Damghan University Press Analytical and Numerical Solutions for Nonlinear Equations 3060-785X 9 1 2025 07 10 System of Volterra Fredholm Integro-Fractional Differential Equations: Application of Fibonacci Polynomials 89 101 1859 10.22128/ansne.2025.993.1135 EN Zahra Gilani Department of Mathematics, Faculty of Basic Science, Babol Noshirvani University of Technology, Babol, Iran Mohsen Alipour Department of Mathematics, Faculty of Basic Science, Babol Noshirvani University of Technology, Babol, Iran Sanaz Rivaz Department of Mathematics, Faculty of Basic Science, Babol Noshirvani University of Technology, Babol, Iran Journal Article 2025 04 18 In this paper, we introduce the Fibonacci polynomials (FPs) and approximate functions using them. Furthermore, several lemmas and corollaries present the properties of FPs. Also, we derive the Fibonacci polynomials operational matrix for the fractional derivative in the Caputo sense, which has not been undertaken before.  As applications of the Fibonacci polynomials operational matrix, we solve the system of Volterra Fredholm integro-fractional differential equations. In this scheme, we approximate one and two variable functions based on Fibonacci basis. Then by applying Fibonacci polynomials operational matrix, the system of Volterra Fredholm integro-fractional differential equations is reduced to a system of algebraic equations that is easily solvable with the help of a software (version 13 of the Mathematica software). The obtained results are in good agreement with the exact solutions and with those in literature. As anticipated, the solutions converge to classical solutions as the fractional derivative order approaches integer values. https://ansne.du.ac.ir/article_1859_207062d32665a511549c0cd2470fafb6.pdf