A NEW SOLUTION METHOD FOR NONLINEAR FRACTIONAL INTEGRO-DIFFERENTIAL EQUATIONS

. The aim of this paper is to obtain approximate solution of a class of nonlinear fractional Fredholm integro-diﬀerential equations by means of sinc- collocation method which is not used for solving them in the literature before. The fractional derivatives are deﬁned in the Caputo sense often used in frac- tional calculus. The important feature of the present study is that obtained results are stated as two new theorems. The introduced method is tested on some nonlinear problems and it seems that the method is a very eﬃcient and powerful tool to obtain numerical solutions of nonlinear fractional integro- diﬀerential equations.


1.
Introduction. Fractional calculus is one of the most popular calculus types having a vast range of applications in many different area of scientific and engineering [18]. Fractional calculus and fractional differential equations are frequently used in the modeling of many sorts of scientific phenomena including image processing, earthquake engineering, physics, viscoelasticity, electrochemistry, diffusion processes, control theory, heat conduction, electricity, mechanics (see some references at the end, e.g., [2,7,15,17]).
Since finding the exact or analytical solutions of fractional order differential equations is not an easy task, several different numerical solution techniques have been developed for the approximate solutions of these types of equations. Some wellknown methods are summarized as follows, but not limited to: Adomian decomposition method [11,12], Taylor expansion method [5], fractional differential transform method [14,1] and homotopy perturbation method [19,13].
Since sinc methods give a much better rate of convergence and more efficient results than classical polynomial methods in the presence of singularities [8], they are used for solving (systems of) the integer order integro-differential eqautions with Fredholm and Volterra type by several authors in the literature (see [9,10,21,4,22]).
The main purpose of the present paper is to obtain approximate solutions of the fractional order nonlinear Fredholm integro-differential equation with homogeneous boundary conditions, for m, s ∈ Z where C a D x is the Caputo fractional derivative operator, p(x), q(x), r(x) and f (x) are analytic functions in an open interval (a, b) and may be singular in a or b or both. Also K(x, t) is analytic function, λ is a parameter, a and b are real constant.
The rest of this paper is organized as follows. In section 2, we have given some difinition and theorems for fractional calculus and sinc-collocation method. In section 3, we present two new theorems and sinc-collocation method to obtain an approximate solution of a class of nonlinear fractional integro-differential equation. In section 4, some test problems are given to show the ability of present method by using tables and graphics. Finally, in section 5, we have completed the paper with a conclusion.
2. Fractional derivative and sinc functions. In this section, we recall notations and definitions of the sinc function and Caputo fractional derivative, also we derive useful formulas that are important for this paper.
is called the Whittaker cardinal expansion of f whenever this series converges.
In general, approximations can be constructed for infinite, semi-infinite and finite intervals. To construct approximation on the interval (a, b) the conformal map is employed. This map carries D E the eye-shaped domain in the z-plane onto the infinite strip D S A NEW SOLUTION METHOD FOR NFIDE.

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The basis functions on the interval (a, b) are derived from the composite translated sinc functions The sinc grid points z k ∈ (a, b) in D E will be denoted by x k because they are real. For the evenly spaced nodes {kh} ∞ k=−∞ on the real line, the image which corresponds to these nodes is denoted by and those on the boundary of D E satisfy For the term of fractional in 1, the infinite quadrature rule must be truncated to a finite sum. The following theorem indicates the conditions under which an exponential convergence results.
then the error bound for the quadrature rule 4 is The infinite sum in 4 is truncated with the use of 5 to arrive at the inequality 6. Making the selections is floor function that gives an integer part of the statement and M is the integer value which specifies the grid size, then We used these theorems to approximate the kernel integral and the arising integral in the formulation of the term fractional in 1.
3. The sinc-collocation method. We assume an approximate solution for y(x) in problem 1 by the finite expansion of sinc basis functions The unknown coefficients c k in 8 are determined by sinc-collocation method. For this purpose, the first and second derivatives of y n (x) are given by Similarly, α order derivative of y n (x) for 0 < α < 1 is given by the following theorem.
Theorem 3.1. If ξ is a conformal map for the interval [a, x], then α order derivative of y n (x) for 0 < α < 1 is given by where Proof of Theorem 3.1. If we use the definition of Caputo fractional derivative given in 2, it is written that Now we use quadrature rule given in 7 to compute the above integral which is divergent on the interval [a, x]. For this purpose, a conformal map and its inverse image that denotes the sinc grid points are given by Then, according to equality 7, we write This completes the proof.
Application of 7 to the kernel integral in 1 gives the following lemma.
where y k denotes an approximate value of y(t k ).
Replacing each term of 1 with the approximation given in 8, 9, 10, 12 and 13 multiplying the resulting equation by {(1/φ ) 2 } and setting x = x j , we obtain the following nonlinear system By using Lemma 2.10, we know that then we obtain the following theorem.
Now we define some notations to represent in the matrix-vector form for system 14. Let D(y) denotes a diagonal matrix whose diagonal elements are y(x −M ), y(x −M +1 ),..., y(x N ) and non-diagonal elements are zero, let G = R(x j ) and denote a matrix and also let I (i) denotes the matrices where D, G, E, I (0) , I (1) and I (2) are square matrices of order nn. Particularly, I (0) , I (1) and I (2) are the identity matrix, the skew-symmetric matrix and the symmetric matrix, respectively. In order to calculate unknown coefficients c k in nonlinear system 14, we rewrite this system by using the above notations in matrix-vector form as where .., c s N ) T Now we have nonlinear system of n equations in the n unknown coefficients given by 15. When it is solved by Newton's method, we can obtain the unknown coefficients c k that are necessary for approximate solution in 8.

Computational examples.
In this section, three problems that have homogeneous boundary conditions will be tested by using the present method via Mathe-matica10 on a personal computer. In all the examples, we take d = π/2, α = β = 1/2, N = M . Example 1. Consider nonlinear fractional Fredholm integro-differential equation in the following form subject to the homogeneous boundary conditions where f (x) = x 2 (e−x) 2 + e 9 252 (x−1)+ 2 Γ(2.5) x 1.5 − e Γ(1.5) x 0.5 −2 and K(x, t) = t 2 (x−1). The exact solution of this problem is y(x) = x(e−x). The numerical solutions which are obtained by using the present method for this problem are presented in Table 1 and Table 2. Additionally, the graphics of the exact and approximate solutions for different values of L and M are given in Figure 1 and Figure 2.
Example 2. Consider the following nonlinear fractional Fredholm integro-differential equation   Table 3 and Table 4. Additionally, the graphics of the exact and approximate solutions for different values of L and M are given in Figure 3 and Figure 4.
The exact solution of this problem is y(x) = x 2 (1 − x 2 ). The numerical solutions which are obtained by using the present method for this problem are presented in Table 5 and Table 6. Additionally, the graphics of the exact and approximate solutions for different values of L and M are given in Figure 5 and Figure 6.

5.
Conclusion. In this paper, sinc-collocation method is used to obtain approximate solution of a general nonlinear fractional integro differential equation with boundary conditions. In order to illustrate the accuracy and effective of the method, it is applied to some examples and obtained results are compared with the exact ones. The comparisons in table and graphical forms show that the approximate solutions converge the exact ones when it is increased that the number of sinc grid points N and the present method is a powerful tool for solving nonlinear fractional integro-differential equations with boundary conditions.