Abstract
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.
1. 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 , , and , , are given functions. indicates that is the Caputo fractional derivative defined by , .
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 [1–5]. 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 [6–15]. 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.
2. Preliminaries
Definition 1. The Caputo fractional derivative operator of order is defined as
Definition 2. Let be the real Hilbert spaces of functions . A function is called reproducing kernel for if (i) for all (ii) for all and all
2.1. The Shifted Jacobi Polynomials
The shifted Jacobi polynomials of degree is given [16] by where
The shifted Jacobi polynomials on the interval are orthogonal with the orthogonality condition which is where is a weight function, and
2.2. Reproducing Kernel Space
Definition 3. Let be the weighted inner product space of the shifted Jacobi polynomials on . The inner product and norm are defined as
Let . From [17–20], we can prove that is a reproducing kernel Hilbert space. Its reproducing kernel is where . Reproducing kernel is shown in Figures 1–4.
Definition 4. Let
Its norm is the same as the norm of . It can easily be shown that is a reproducing kernel Hilbert space. According to [18–22], the reproducing kernel of is
Definition 5. The inner product space is defined as Its inner product and norm are defined by
It is easy to verify that is a Hilbert space with the definition of inner product (13). Similarly, is also a Hilbert space.
3. The Reproducing Kernel Interpolation Collocation Method
To solve equation (1), let
So, equation (1) can be turned into where
The operator is a bounded linear operator.
Assuming that is dense on the interval , put , where is the adjoint operator of . From [23–25], we have
Putting
Theorem 6. For each fixed , is linearly independent in .
Proof. Letting
where , when . But , when take other value, .
When , we have . So,
So, . Similarly, we have .
Theorem 7. is complete in space in .
Proof. For each it follows that for every . Since equation (1) has a unique solution, it follows that .
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).
Theorem 8. Let be the exact solution of equation (1), be the approximate solution of , then converges uniformly to .
Proof. Similarly,
If we can obtain the coefficients of each , the approximate solution can be obtained as well. Using to do the inner products with both sides of equation (24), we have
Letting
It is obvious that the inverse of exists by Theorem 6. So, we have
4. Numerical Experiment
Example 1. We consider the following linear integrodifferential equations of fractional order [5]: where the exact solution . The numerical results are given in Tables 1 and 2, and the absolute errors of Example 1 for are plotted in Figures 5 and 6. Comparisons are made between the approximate and the exact solution for in Figures 7 and 8. Errors of and for are plotted in Figures 9 and 10.
Example 2. We consider the following linear integrodifferential equations of fractional order [5]: where the exact solution is . We obtain the numerical results which are given in Tables 3 and 4, and the absolute errors of Example 2 for are plotted in Figures 11 and 12. Comparisons are made between the approximate and exact solutions for in Figures 13 and 14. Absolute errors of for are plotted in Figure 15. Absolute errors of for are showed in Figure 16.
Example 3. We consider the following linear integrodifferential equations of fractional order [4]. (i)where , the exact solution is . By the proposed algorithm, we obtain the numerical results which are given in Tables 5 and 6, and the absolute errors of this example for are plotted in Figures 17 and 18. Comparisons are made between the approximate and exact solutions in Figures 19 and 20. When taking different values of , errors of and for are plotted in Figures 21–24. Figures 25 and 26 illustrate the approximate solutions of and using the present method for different values of which are in agreement with the exact solution. Figures 27 and 28 illustrate the approximate solutions for different values of compared with the exact solution in Ref. [4](ii)where , , the exact solution is . The absolute errors of this example for are given in Figures 29 and 30
5. 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 parameters . From all tables and figures, we obtain that the algorithm is remarkably accurate and effective.
All computations are performed by the Mathematica 7.0 software package.
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.
Acknowledgments
This paper is supported by the Natural Science Foundation of Inner Mongolia (2017MS0103) and the National Natural Science Foundation of China (11361037).