A numerical approach for a category of piecewise fractional variational problems depending on an indefinite integral

The primary focus of this study is to introduce some kinds of piecewise fractional derivatives (PFDs). These derivatives are defined using fractional derivatives in both the Atangana-Baleanu and Caputo senses. They are considered to generate a novel collection of fractional variational problems that rely on an indefinite integral. A numerical method established on the piecewise Chebyshev cardinal functions (as a suitable family of basis functions for such situations) is utilized to solve these problems. To this end, some operational matrices for PFDs of the expressed cardinal functions are derived and used to generate the presented method. Using the proposed technique, solving the desired problems is converted into solving associated algebraic systems. The effectiveness of the procedure is checked by solving some illustrative examples.