Numerical Analysis of Nonlinear Differential Equations in Complex Physical Systems

Abstract

Nonlinear differential equations are widely used to describe complex phenomena in physics, engineering, biology, and environmental science. This study investigates numerical methods for solving nonlinear differential equations when analytical solutions are difficult or impossible to obtain. It examines finite difference methods, finite element techniques, iterative algorithms, and adaptive time-stepping procedures for improving numerical stability and computational efficiency. Particular attention is given to convergence, approximation error, sensitivity to initial conditions, and the treatment of nonlinear boundary conditions. Applications in fluid dynamics, heat transfer, population modeling, and wave propagation are also considered. Improved numerical approaches can support more accurate simulations and reliable predictions of complex dynamical systems.