Application of New Orthogonal Shifted Vieta-Lucas Polynomials in Solving Integro-Differential Equations: Methods and Results

This article presents an innovative method for solving linear and nonlinear integro-differential equations using Vieta-Lucas polynomials as basis functions, combined with the Galerkin method. Initially, these polynomials are transformed within an arbitrary interval, and their orthogonality is utilized to approximate each function in the equation. A key aspect of this study is the detailed expression of the weight function and orthogonality conditions of these polynomials across any interval. Leveraging these properties and the Galerkin method, the integro-differential equation is converted into a system of algebraic equations. Error estimation is thoroughly investigated through several lemmas and theorems, and the existence and uniqueness of the solution are proven. Finally, numerical tests are conducted using Maple software to validate the accuracy and effectiveness of the proposed method, with comparative analyses demonstrating its superiority over existing techniques.