Solutions of Volterra–Fredholm type fractional integro-differential equations in terms of shifted Gegenbauer wavelets compared with the solutions by Genocchi polynomial method

Abstract

This research introduces a novel numerical technique based on shifted Gegenbauer wavelets for solving Fredholm–Volterra fractional integro-differential equations (FVFIDEs), a class characterized by the presence of both Fredholm and Volterra integral parts. By assuming properties of the fractional derivative and applying the wavelet solution directly to the equation, the problem is transferred to finding the family of solutions of the system of algebraic equations, whose solutions are the coefficients of the series of wavelet solutions. The accuracy and efficiency of the Gegenbauer wavelet approach are primarily evaluated through a direct comparison against solutions generated using the Genocchi polynomials method for established test problems. The study demonstrates that the shifted Gegenbauer wavelet method provides precise and effective solutions, which were analyzed under varying resolution parameters and degrees of Gegenbauer polynomials.