Reproducing Kernel Method for Solving Nonlinear Fractional Fredholm Integrodifferential Equation

This article is devoted to both theoretical and numerical studies of nonlinear fractional Fredholm integrodifferential equations. In this paper, we implement the reproducing kernel method (RKM) to approximate the solution of nonlinear fractional Fredholm integrodifferential equations. Numerical results demonstrate the accuracy of the present algorithm. In addition, we prove the existence of the solution of the nonlinear fractional Fredholm integrodifferential equation. Uniformly convergence of the approximate solution produced by the RKM to the exact solution is proven.


Introduction
Fractional Fredholm integrodifferential equations have various applications in sciences and engineering.Most of these problems cannot be solved analytically, and hence finding accurate numerical solution for these problems will be very useful.Wazwaz [1,2] studied the Fredholm integral equations of the form where  and  are constants,  is a parameter, () is the data function, (, ) is the kernel of the integral equation, and () is the unknown function that will determined.In this paper, we study the generalization of the above problem of the form    () =  () +  ∫    (, )   () , 0 <  ≤ 1 (2) subject to Note that   in (2) is in the Caputo derivative.Equation ( 2) is called the nonlinear fractional Fredholm integrodifferential equations of the second kind characterized by the occurrence of the unknown function () inside and outside the integral sign.To homogenize the initial condition, we assume () = () −  0 .Then, subject to In the following definition and theorem, we write the definition of Caputo derivative as well as the power rule which we are used in this paper.For more details on the geometric and physical interpretation for Caputo fractional derivatives, see [3].

Complexity
Definition .For  to be the smallest integer that exceeds , the Caputo fractional derivatives of order  > 0 is defined as Theorem 2. e Caputo fractional derivative of the power function satisfies The reproducing kernel Hilbert space method is a useful numerical technique to solve nonlinear problems [4][5][6].The reproducing kernel is given by this definition.
The second condition is called the reproducing property and a Hilbert space which possesses a reproducing kernel is called a reproducing kernel Hilbert space (RKHS).More details can be in [7][8][9][10][11][12][13][14].A description of the RKM for discretization of the linear fractional Fredholm integrodifferential equations problem (4)-( 5) is presented in Section 2. In Section 3, we study the nonlinear fractional Fredholm integrodifferential equations.Several numerical examples and conclusions are discussed in Section 4. Conclusions and closing remarks are given in Section 5.

Analysis of RKHSM for Linear Fractional Fredholm Integrodifferential Equations
In this section, we discuss how to solve the following linear fractional Fredholm integrodifferential equation using RKHSM: subject to In order to solve problem ( 8)-( 9), we construct the kernel Hilbert spaces The inner product in  2 2 [, ] is defined as and the norm ‖‖  2 2 [,] is given by where , V ∈ In this case, (, ) is given by where Proof.Using the integration by parts, one can get Since () and (, ) ∈ Thus, where  is the Dirac-delta function and Since the characteristic equation of ( 2 / 2 )(, ) = ( − ) is  2 = 0 and its characteristic value is  = 0 with 2 multiplicity roots, we write (, ) as Using conditions (18) and ( 22)-(25), we get the following system of equations: We solved the last system using Mathematica to get  are absolutely continuous real value f unctions, The inner product in  1 2 [, ] is defined as and the norm ‖‖  1 2 [,] is given by where , V ∈ In this case, (, ) is given by Hence, Then, Now, we present how to solve problem ( 8)-( 9) using the reproducing kernel method.Let where (  ()) =   () −  ∫  0 (, )(() −  0 ) and  * is the adjoint operator of .Using Gram-Schmidt orthonormalization to generate orthonormal set of functions {  ()} ∞ =1 where and   are coefficients of Gram-Schmidt orthonormalization.
In the next theorem, we show the existence of the solution of Problem ( 8)- (9).

Conclusions and Closing Remarks
In this paper, we investigate the nonlinear fractional Fredholm integrodifferential equations where 0 <  ≤ 1.We implement the reproducing kernel method to approximate the solution of the proposed problem.Numerical results demonstrate the accuracy of the present algorithm.In addition, we prove the existence of the solution of the nonlinear fractional Fredholm integrodifferential equation.Uniformly convergence of the approximate solution produced by the RKM to the exact solution is proven.We noted the following: (i) The proposed method is very accurate.We get the exact solution in Examples 1 and 3.
(ii) Form Table 1, we note that the error is very small in Example 2.
(iii) Figure 1 shows that the approximate solution and the exact solution are identical.
(iv) The proposed method can be generalized for more models in Physics and Engineering.

Table 1 :
The error for Example 1.