A fifth order block methods for solving second-order stiff ordinary differential equations using trigonometric functions and polynomial function as the basis function

The numerical solution of second-order ordinary differential equations (ODEs) is examined in this work through a four-step linear multistep method. It employs a combination of trigonometric and polynomial functions as the approximate solution to the general second-order ordinary differential equations (ODEs). The method was developed using interpolation and collocation techniques. This methodology involves interpolating the basis function at two points and subsequently collocating it across all points, ensuring a robust scheme. To assess its efficacy, we solved three initial value problems (IVPs) associated with stiff differential equations. Through this examination, we established the method’s core characteristics: consistency, zero stability, and consequently convergence. This thorough analysis demonstrates its reliability and suitability for resolving second-order ordinary differential equations. The comparison of our newly derived block method against existing approaches reveals its superiority. Our method’s performance, evaluated across a spectrum of stiff second-order ordinary differential equations, surpasses the outcomes obtained from established authors. This substantiates its efficiency and effectiveness in addressing these mathematical challenges. This study marks a significant advancement by introducing a robust approach that not only accurately solves second-order ordinary differential equations but also streamlines the computational process. By integrating trigonometric and polynomial functions and leveraging interpolation and collocation techniques, our method stands out for its accuracy, stability, and convergence properties, offering a promising avenue for future research in this domain.