A Survey of Contemporary Numerical Methods for Differential Equations: Theory, Applications, and Challenges

Abstract

Differential equations play a fundamental role in modeling physical, engineering, biological, and economic systems, making numerical methods essential for solving problems that cannot be addressed through exact analytical techniques. This survey examines contemporary numerical methods for ordinary and partial differential equations, focusing on their theoretical foundations, computational efficiency, practical applications, and emerging challenges. The study reviews widely used approaches such as finite difference methods, finite element methods, Runge–Kutta techniques, spectral methods, and meshless algorithms, highlighting their strengths in solving linear, nonlinear, and multidimensional problems. Special attention is given to the accuracy, stability, convergence, and error control mechanisms that determine the reliability of numerical solutions. The survey also explores real-world applications of these methods in fluid dynamics, structural analysis, climate modeling, biomedical engineering, signal processing, and artificial intelligence. Furthermore, the paper discusses current challenges including computational complexity, numerical instability, high-dimensional modeling, and the need for scalable algorithms suitable for modern high-performance computing environments. Recent advancements involving adaptive algorithms, parallel computing, and machine learning-assisted numerical techniques are also analyzed. The study concludes that contemporary numerical methods continue to evolve rapidly, providing powerful and flexible tools for addressing increasingly complex differential equation models across scientific and industrial domains.