Comparative Analysis of Numerical Methods for Solving Nonlinear Differential Equations

Thesis Abstract

Nonlinear differential equations are fundamental to modeling complex phenomena across various scientific and engineering disciplines, yet the lack of universal analytical solutions necessitates efficient and accurate numerical methods. This study aims to conduct a comprehensive comparative analysis of prominent numerical techniques—namely, the Runge-Kutta methods, finite difference methods, and adaptive step-size algorithms—for solving nonlinear differential equations of various classes. The specific objectives include evaluating the accuracy, stability, computational efficiency, and convergence properties of these methods, as well as identifying their relative strengths and limitations in different problem contexts. The research adopts an empirical research design, integrating quantitative assessment through computational simulations. The population of the study comprises a selection of commonly encountered nonlinear differential equations derived from real-world models in physics, biology, and finance, such as the Lorenz system, predator-prey models, and option pricing equations. A purposive sampling approach was employed to select twenty representative equations, ensuring diversity in nonlinearity levels and stiffness characteristics. The primary data collection instrument entails custom-developed MATLAB-based simulation programs capable of implementing the respective numerical methods uniformly across problems. Validation of computational codes was achieved through benchmarking against known analytical solutions where available, and against high-precision solutions obtained via symbolic computation. Data analysis involved the application of descriptive statistics to compare solution accuracy (measured by L2 norm and maximum error), computational time, and stability across methods. Inferential statistical analysis utilized repeated-measures ANOVA to determine significant differences in performance metrics under varying problem parameters such as step size and nonlinearity degree. A qualitative evaluation based on computational complexity, ease of implementation, and robustness complemented quantitative findings. The theoretical framework integrates the stability theory of numerical methods, including Dahlquist's stability concepts, and the Runge-Kutta order conditions, to interpret the empirical results. Expected findings suggest that adaptive step-size Runge-Kutta methods exhibit superior accuracy and stability for stiff and highly nonlinear problems, albeit at a higher computational cost, whereas finite difference methods perform adequately for smooth, less stiff equations with significant efficiency gains. The study anticipates identifying critical thresholds in problem parameters where method performance diverges, thereby providing a practical decision framework for selecting suitable numerical techniques based on problem characteristics. The contribution to knowledge lies in providing a systematic, empirically validated comparison of numerical methods specifically tailored for nonlinear differential equations, extending existing literature predominantly focused on linear problems. The insights derived are expected to guide practitioners and researchers in optimizing computational strategies, especially in complex modeling scenarios where analytical solutions are infeasible. Additionally, the study proposes a set of criteria and decision matrices for method selection aligned with problem features and resource constraints. The main conclusion underscores that no single method uniformly outperforms others across all problem types, emphasizing the importance of contextual method selection. The study recommends adopting adaptive algorithms for stiff and highly nonlinear equations, promoting further investigations into hybrid approaches that integrate multiple methods, and extending comparative analyses to include emerging techniques such as machine learning-enhanced solvers. Future research avenues include exploring real-time applications in dynamic systems modeling and assessing the impact of parallel computing frameworks on the efficiency of numerical methods for nonlinear differential equations.