Numerical solution for a class of space-time fractional equation by the piecewise reproducing kernel method

ABSTRACT

Due to the non-locality of fractional derivative, the analytical solution and good approximate solution of fractional partial differential equations are usually difficult to get. Reproducing kernel space is a perfect space in studying this type of equations, however the numerical results of equations by using the traditional reproducing kernel method (RKM) isn't very good. Based on this problem, we present the piecewise technique in the reproducing kernel space to solve this type of equations. The focus of this paper is to verify the stability and high accuracy of the present method by comparing the absolute error with traditional RKM and study the effect on absolute error for different values of α. Furthermore, we can study the distribution of entire space at a particular time period. Three numerical experiments are provided to verify the efficiency and stability of the proposed method. Meanwhile, it is tested by experiments that the change of the value of α has little effect on its accuracy.

KEYWORDS:

• Piecewise technique
• reproducing kernel space
• space-time fractional equation
• Caputo fractional derivative
• approximate solution

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Partial differential equations

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This paper is supported by the Natural Science Foundation of Inner Mongolia (No. 2017MS0103), the National Natural Science Foundation of China (No. 11361037), Numerical analysis of graduate course construction project of Inner Mongolia University of Technology (No. KC2014001).