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

Zhiyuan Li,Meichun Wang,Jing Pang,Yulan Wang

doi:10.1155/2020/8101843

Zhiyuan Li, Meichun Wang + Show 2 more

Open Access

https://doi.org/10.1155/2020/8101843

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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.

Highlights

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): 8 >>>>>>< Dμu1 ðxÞ + ð ð t0 t k11 ðx, tÞu1 ðtÞ k12 ðx, tÞu2 ðt Þdt = f ðxÞ
Fractional order integrodifferential equation appears in the formulation process of applied science, such as physics and finance
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]

Summary

Introduction

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):. Fractional order integrodifferential equation appears in the formulation process of applied science, such as physics and finance. 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]. There are no scholars that use the reproducing kernel interpolation collocation method to solve the linear integrodifferential equations of fractional order. 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.

Preliminaries

Reproducing Kernel Space

The Reproducing Kernel Interpolation Collocation Method

Numerical Experiment

Conclusions and Remarks