This paper focuses on computational technique to solve linear systems of Volterra integro-fractional differential equations (LSVIFDEs) in the Caputo sense for all fractional order $\mathrm{lins}\mathrm{in}\left(0,1\right]$ using two and three order block-by-block approach with explicit finite difference approximation. With this method, we aim to use an appropriate process to transform our problem into an analogous piecewise iterative linear algebraic system. Moreover, algorithms for treating LSVIFDEs using the above process have been developed, in order to express these solutions. In addition, numerical examples for our scheme are presented based on various kernels, including symmetry kernel and other sorts of separate kernels, are used to illustrate the validity, effectiveness and applicability of the suggested approach. Consequently, comparisons are performed with exact results using this technique, to communicate these approaches most general programs are written in Python V 3.8.8 software $\left(2021\right)$.

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.