Comparative Analysis of Numerical Methods for Solving Nonlinear Differential Equations

Thesis Abstract

Nonlinear differential equations are fundamental to modeling complex systems across various scientific and engineering disciplines, yet their analytical solutions are often unattainable, necessitating reliable numerical methods for practical problem-solving. This study aims to conduct a comprehensive comparative analysis of prominent numerical techniques—namely the Runge-Kutta methods, Adams-Bashforth-Moulton methods, multistep methods, and finite difference approaches—focusing on their accuracy, stability, computational efficiency, and convergence properties when applied to nonlinear differential equations. The specific objectives include evaluating the performance of these methods on benchmark nonlinear problems such as the Lorenz system, the Van der Pol oscillator, and the logistic growth model; establishing criteria for method selection based on accuracy and computational cost; and providing recommendations for practitioners in applied mathematics, physics, and engineering. The research adopts a quantitative, experimental research design, employing a cross-sectional analysis paradigm. The population encompasses a set of 120 nonlinear differential equations derived from real-world applications, randomly selected from established mathematical problem repositories, ensuring a broad representation of problem types and complexity levels. A stratified sampling technique is utilized to select a representative sample of 30 equations for detailed numerical testing. Data collection involves the implementation of the numerical methods within MATLAB and Python programming environments, using standardized algorithmic codes validated through benchmark tests. The key performance metrics—mean squared error (MSE), stability regions, computational time, and convergence rate—are recorded for each method in solving the selected problems. Data analysis employs descriptive statistics to summarize the performance characteristics of each numerical technique, while inferential statistical procedures—including one-way Analysis of Variance (ANOVA)—are applied to determine statistically significant differences in accuracy and efficiency among the methods at a 95% confidence level. Post hoc analyses further identify specific pairs of methods with significant performance variations. Additionally, graphical analysis through convergence plots and stability diagrams complements quantitative findings, facilitating visual interpretation of comparative performance. The anticipated findings suggest that while explicit Runge-Kutta methods offer high accuracy for moderate-sized problems, their stability limitations restrict their applicability in stiff equations, where implicit methods like the backward differentiation formulas outperform in terms of stability and computational efficiency. Multistep methods are expected to demonstrate superior efficiency in large-scale problems with smooth solutions, whereas finite difference methods may excel in boundary value problems. The study aims to establish a performance hierarchy and develop a decision framework for method selection based on problem type, required precision, and available computational resources. This research contributes significantly to the body of knowledge by providing a systematic, empirical comparison of numerical techniques tailored specifically to nonlinear differential equations, filling existing gaps where prior studies primarily focus on linear or specific subclass problems. The practical implications include offering guidelines for engineers, physicists, and applied mathematicians to optimize their computational strategies when tackling nonlinear models. The study concludes with recommendations for the selection and development of hybrid methods to harness the strengths of multiple approaches and suggests avenues for future research, such as the integration of machine learning algorithms for adaptive method selection based on problem features.