This paper presents an efficient method for solving systems of fractional differential equations by using the fractional-order Chelyshkov functions (FCHFs). The fractional derivative and the fractional integral are considered in the Caputo sense and the Riemann–Liouville sense, respectively. The proposed method is based on using the operational matrix of fractional integration for FCHFs, together with the spectral collocation method to transform the system of fractional differential equations into a system of algebraic equations. The convergence of the presented method is demonstrated. The performance of the presented method is tested through various examples of systems of fractional differential equations. A comparison with some existing methods shows that the presented method can be successfully used to solve systems of fractional differential equations with accuracy and efficiency.

<abstract><p>In this paper, we present a new numerical method based on the fractional-order Chelyshkov functions (FCHFs) for solving fractional variational problems (FVPs) and fractional optimal control problems (FOCPs). The fractional derivatives are considered in the Caputo sense. The operational matrix of fractional integral for FCHFs, together with the Lagrange multiplier method, are used to reduce the fractional optimization problem into a system of algebraic equations. Some results concerning the approximation errors are discussed and the convergence of the presented method is also demonstrated. The performance of the introduced method is tested through several examples. Some comparisons with recent numerical methods are introduced to show the accuracy and effectiveness of the presented method.</p></abstract>