ANALYTICAL AND NUMERICAL APPROACHES TO NONLINEAR DIFFERENTIAL EQUATIONS IN MODERN PHYSICAL MODELLING: CHALLENGES AND INNOVATIONS

Abstract

This article provides a comprehensive examination of modern analytical and numerical methods used in solving nonlinear differential equations, which remain central to modeling complex physical systems. The study critically evaluates classical techniques such as perturbation theory and Fourier analysis, as well as contemporary computational algorithms including finite difference methods, Runge-Kutta schemes, and spectral techniques. Special attention is given to their implementation in modeling phenomena such as wave propagation, thermal diffusion, and nonlinear oscillations. The article aims to reveal the comparative strengths and limitations of each method in terms of accuracy, convergence, and computational efficiency, thereby offering practical guidance for researchers and engineers. Moreover, it explores the hybridization of analytical and numerical approaches and their applications in modern physical modeling. The research contributes to enhancing the theoretical foundations of mathematical physics and improving computational practices in applied science.