Abstract
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.
1. 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 formsubject toNote 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 Then,subject toIn 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].
Definition 1. For to be the smallest integer that exceeds , the Caputo fractional derivatives of order is defined as
Theorem 2. The Caputo fractional derivative of the power function satisfies
The reproducing kernel Hilbert space method is a useful numerical technique to solve nonlinear problems [4–6]. The reproducing kernel is given by this definition.
Definition 3. Let . A function is a kernel of i if(i) for all ,(ii) for all and
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 [7–14]. 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.
2. 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 and in which every function satisfy the boundary conditions (9). Let The inner product in is defined as and the norm is given by where .
Theorem 4. There exists such that, for any and each fixed , we have In this case, is given by where
Proof. Using the integration by parts, one can get Since and , andThus, Since is a reproducing kernel of ,Thus,where is the Dirac-delta function andSince the characteristic equation of is and its characteristic value is with 2 multiplicity roots, we write asSince , we haveOn the other hand, integrate from to with respect to and let to getUsing conditions (18) and (22)-(25), we get the following system of equations: We solved the last system using Mathematica to getNext, we study the space Let The inner product in is defined as and the norm is given by where
Theorem 5. There exists such that, for any and each fixed , we have In this case, is given bywhere
Proof. Using the integration by parts, one can get Since is a reproducing kernel of ,Then,andSince the characteristic equation of is and its characteristic value is with 2 multiplicity roots, we write asSince , we haveHence,Then,Now, we present how to solve problem (8)-(9) using the reproducing kernel method. Let for . It is clear that is bounded. Let where and is the adjoint operator of . Using Gram-Schmidt orthonormalization to generate orthonormal set of functions whereand are coefficients of Gram-Schmidt orthonormalization. In the next theorem, we show the existence of the solution of Problem (8)-(9).
Theorem 6. If is dense on , then
Proof. First, we want to prove that is the complete system of and It is clear that for . Simple calculations implies thatFor each fixed , letThen,Since is dense on Since exists, . Thus, is the complete system of Second, we prove (47). Simple calculations imply that Let the approximate solution of problem (8)-(9) be given by
In the next theorem, we show the uniformly convergent of the to
Theorem 7. If and are given as in (47) and (52), then converges uniformly to
Proof. For any , Hence, From Theorem 6, one can see that converges uniformly to Hence, which implies that converges uniformly to
3. Analysis of RKHSM for Nonlinear Fractional Fredholm Integrodifferential Equations
In this section, we discuss how to solve the following the following problem using RKHSM:subject toLet We construct a homotopy as follows:where is an embedding parameter. For , we get a linear equation which can be solved by using RKHSM as we described in the pervious section. If , we turn out to be problem (56). Following the Homotopy Perturbation method [15], we expand the solution in terms of the Homotopy parameter asSubstitute (61) into (59) and equate the coefficients of the identical powers of to get the following system:
From (61), it is easy to see that the solution to problem (56)-(57) is given by We approximate the solution of problem (56)-(57) by
4. Numerical Results
In this section, we present three numerical examples to show the efficiency of the proposed method.
Example 1. Consider the following fractional Fredholm integrodifferential equation: subject towhere Using , the approximate solution is Thus, is the exact solution.
Example 2. Consider the following fractional Fredholm integrodifferential equation: subject towhere and is the two-parameter of Mittag-Leffler function. The exact solution is . Using , the approximate solution is given in Figure 1. The error is given in Table 1.
Example 3. Consider the following fractional Fredholm integrodifferential equation: subject to where and is the gamma function. The exact solution is . Using , the approximate solution is Hence, is the exact solution.
5. Conclusions and Closing Remarks
In this paper, we investigate the nonlinear fractional Fredholm integrodifferential equations where . 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.
Data Availability
The data and results used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. (D-001-363-1439). The authors, therefore, gratefully acknowledge the DSR technical and financial support.