A Lucas Operational Matrix Method for Solving Nonlinear Fractional Two-Dimensional Partial Volterra Integral Equations

This paper introduces a new numerical method for solving a class of two-dimensional fractional partial Volterra integral equations (2DFPVIEs). Our approach uses Lucas polynomials (LPs) to construct operational matrices (OMs) that effectively transform the complex fractional-order equations into a more manageable system of algebraic equations. This conversion facilitates efficient numerical solutions. We derive both 1D and 2D OMs for fractional integration, differentiation, and other operations, providing a comprehensive computational framework. The proposed method is validated through several illustrative examples. The results demonstrate its high accuracy and computational efficiency, as evidenced by the rapid convergence and low absolute errors (AEs) achieved.