A Novel Algorithm for Solving Fredholm Integro-Differential Equations Using Lucas Polynomials

Fredholm integro-differential equations play a crucial role in mathematical modelling across various disciplines, including physics, biology, and finance. In this paper, Fredholm integro-differential equations are solved using the derivative of the Lucas polynomials in matrix form. The equation is first transformed into systems of nonlinear algebraic equations using the Lucas polynomials. The unknown parameters required for approximating the solution of Fredholm integro-differential equations are then determined using Gaussian elimination. The method has proven to be an active and dependable technique for solving the Fredholm integro-differential equation of any order by updating the matrix of Lucas polynomials. Additionally, the technique is successfully applied to a mixed Fredholm-Volterra integro differential equation demonstrating its versatility. Comparative analysis with some existing methods highlights the improved accuracy and efficiency of the proposed approach. Numerical experiments, including benchmark problems from the literature, confirm the validity and applicability of the technique, achieving lower error margins than conventional methods.