Derivation and Implementation of an Equally Spaced Numerical Scheme (ESNS) for Solving Nonlinear Problems Modeled Differential Equations

Abstract

This paper presents the development and comprehensive analysis of an Equally Spaced Numerical Scheme (ESNS) for solving nonlinear modeled oscillatory differential equations, which frequently arise in engineering and biological systems where closed-form analytical solutions are unavailable. The ESNS is constructed using a continuous power series approximation evaluated on uniformly distributed grid points, a strategy that significantly enhances computational simplicity while improving stability and convergence characteristics. The derivation employs interpolation and collocation techniques, yielding a linear multistep method with a continuous form that is subsequently discretized to produce an efficient step-by-step numerical solver. A rigorous theoretical investigation of the scheme is conducted, including determination of its order and error constant, demonstration of consistency and zero-stability, proof of convergence, and delineation of its absolute stability region, which confirms that the method is A-stable and therefore suitable for stiff oscillatory problems. The practical performance of the ESNS is evaluated through numerical simulations on three classes of oscillatory differential equations, including a susceptible-infected-recovered (SIR) epidemiological model that computes disease transmission dynamics within a population over time. The numerical results are systematically compared with established methods, and both tabular and graphical comparisons demonstrate that the ESNS yields approximate solutions of remarkably high accuracy, characterized by extremely small error magnitudes. The findings confirm that the ESNS is a reliable, computationally efficient and robust numerical tool for solving nonlinear oscillatory initial value problems, offering a valuable alternative for researchers and practitioners in applied mathematics and related disciplines.