In this article, the numerical solution of the mixed Volterra–Fredholm integro-differential equations of multi-fractional order less than or equal to one in the Caputo sense (V-FIFDEs) under the initial conditions is presented with powerful algorithms. The method is based upon the quadrature rule with the aid of finite difference approximation to Caputo derivative usage collocation points. For treatments, our technique converts the V-FIFDEs into algebraic equations with operational matrices, some of which have the symmetry property, which is simple for evaluating. Furthermore, numerical examples are presented to show the technique’s validity and usefulness as well comparisons with previous results. The majority of programs are performed using MATLAB v. 9.7.

This paper introduces a new computational method for solving high fractional-order multi-term linear Fredholm–Volterra integro-differential equations (F-VIFDEs) with mixed boundary conditions. The approach utilizes the Caputo fractional derivative and shifted Lucas polynomial collocation technique. The core of the method involves transforming the F-VIFDE and its constraints into a system of algebraic equations by computing generalized Lucas coefficients. This matrix-based formulation provides an efficient framework for numerical solutions. The accuracy of the method is validated through numerical examples, which compare the approximate solutions with exact solutions. A least-squares error minimization approach across the domain further ensures precision. The presented method is computationally robust and reproducible, with implementations available for broader research use. The results demonstrate its effectiveness and potential as a reliable tool for theoretical and applied problems in fields such as physics, engineering, and mathematical modeling.