The Fractional Residual Power Series Method for Solving a System of Linear Fractional Fredholm Integro-differential Equations

In this manuscript, the fractional residual power series (FRPS) method is employed in solving a system of linear fractional Fredholm integro-differential equations. The signiﬁcant role of this system in various ﬁelds has attracted the attention of researchers for a decade. The deﬁnition of fractional derivative here is described in the Caputo sense. The proposed method relies on the generalized Taylor series expansion as well as the fact that the fractional derivative of stationary is zero. The process starts by constructing a residual function by supposing the ﬁnite order of an approximate power series solution that prescribes the initial conditions. Then, utilizing some conditions, the residual functions are converted to a linear system for the power series coefﬁcients. Solving the linear system reveals the coefﬁcients of the fractional power series solution. Finally, by substituting these coefﬁcients into the supposed form of a solution, the approximate fractional power series solutions are derived. This technique has the advantage of being able to be applied directly to the problem and spending less time on computation. It is not only an easy method for implementation of the problem, but also provides productive results after a few iterations. Some problems with known solutions emphasize the procedure’s simplicity and reliability. Moreover, the obtained exact solution demonstrated the efﬁciency and accuracy of the presented method.


Introduction
Fractional calculus came into existence from a question posed by L'Hôpital in a message to Leibniz in 1695 [20]. The essential question is about what a derivative of order 1/2 is. Since then, an investigation of scholars has been started and fractional derivatives have been defined in many aspects by several mathematicians such as Riemann-Liouville, Caputo, Grunwald-Letnikov, Hadamard. These developments lead to the widespread use of fractional calculus in various areas.
In recent years, a fractional integro-differential equation system, one of the fractional calculus applications, has become a trending topic for investigation as it has been used as a mathematical model for a variety of phenomena. Unfortunately, solving this sort of system and related topics is extremely difficult and challenging because most of them do not have a precise solution. Many strategies have been devised to help estimate the solution to defeating the system, such as the Adomian decomposition method (ADM) [19], Homotopy analysis [26], B-spin method [2], Sadik decomposition method [21], Genocchi polynomial method [18], Chebyshev spectral method [25], Chebyshev pseudo-spectral method [11], Chebyshev wavelet method [12], approximation method based on Taylor expansion [5] and reference therein.
The system of linear fractional Fredholm integro-differential equations is one of the most interesting integro-differential systems that has attracted attention from many academics. This type of system is crucial in the fields of research and engineering. Some techniques, such as ADM [23] and Homotopy Perturbation Method (HPM) [22], have been employed to 793 solve the system. Even if these approximation approaches are achieved, some restriction is required. As a result, researchers have been eager to find more practical, less constrained methods of searching for solutions.
The fractional residual power series (FRPS) method is a semi-analytic, powerful procedure based on the generalized Taylor series and a residual error function. Because no linearization, discretization, or perturbation is required, this method is efficient in addressing critical scientific and engineering models, such as the fractional Fisher equation [3], fractional stiff system [7], fractional Sharma-Tasso-Olever equation [17], fractional cancer tumor model [16], fractional fluids flow model [4], the fractional vibration model of large membranes [10], and fractional SIR Epidemic model [8]. Despite the fact that the FRPS method has been developed for various problems, there has not been much research on the fractional system of integro-differential equations.
This study aims to use the FRPS algorithm to solve a linear system of fractional Fredholm integro-differential equations (FIDEs), with initial condition u i (0) = c i,0 , i = 1, 2, 3, . . . , n, where D γi is fractional derivative operator in Caputo sense, g i (x) is a real function, κ is kernel, F i , R i are linear functions, a, b, λ i and γ i are constants, 0 < γ i < 1. This work is considered as an extension of [24]. The rest of the paper is managed as follows. Section 2 presents a basic definition of fractional integrals, fractional Caputo derivatives, and fractional power series. Section 3 describes the construction of fractional power series solutions for the system of fractional FIDEs. Illustrative examples are shown in the last section.
2 The basic concept of fractional integral, fractional derivative and fractional power series This section provides a fundamental idea of fractional calculus. The Riemann-Liouville fractional integral and the Caputo fractional derivative are presented. In addition, the primary notion and the facts related to fractional power series are mentioned.
Definition 2.2 [13] For n − 1 < γ < n, n ∈ N. The Caputo fractional derivative operator of order γ is defined by where the function φ(x) has absolutely continuous derivatives up to order n−1.
The operators D γ a and J γ a satisfy the following properties: for n−1 < γ < n, p > n − 1, and it is equal to zero otherwise.
One can note that D γ a and J γ a are linear operators, that is for

Definition 2.3 [14] A power series expansion at
for n − 1 < β ≤ n, n ∈ N and x ≤ x 0 , is called the fractional power series (FPS).
Theorem 2.1 [6] There are only three possibilities for the FPS ∞ m=0 a m (x − x 0 ) mβ , which are: 1. The series converges only for x = x 0 . That is; the radius of convergence equals zero.
2. The series converges for all x ≥ x 0 . That is; the radius of convergence equals ∞.
3. The series converges for x ∈ [x 0 , x 0 + R), for some positive real number R and diverges for x > x 0 + R. Here, R is the radius of convergence for the FPS.

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The Fractional Residual Power Series Method for Solving a System of Linear Fractional Fredholm Integro-differential Equations Theorem 2.2 [6] Suppose that u(x) has a FPS representation at x = x 0 of the form 3 Application of FRPS method to the system of fractional Fredholm integro-differential equations Assume that the FPS solution of the system (1) with initial conditions of u i (0) = c i at x = 0 has the following form: Using the initial condition, the solution can be written as To follow the fractional power series method, let's suppose the approximate solution of the system (1)is in the form a kthtruncated series: According to the RPS algorithm, the residual function is defined as Hence, the kth-residual function Res u i,k (x), for k = 1, 2, . . . , n, are given by As the results in [6], [14], [15], we have Res u i (x) = 0, and lim k→∞ Res u i,k (x) = Res u i (x), for any x ≥ 0. These implies that D mγi Res u i,k (x) = 0 for m = 0, 1, 2, . . . , k, i = 1, 2, . . . , n, and D nγi Res u i (0) = D nγi Res u i,k (0) = 0. Therefore, the coefficients of (3) can be found by solving the following equation: To determine the coefficient c i,1 in (3), one substitutes the 1-st residual power series approximate solution, into equation (4) with k = 1, to obtain . . . , c n + c n,1 . . . , c n + c n,1 τ γn Γ(1 + γ n ) )dτ.

Illustrative Examples
In this section, we verify the efficiency and accuracy of the proposed method by employing them to solve some fractional systems of linear FIDEs with a known solution. Some derived steps are simplified via the properties of the Gamma function.

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The Fractional Residual Power Series Method for Solving a System of Linear Fractional Fredholm Integro-differential Equations We substitute (15) into (13)- (14), then the 1st-residual functions are observed By the condition (5), it deduces that Res u 1 (0) = 0, Res v 1 (0) = 0 and conducts to a 1 = 0, b 1 = 0. For k = 2, we can write the 2nd-truncated FPS approximation in the form x 2 3 and the 2nd-residual functions are obtained Apply the conditions, D Res v 2 (0) = 0, the unknown coefficients are found, a 2 = 0, b 2 = 0. Next, suppose the 3rd-truncated FPS approximate solution in the form x.

Conclusions
For applying the FRPS method to the linear system of FIDEs, we start by assuming the approximate solution as a truncated fractional power series that satisfies the initial condition. Later, define a residual function by substituting the truncated fractional power series into the original system. An easily solved linear system of an algebraic equation is addressed by requiring conditions (5). One can note that solving the linear algebraic system may not need a standard method. Each equation can be solved separately. Finally, the unknown coefficients of the fractional series solution are determined and the approximate fractional power series solution is obtained. As demonstrated in the preceding examples, when the exact solution is a fractional polynomial function of degree up to kγ, the derived kγ approximate power series is exact.
In conclusion, the FRPS method is an analytical, powerful method for constructing the fractional power series solution of the linear system of FIDEs. When compared to the procedure in [23], [22] the proposed method is simpler, more convenient, and more accurate. Testing examples with known solutions demonstrate the efficiency and accuracy of this technique.

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The Fractional Residual Power Series Method for Solving a System of Linear Fractional Fredholm Integro-differential Equations