Comparative Analysis of Numerical Methods for Solving Nonlinear Differential Equations

Thesis Abstract

In the realm of mathematical modeling, nonlinear differential equations serve as fundamental tools for describing complex dynamical systems across sciences and engineering; however, their analytical solutions are often elusive, necessitating the use of numerical methods for approximate solutions. This study addresses the challenge of selecting optimal numerical techniques by conducting a comprehensive comparative analysis of their accuracy, stability, computational efficiency, and applicability to a variety of nonlinear differential equations encountered in real-world scenarios. The primary aim is to evaluate and contrast the performance of explicit methods such as Euler’s and Runge-Kutta, implicit methods like Backward Euler, and specialized algorithms including the Adams–Moulton and Fehlberg methods, against a diverse set of nonlinear differential problems representative of physical, biological, and engineering systems. The specific objectives include (1) implementing the selected numerical schemes to solve a curated set of nonlinear differential equations with variable complexity; (2) establishing quantitative criteria for evaluating solution accuracy, stability, and computational resource consumption; (3) analyzing the influence of method parameters such as step size and convergence criteria on solution quality; and (4) providing practical recommendations for method selection based on problem characteristics. The research adopts a quantitative, experimental design utilizing simulated data, where the population comprises a comprehensive suite of fifty nonlinear differential equations sourced from established mathematical modeling cases, including the Lorenz system, Van der Pol oscillator, and predator-prey models. A purposive sampling technique ensures a diverse representation of problem types, enabling robust comparison across different system behaviors. Data collection involves executing each numerical method within MATLAB and Python environments, utilizing built-in and custom-developed algorithms, with performance metrics recorded through automated scripting. Methodologically, the study employs descriptive statistical analysis complemented by advanced inferential techniques such as Analysis of Variance (ANOVA) to test for significant differences in accuracy, stability, and efficiency among the methods across varied problem sets. The evaluation metrics include mean squared error (MSE), stability thresholds, execution time, and convergence rate. A series of sensitivity analyses are conducted to investigate parameter influence, while the theoretical framework anchors on established stability theory (Lyapunov stability) and convergence analysis, complemented by the application of the Local Truncation Error concept to evaluate numerical accuracy. Expected findings include identifying specific methods that excel in stability and accuracy for stiff versus non-stiff equations, clarifying the trade-offs between computational resource utilization and precision, and developing decision matrices to guide method selection based on problem parameters. Anticipated contributions to knowledge involve filling gaps in empirical data comparing classical and modern numerical schemes across diverse nonlinear systems, and providing a practical framework for researchers and practitioners in selecting suitable algorithms. The study concludes that hybrid approaches or adaptive step-size strategies may offer enhanced performance for complex nonlinear models. The primary recommendation emphasizes context-dependent method selection and advocates for integrating stability and error control mechanisms in algorithm implementation. Furthermore, the research suggests avenues for future investigations into parallel computation techniques and machine learning-based adaptive methods to enhance solving efficiency. This study thereby makes a significant contribution to numerical analysis literature and supports practical applications requiring reliable solutions to nonlinear differential equations in scientific and engineering domains.