ADVANCED NUMERICAL TECHNIQUES FOR SOLVING DIFFERENTIAL EQUATIONS: A REVIEW-BASED EXPLORATION

Abstract

This review explores advanced numerical techniques used to solve differential equations, focusing on methods that have demonstrated significant improvements in accuracy, stability, and computational efficiency. Differential equations play a crucial role in modeling various physical, biological, and engineering systems, but their solutions are often challenging, particularly for nonlinear and complex systems. Traditional methods, such as finite difference and finite element techniques, while widely applied, may fall short in addressing issues like convergence and handling high-dimensional problems. This review delves into more sophisticated approaches, including spectral methods, adaptive mesh refinement, and multigrid techniques, which provide enhanced capabilities for tackling such challenges.