A new method of solving the best approximate solution for a nonlinear fractional equation

Abstract

A new method of solving the best approximate solution for nonlinear fractional equations with smooth and nonsmooth solutions in reproducing kernel space is proposed in the paper. The nonlinear equation outlines some important equations, such as fractional diffusion-wave equation, nonlinear Klein–Gordon equation and time-fractional sine-Gordon equation. By constructing orthonormal bases in reproducing kernel space using Legendre orthonormal polynomials and Jacobi fractional orthonormal polynomials, the best approximate solution is obtained by searching the minimum of residue in the sense of $\Vert \cdot {\Vert }_C$. Numerical experiments verify that the method has higher accuracy.

Keywords:

• Nonlinear fractional equation
• reproducing kernel space
• the best approximate solution
• nonsmooth solution

YYYY AMS SUBJECT CLASSIFICATIONS:

• 65M12
• 65N12

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by Guangdong Basic and Applied Basic Research Foundation [grant no. 2022A1515010022] and GuangDong Ocean University [grant no. R20050].