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<p>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.</p>

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A novel Computational Method using Lucas Polynomials for Fractional Fredholm–Volterra Integro-Differential Equations

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2025年12月27日

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  • Shabaz Jalil Mohammedfaeq,
  • Shokhan Ahmed Hamasalih,
  • Hiwa Abdullah Rasol,
  • Shazad Shawki Ahmed

摘要

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.