Application of fractional-order hybrid Chelyshkov functions for solving a general class of fractional integro-differential equations with weakly singular kernels

In recent years, various fractional-order basis functions have been developed to tackle different types of fractional problems. This paper introduces a new class of fractional-order hybrid functions constructed from block-pulse functions and Chelyshkov polynomials. By applying the Riemann–Liouville integral operator, we derive explicit results using the closed form of Chelyshkov polynomials. These results are then employed to develop a numerical scheme for solving a general class of fractional integro-differential equations with weakly singular kernels. The approach, combined with the essential properties of the Caputo derivative and the Riemann–Liouville operator, leads to the definition of remainders associated with the main problem. By selecting suitable collocation points, the problem is transformed into a solvable system of equations. An error bound is established for the proposed approximation, and the effectiveness of the method is demonstrated through several illustrative examples.