new solver by analysing the fractional impulsive Rössler and SEI systems. To sum up, Section 4 proffers the concluding remarks and statements.
2. Theoretical Argument
The proposed benefits of this section are twofold: (1)It gives a fractional order approximation of the integral nonlocal operators(2)It provides an accurate and computationally efficient technique for solving FIDE (1)
Thereafter, we consider that , , and means the uniform step size, and
Proposition 1. Assume that be a function, and where . The approximation of the nonlocal integral, , using the B-spline interpolation can be stated as follows: where In addition, the truncation error of (3) is
Proof. The approximation function, , in , by considering the B-spline interpolation is stated as
Substituting (6) into (2), we obtain the time discretization form of (2) as follows:
After rearranging and simplifying the above equation, it leads to (3) where the coefficients are given by (4).
Subsequently, the B-spline interpolation polynomial satisfies
where and denote error function. Therefore, we have
In the rest of this section, we designate a fast-running technique for solving FIDE (1) by means of Proposition 1. FIDE (1) is able to state the following two equivalent equations with the same solutions: or
By using the presented approximation in Proposition 1, we get the following approximation:
Therefore, by replacing (12) with (10) (or (11)), the following equation derives where is given by (4). Due to the nonlinear source term , we have where
Ultimately, replacing in the righthand side of (13) yields
3. Numerical Application and Discussion
This section evaluates the accuracy and computational efficiency of the proposed numerical technique. To evaluate the computational impact of this solver, the mean absolute error (), where and represents the number of interior mesh points, and the convergence order () is considered evaluation criteria. All the computational results are implemented with MATLAB R2019a on an AMD Ryzen 7 5700 U @ 1.80 GHz machine. Furthermore, a comparison is made with the IM algorithm that was formulated and investigated in [45, 46].
Example 2. Let . Then, we get where and define the Lommel function as where defines the generalized hypergeometric function.
The performance of the presented method is described by in Example 2 which is shown in Table 1. Table 1 shows the values of , , and computational times of Equation (19) with and in the interval . The numerical results display the improved accuracy of the presented scheme compared to the IM scheme [45] in the viewpoint of the , , and computational times. Figure 1 depicts the curves of Equation (19) for with step size . The outcomes in Figure 1 and Table 1 show that the proposed scheme is more accurate and has less computational time than the IM scheme [45].
3.1. Application of the Suggested Solver
In this section, the performance of the suggested solver is investigated for FIDEs.
Application 3. The fractional Rössler system is stated as where and and .
In Figure 2, we plot the phase curves of the integer-order and fractional Rössler chaotic system (21) by means of the suggested scheme with initial conditions , , and for and , , and with step size and .
We can rewrite system (21) into the following system: where , , and .
Hence, the fractional impulsive control of chaotic system (22) is defined as where with initial conditions
System (24) with the nonfractional term, i.e., for , and fractional term was studied in [47–49].
In Figure 3, we depict the numerical approximations of system (24) by using the suggested method with the impulsive intervals , in the interval and step size for , , and . We can view the effects of the impulsive behaviors on this system for in these figures.
Application 4. Assume that the functions , , and denote susceptible, exposed, and infectious pests densities at time , respectively. Furthermore, the defines the death rate of exposed and infectious pests. The fractional susceptible-exposed-infectious (SEI) chaotic system is stated as where and . Moreover, grows logistically with a carrying capacity in the absence of and with an intrinsic birth rate constant .
In Figure 4, we plot the phase curves of the integer-order and fractional SEI system (26) by means of the suggested scheme with initial conditions , , and , plus , and for , , and with step size and .
We can rewrite system (26) into the following system: where , ,
Hence, the fractional impulsive control of chaotic system (27) is defined as where with initial conditions
System 24 with the nonfractional term, i.e., for , was studied in [50].
In Figure 5, we depict the numerical approximations of systems (26) and (29) by using the suggested method with the impulsive intervals , in the interval and step size for , , and . We can view the effects of the impulsive behaviors on this system in these figures.
4. Conclusion
In the framework of this study, an implicit numerical algorithm for computing the approximate solutions of fractional impulsive differential equations was presented. This numerical solver relies on the B-spline interpolation to reasonably approximate the nonlocal integral operators. An illustrative example showed the accuracy of the comparison of the results obtained by the IM scheme and proposed numerical technique. The results confirmed the superiority of the presented scheme. Then, the proposed algorithm for solving the fractional chaotic dynamic Rössler and SEI systems was applied, and the results were studied using phase figures. To top it all off, the fractional impulsive systems were approximated by the presented method, and the achievement results of the impulsive behavior were analyzed.
Data Availability
There is no underlying data supporting the results in this study.
Conflicts of Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.