A Versatile Spectral Matrix Collocation Method for Tackling Bratu, Riccati, and Higher Order Linear and Nonlinear Differential Equations Containing Delay Variables

This study presents a numerical matrix collocation method for solving Bratu equations, Riccati equations, and high‐order linear/nonlinear differential equations with delay variables—all of which hold significant importance in the literature. Addressing these widely studied problems from a novel perspective, we develop a new technique that transforms each problem into an algebraic matrix equation based on Lucas polynomials. The proposed method constructs approximate solutions using Lucas polynomials with conventional ordering points, which form the foundational basis of our approach. For each problem type, we explore different solution scenarios and provide tailored computational strategies. Numerical results, presented in tables and figures, demonstrate consistency and effectiveness by comparing our method with existing approaches for various ranking points.