Numerical Simulation of Coupled Fractional Differential-Integral Equations Utilizing the Second Kind Chebyshev Wavelets

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Introduction
Fractional diferential-integral belongs to the feld of applied mathematics and can handle integrals and derivatives of any order. It has the advantage of being a practical tool for reasonably explaining the memory and genetic quality of various materials and processes. With the development of natural science and social economy, it is found that integer order diferential equation cannot be used to solve complex problems, but these problems can be solved by fractional calculus equation [1]. Terefore, diferential-integral equation plays a very important role in various problems of natural science and related felds of engineering technology [2]. Tere are diferent types of integral equations, such as Fredholm integral equations, Volterra integral equations, and Volterra-Fredholm integral equations, depending on the structure of the integral [3][4][5]. Nonlinear integral equations have been widely studied in many diferent disciplines, such as vehicular trafc, biology, optimal control theory, economics. [6,7]. In this paper, numerical calculations are carried out for a class of system of coupled fractional Fredholm diferential-integral equations. Te general form of the research question is as follows: where p 1 , p 2 , q 1 , q 2 > 1, and the initial conditions are given as follows: where g 1 , g 2 ∈ L 2 ([0, 1]), k 11 , k 12 , k 21 , k 22 ∈ L 2 ([0, 1] × [0, 1]) are given functions, Y 1 (t), Y 2 (t) are the solutions to be determined, D α t , D β t refer to fractional derivatives in the Caputo sense, and p, q are positive integers.
In recent years, diferent types of numerical methods have been proposed for solving fractional diferential and integral equations. Tese methods mainly include the variation of parameters method [8], ADM [9,10], VIM and HPM [11], CAS wavelet [12], and Taylor expansion method [13]. Te wavelet methods stand out among these methods and have been adopted in various scientifc or engineering applications. Te wavelets method makes the accurate representation of diferent functions and operators and the establishment of the connection with fast numerical algorithms possible [14]. Wavelet methods, including CAS wavelet [15], Legendre wavelet [16], Haar wavelet [17], Chebyshev wavelet [18], and others, are widely used in solving linear or nonlinear diferential equations, integral equations, and integro-diferential equations. In [12], the numerical method of CAS wavelet is applied to solve the fractional integral and diferential equations, and the numerical results are compared with the analytical results. In [19], the second kind of Chebyshev wavelet method is utilized to obtain the numerical solutions of fractional nonlinear Fredholm integral-diferential equations. In recent years, wavelet methods develop rapidly [20][21][22]. At present, some scholars have done in-depth research on the time delay and the existence and controllability of equations for fractional diferential-integral system [23][24][25][26][27]. In view of the problem, the huge algebraic equations will lead to considerable computational complexity and large data storage requirements in the calculation process. Because of the structural redundancy of the second kind of Chebyshev wavelet, the computational complexity of the algebraic system can be reduced. In order to more efectively solve the coupled fractional diferential-integral equations [28,29], the second kind of Chebyshev wavelet method is introduced in this paper, and the original problem can be reduced to a system of linear algebraic equations, which can be easily solved by some mathematical techniques. In the study, the main system of fractional Fredholm integral-diferential equations is discussed in detail by the second kind Chebyshev wavelet, and the convergence analysis of the system is investigated.
Te main structure of the article is as follows: in Section 2, some necessary defnitions of the fractional integral-differential are introduced. In Section 3, the second kind Chebyshev wavelet and its operational matrix of the fractional integration are derived. In Section 4, the main computing steps are described in detail. In Section 5, the convergence analysis of the system is investigated. In Section 6, the efectiveness of the proposed method is verifed by several test problems. Finally, a conclusion is drawn in Section 7.

Fractional Calculus
In the section, some necessary defnitions and mathematical preliminaries about the fractional integral-diferential theory are given [30,31]. Defnition 1. Riemann-Liouville fractional integral operator I α of order α defned in the region [a, b] is given by (3) Defnition 2. Caputo defnition of a fractional diferential operator is given by Caputo fractional diferential operator has some basic properties: and

Fractional Integration Operational Matrix.
In the interval [0, 1), a m− set of block-pulse functions (BPFs) is defned Te functions B i (t) are disjoint and orthogonal Te second kind Chebyshev wavelet vector can be expressed by an m-term block-pulse functions as In the previous study, the fractional integral operational matrix of block-pulse functions can be expressed as where and Te fractional integration of the second kind Chebyshev wavelet can be given as where P α m×m is called fractional integral operational matrix of the second kind Chebyshev wavelet, and it can be given by For more details, see [32].

Numerical Implementation
In the section, the second kind Chebyshev wavelet method is presented to numerically solve the system of nonlinear fractional Fredholm integral-diferential [34][35][36] equation (1). Firstly, the functions k 11 (x, t), k 12 (x, t), k 21 (x, t) and k 22 (x, t) can be approximated by the second kind Chebyshev as Mathematical Problems in Engineering where and Moreover, For simplifcation, assuming δ i � c j � 0, then by equations (2), (7), (24), and (28), we can obtain According to the property of BPFs, we have where . Trough induction, we can easily obtain Ten, we have 4 Mathematical Problems in Engineering and Mathematical Problems in Engineering 5 Ten, Substituting equations (28), (29), and (43) into (1), we have Obviously, (39) is a system of linear algebraic equations. Dispersing the unknown variable t using Matlab software, we can obtain [r 0 , r 1 , . . . , r m−1 ] and [s 0 , s 1 , . . . , s m−1 ], and then, by (29), the solutions of (1) can be obtained.

Convergence Analysis
In this section, we focus on the convergence analysis of the system. In fact, by (9), Y(t) converge (in the mean) to Y(t) as k approaches to ∞. If the function Y(t) is several times diferentiable, we can bound the error, as established by the following theorem.

Mathematical Problems in Engineering
where Y 1 (t), Y 2 (t) are the m times continuous diferentiable analytical functions of the system, and Y 1 (t), Y 2 (t) are the approximate functions of the system, respectively. Ten, we have and

Numerical Experiments
In order to verify the efectiveness of the proposed method via the following examples, we defned the 2-norm error as where Y i (t) is the analytical solution of this system, and Y i (t) is the approximate solution of this system.
Test Problem 1. Considering the following nonlinear system of fractional-order Fredholm diferential-integral equations with the zero initial conditions: Te analytical solutions of the system are Y 1 (t) � t 3 and Y 2 (t) � −t 3 .
Te numerical results of the system with some value of k, M are shown in Tables 1 and 2. From Tables 1 and 2, it can be concluded that the numerical solutions approach to the analytical solutions as k, M increases. Tables 3 and 4 show the absolute error tables of Y 1 and Y 2 , respectively. It can be        seen from the error that the error decreases with the increase of the discrete term.
Test Problem 2. Considering the following nonlinear system of fractional-order Fredholm diferential-integral equations with the initial conditions: Te analytical solutions of the system are Y 1 (t) � t 3/2 and Y 2 (t) � t 2 . When M � 4, k � 3, 4, 5, the obtained numerical results are shown in Tables 5 and 6. From Tables 5   and 6, it can be concluded that the numerical solutions approach to the analytical solutions as k, M increases.
Trough the analysis of the above two experimental examples, it can be concluded that the numerical results approach to the analytical results as m grows. Te results show that the absolute error between the analytical and numerical results can achieve a high convergence precision. Figures 1-3 show the absolute error of Y 1 and Y 2 under diferent fractional orders. It can be seen from the fgure that the error is increasing with the increase of the order, but it is not an absolute increase. Due to some calculation factors, the error will also fuctuate to a certain extent.

Conclusions
In this paper, in view of the problem that the huge algebraic equations will lead to considerable computational complexity and large data storage requirements in the calculation process, the second kind Chebyshev wavelet method is applied to obtain the numerical solutions of nonlinear system of fractional Fredholm integral-diferential equations. Using this method, the system of diferential-integral equations has been reduced to a system of algebraic equations. Te results show that the second Chebyshev wavelet method is used; the accuracy is improved with the increase of K and M values. In addition, the convergence analysis of the system based on the second kind of Chebyshev wavelet is studied. Te analysis is carried out by several numerical experiments, and the absolute error values under diferent fractional orders are given, which proves the superiority and efectiveness of the proposed method. It provides support for improving the precision and reliability of the system.

Data Availability
All data generated or analysed during this study are included in this published article.