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.
Al-Sharif et al. (Thu,) studied this question.