Solving Complex Nonlinear Differential Equations: A Survey of Analytical, Semi-Analytical and Numerical Methods

Nonlinear differential equations (NLDEs) constitute the mathematical backbone of an enormous range of physical, biological, chemical, and engineering phenomena, yet their analytic intractability has historically limited rigorous study to narrow special cases. This paper presents a comprehensive survey of methods for solving complex nonlinear ordinary differential equations (NODEs) and nonlinear partial differential equations (NPDEs), encompassing classical analytical approaches, modern semi-analytical techniques, and numerical algorithms. The survey systematically covers: the classification and sources of nonlinearity in differential equations; exact methods including separation of variables, integrating factors, and Bernoulli substitution for specific equation types; approximate analytical methods including the Regular Perturbation Method (RPM), Multiple-Scale Analysis (MSA), and Poincaré-Lindstedt Method; semi-analytical iterative methods including the Adomian Decomposition Method (ADM), Homotopy Perturbation Method (HPM), Homotopy Analysis Method (HAM), and Variational Iteration Method (VIM); and numerical methods including Euler's method, the classical fourth-order Runge-Kutta method (RK4), predictor-corrector algorithms, finite difference methods, and stiff-equation solvers. Worked examples are provided for the Bernoulli equation, the Duffing oscillator via perturbation theory, the Van der Pol equation via Runge-Kutta simulation, a nonlinear heat equation via ADM, and Burgers' equation via HPM. Comparative performance analyses from reviewed literature are synthesised in tabular form, covering accuracy, convergence, computational cost, and domain of applicability for each method. Applications of NLDEs in fluid mechanics, population dynamics, electrical circuits, epidemiology, and structural engineering are surveyed. Research frontiers including fractional nonlinear differential equations, machine learning-assisted solvers, and Physics-Informed Neural Networks (PINNs) are identified and discussed. The review concludes that no single method dominates across all problem classes and presents evidence-based guidance for method selection given problem characteristics.