A numerical method based on shifted fractional-order Chelyshkov functions for fractional delay integro-differential equations

This paper introduces a new operational matrix of the Riemann–Liouville fractional integral using shifted fractional-order Chelyshkov functions (SFCHFs) for solving fractional delay integro-differential equations. The operational matrix is derived using the Laplace transform and combined with the collocation method to reduce the fractional delay integro-differential equation to a system of algebraic equations, where the coefficients of the SFCHFs are treated as unknowns. The convergence of the proposed method is established, and an error estimate for the SFCHFs approximation is provided. The efficiency and accuracy of the introduced method are verified using different examples and comparisons are made with existing results from the literature. These numerical results confirm that the proposed method is effective and can achieve high accuracy using only a small number of SFCHF terms.