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December 11, 2025Open Access

Fractional-order Chelyshkov collocation method for solving variable-order fractional differential equations

Key Points

To present a numerical method for solving variable-order fractional differential equations (VO-FDEs).
Utilized fractional-order Chelyshkov functions to construct numerical methods.
Considered derivative and integral in Caputo and Riemann-Liouville senses respectively.
Transformed VO-FDEs into algebraic equations using spectral collocation method.
Demonstrated convergence of the proposed method through various examples.
The method shows high efficiency and accuracy compared to recent numerical methods.
Successful applications to various examples of VO-FDEs were achieved.

Abstract

Abstract This paper presents an efficient numerical method for solving variable-order fractional differential equations (VO-FDEs) by using the fractional-order Chelyshkov functions (FCHFs). The variable-order fractional derivative and the variable-order fractional integral are considered in the Caputo sense and the Riemann-Liouville sense, respectively. The exact formula for the Riemann-Liouville variable-order fractional integral of the FCHFs is obtained. This value, together with the spectral collocation method are used to transform the VO-FDE into a system of algebraic equations. The convergence of the proposed method is demonstrated. The performance of the introduced method is tested through various examples of VO-FDEs. A comparison with recent numerical methods shows that the proposed method can be successfully used to solve VO-FDEs with accuracy and efficiency.

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