An hp-version spectral collocation method for multi-term nonlinear fractional initial value problems with variable-order fractional derivatives

Abstract

A general class of multi-term nonlinear fractional initial value problems involving variable-order fractional derivatives are considered. Some sufficient conditions for the existence and uniqueness of exact solution are established. An hp-version spectral collocation method is presented for solving the problem in numerical frames. It employs the shifted Legendre-Gauss interpolations to conquer the influence of the nonlinear term and the variable-order derivatives. The most remarkable feature of the method is its capability to achieve higher accuracy by refining the mesh and/or increasing the degree of the polynomial. The rigorous error estimates are derived for the problem with smooth solutions on arbitrary meshes and weakly singular solutions on quasi-uniform meshes. Numerical results are given to support the theoretical conclusions.

Keywords:

• Spectral collocation method
• variable fractional order
• initial value problem
• convergence analysis

2010 Mathematics Subject Classifications:

• 65L60
• 41A05
• 41A10
• 41A25

Acknowledgments

The authors sincerely thank the editors and reviewers for their valuable suggestions and useful comments that have led to the present improved version of the original manuscript.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by the National Key R&D Program of China [grant number 2017YFC1405600], the National Natural Science Foundation of China [grant number 11701124].