New recursive approximations for variable-order fractional operators with applications

TL;DR

This paper introduces Laguerre spectral collocation methods for solving variable-order fractional differential equations, providing efficient calculations and demonstrating superior accuracy over existing approaches.

Contribution

The paper develops new recursive formulas for variable-order fractional operators and applies them to spectral collocation methods, enhancing numerical solution accuracy.

Findings

Spectral collocation methods outperform existing techniques in accuracy.

Derived recurrence relations enable efficient computation of fractional operators.

Methods are applicable to variable-order fractional initial value problems.

Abstract

To broaden the range of applicability of variable-order fractional differential models, reliable numerical approaches are needed to solve the model equation. In this paper, we develop Laguerre spectral collocation methods for solving variable-order fractional initial value problems on the half line. Specifically, we derive three-term recurrence relations to efficiently calculate the variable-order fractional integrals and derivatives of the modified generalized Laguerre polynomials, which lead to the corresponding fractional differentiation matrices that will be used to construct the collocation methods. Comparison with other existing methods shows the superior accuracy of the proposed spectral collocation methods.