Solutions of Linear and Nonlinear Fractional Fredholm Integro-Differential Equations

The present paper analyzes a class of first-order fractional Fredholm integro differential equations in terms of Caputo fractional derivative. In the literature, such kind of fractional integrodifferential equations have been solved using several numerical methods, while the exact solutions were not obtained. However, the exact solutions are obtained in this paper for various linear and nonlinear examples. It is shown that the exact solution of the linear problems is unique, while multiple exact solutions exist for the nonlinear ones. Moreover, the obtained results reduce to the classical ones in the relevant literature as the fractional order becomes unity. The obtained exact solutions can be further invested by other researchers to validate their numerical/approximation methods.


Introduction
The fractional calculus (FC) has gained observable interest in recent years due to its applications several fields [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. The FC has been also extended to integro-differential equations (FIDEs) as observed in the literature [15][16][17][18][19][20][21][22][23][24][25][26][27][28], where various numerical and analytical methods were applied to solve for approximate solutions. We are concerned here with fractional Fredholm integro-differential equations (FFIDEs) of first-order. Although important results were reported [15][16][17][18][19][20][21][22][23][24][25][26][27][28] for FIDEs, obtaining the exact solution of FFIDEs is not an easy task, even for simple equations as will be shown later. So, we consider in this paper the following class of FFIDEs: K(x, τ, u(τ )) dτ, 0 < α ≤ 1, (1.1) where h, λ, b 1 and b 2 are given constants, f (x) is a given continuous function on [b 1 , b 2 ]. The objective of this paper is to introduce a direct analytic approach for obtaining exact solutions for the class (1)(2). It will be shown that the solution is unique when K(x, τ, u(τ )) is a linear function in the unknown function u(τ ). In addition, it will be declared that multiple exact solutions exists when The Caputo definition is chosen as a fractional derivative in Eq. (1) and the structure of the paper is as follows. In section 2, we give the main aspects of the FC. In addition, a basic Lemma will be provided for the formal exact solution of the class (1-2). Sections 3 investigates the application of the present approach on several linear and nonlinear problems. Besides, the way of obtaining exact dual solution for the nonlinear case will be demonstrated in section 3. Moreover, it will be shown that the present exact solutions reduce to the classical ones as α → 1. Finally, section 5 outlines the conclusions.

Main aspects of FC
The Riemann-Liouville fractional integral of order α is defined as [1]: The Caputo's FD of order α of a function u(x) is defined by (2. 2) The J α 0 and C 0 D α x are related by: which is useful when solving FDEs/FIEs. A basic property of the J α 0 is J α 0 (x r ) = Γ(r + 1) Γ(α + r + 1) x α+r , r > −1.
The Mittag-Leffler function (MLF) of one-parameter is defined as , z ∈ C, (2.5) while the two-parameter MLF is given as The following properties are also hold: The analytic solution of the first-order FFIDE (1-2) is given by provided that the fractional integral of f (x), i.e., J α 0 (f (x)), exists and a is the constant given by Proof: The bounded integral involved in Eq. (1) can be assumed as a constant. Besides, we assume that such integral is given by the constant a as Operating with J α 0 on Eq. (1) and implementing (2), (5), and (14), it then follows Calculating J α 0 (1) from Eq. (6) at r = 0, we have J α 0 (1) = x α Γ(α+1) . Substituting this last result into Eq. (14) we obtain Eq. (11) which completes the proofs.
(3.24) Therefore, u(x) is finally given by and I is already defined by Eq. (37). The solution given by Eq. (39) reduces, as α → 1, to which is the corresponding solution for the classical form: Assuming that where a 4 and a 5 are constants. Following the same analysis of the previous examples, we obtain .
(3.55) Solving Eq. (68) for the constant a 7 , we get It can be seen from Eq. (71) that there are two different solutions for the present nonlinear example, the first one is given by while the second solution is In order to check these two solutions, we evaluate them as α → 1. In this case, we have from Eqs.
(69) that Hence, and (u 2 (x)) α→1 = − 9 2 x + 5x 2 , (3.62) The solutions (75) and (76) are the same obtained one in Ref. [29] for the classical nonlinear version u (x) = 10x − 5 + Furthermore, as the fractional order is unity, the results agree with to the corresponding classical problems. This study may deserve extensions to further FFIDEs of higher-order.

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Conflicts of Interest:
The author(s) declare that there are no conflicts of interest regarding the publication of this paper.