Using Reproducing Kernel for Solving a Class of Fractional Order Integral Differential Equations

This paper is devoted to the numerical scheme for a class of fractional order integrodifferential equations by reproducing kernel interpolation collocation method with reproducing kernel function in the form of Jacobi polynomials. Reproducing kernel function in the form of Jacobi polynomials is established for the first time. It is implemented as a reproducing kernel method. The numerical solutions obtained by taking the different values of parameter are compared; Schmidt orthogonalization process is avoided. It is proved that this method is feasible and accurate through some numerical examples.


Introduction
In this paper, the reproducing kernel interpolation collocation method with reproducing kernel function in the form of Jacobi polynomials is applied to solve the following linear fractional integrodifferential equations (FIDEs): where 0 < μ ≤ 1, f n ðxÞ, n = 1, 2, and k ij ðx, tÞ, i, j = 1, 2, are given functions. D μ u n ðxÞ indicates that μ is the Caputo fractional derivative defined by u n ðxÞ, n = 1, 2.
Fractional order integrodifferential equation appears in the formulation process of applied science, such as physics and finance. However, it is very difficult to obtain the analytic solution of linear integrodifferential equations of fractional order, so many researchers try their best to study numerical solution of linear FIDEs and system of linear FIDEs in recent years [1][2][3][4][5]. Since the reproducing kernel method can not only obtain the exact solution in the form of series but also obtain the approximate solution with higher accuracy, the method has been widely used in linear and nonlinear problems, integral and differential equations, fractional partial differential equation, and so on [6][7][8][9][10][11][12][13][14][15]. But there are no scholars that use the reproducing kernel interpolation collocation method to solve the linear integrodifferential equations of fractional order. In this paper, linear integrodifferential equations of fractional order are solved by the reproducing kernel interpolation collocation method with reproducing kernel function in the form of Jacobi polynomials for the first time. The fractional derivative is described in the Caputo sense.
Reproducing kernel is shown in Figures 1-4.

Advances in Mathematical Physics
Its norm is the same as the norm of H n ½0, 1. It can easily be shown that H n ½0, 1 is a reproducing kernel Hilbert space. According to [18][19][20][21][22], the reproducing kernel of Definition 5. The inner product space is defined as Its inner product and norm are defined by It is easy to verify that H n ½0, 1 ⊕ H n ½0, 1 is a Hilbert space with the definition of inner product (13). Similarly, L 2 ½0, 1 ⊕ L 2 ½0, 1 is also a Hilbert space.

The Reproducing Kernel Interpolation Collocation Method
To solve equation (1), let So, equation (1) can be turned into where L = l 11 l 12 The operator L : Putting Theorem 6. For each fixed n, fΨ ij g ðn,2Þ ð1,1Þ is linearly independent in H n ½0, 1 ⊕ H n ½0, 1.
Since equation (1) has a unique solution, it follows that UðXÞ = 0.
The exact solution of equation (1) can be expressed as and truncating the infinite series of the analytic solution, we obtain the approximate solution of equation (1).
be the exact solution of equation (1), U m be the approximate solution of U, then U m converges uniformly to U. Proof. Similarly, If we can obtain the coefficients of each Ψ ij ðxÞ, the approximate solution U m ðxÞ can be obtained as well. Using Ψ ij ðxÞ to do the inner products with both sides of equation (24), we have Letting It is obvious that the inverse of A 2m exists by Theorem 6. So, we have Advances in Mathematical Physics

Numerical Experiment
Example 1. We consider the following linear integrodifferential equations of fractional order [5]: where the exact solution UðxÞ = ðx − x 3 , x 2 − xÞ T . The numerical results are given in Tables 1 and 2 Example 2. We consider the following linear integrodifferential equations of fractional order [5]: where the exact solution is UðxÞ = ðx 3 − x 2 , ð15/8Þx 2 Þ T . We obtain the numerical results which are given in Tables 3  and 4 Figure 15. Absolute errors of u 2 for m = 10, n = 3, μ = 3/4 are showed in Figure 16.

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Example 3. We consider the following linear integrodifferential equations of fractional order [4].

Conclusions and Remarks
In this paper, linear integrodifferential equations of fractional order have been solved by the reproducing kernel interpolation collocation method with reproducing kernel function in the form of Jacobi polynomials for the first time. Comparisons are made between the approximate and exact solutions. We verify the feasibility of this method by selecting different

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.