Hybrid functions for numerical solution of fractional Fredholm-Volterra functional integro-differential equations with proportional delays

In this work, we propose an efficient and accurate computational approach based on the Genocchi hybrid functions (GHFs) for solving a class of fractional Fredholm-Volterra functional integro-differential equations (FVFIDEs) with proportional delays. First, the Genocchi hybrid functions are constructed. Then, the operational matrix of the Caputo fractional derivative with some properties of the GHFs is employed to reduce the fractional FVFIDEs with proportional delays to systems of algebraic equations in terms of the unknown coefficients. Ultimately, the upper bound of error and convergence of approximate solution to exact solution are discussed. Moreover, we consider some examples to demonstrate the validity and applicability of our method.