Numerical methods for solving nonlinear differential equations in engineering applications

Abstract

Nonlinear differential equations (NLDEs) frequently emerge in diverse engineering fields such as fluid dynamics, structural mechanics, chemical engineering, and electrical circuits. Traditional analytical methods are typically inadequate for solving these equations due to their inherent complexity. Hence, numerical methods have gained prominence as indispensable tools in handling NLDEs. This research review systematically explores recent numerical techniques employed for solving NLDEs in engineering applications. Specifically, it evaluates the efficacy, computational efficiency, and practical applicability of various numerical methods such as finite difference, finite element, finite volume, spectral methods, and advanced methods like neural-network-based numerical solvers. Key findings indicate that finite element and finite volume methods demonstrate significant flexibility and accuracy, particularly in structural analysis and computational fluid dynamics (CFD). Spectral methods show superior accuracy for smooth problems but exhibit limitations in handling discontinuities. Neural network-based numerical methods have recently emerged as powerful alternatives, offering promising results in solving high-dimensional and complex nonlinear problems. However, challenges such as computational cost, convergence issues, and algorithmic complexity persist. This review concludes by highlighting critical research gaps and recommending areas for future investigation, including adaptive algorithms, hybrid numerical techniques, and improved machine-learning-based approaches for efficiently solving NLDEs.