A numerical framework based on piecewise Chebyshev cardinal functions for fractional integro-differential equations

In this study, we develop an operational matrix technique to address a set of fractional nonlinear integro-differential equations with the Caputo–Hadamard derivative. We utilize a family of the piecewise Chebyshev cardinal functions as basis functions in this regard. Some formulas are introduced for calculating the classical and Hadamard fractional integrals of these functions. In the established strategy, the fractional expression is approximated utilizing these piecewise functions. By employing the aforementioned operational matrices and leveraging the cardinal property of the basis functions, we solve an algebraic system to obtain the solution. Convergence is both analytically demonstrated and confirmed through numerical experiments. Additionally, we compare the results obtained using this method with those derived from the hat functions method.