Application of the Natural Decomposition Method to Strongly Coupled Partial Differential Equations: The Gray-Scott Model. - Nsingwire, Godwin (Kabale University, 2024)
This study investigated the application of the natural decomposition method for solving the Gray-Scott reaction-diffusion model, which is known for its complex nonlinearities and strong coupling between chemical species. A general form of the natural decomposition method was developed to address these complexities, providing a systematic framework for analyzing the model's behavior. By decomposing the nonlinear partial differential equations into a series of simpler equations, the method facilitates both analytical and numerical solutions, allowing for a detailed examination of the spatiotemporal patterns that emerge. Numerical techniques, such as the Newton-Raphson method, were integrated to handle the nonlinear algebraic equations, particularly those involving the term𝑢𝑣2. The results show that the natural decomposition method effectively captures the intricate dynamics of the Gray-Scott model, demonstrating good convergence for specified initial conditions and parameter values. Sensitivity analysis further revealed that even small changes in initial conditions could significantly affect chemical concentration patterns, underscoring the need for accurate parameter settings. While this study employed a fixed parameter approach, it also recommends exploring the infinite series solution to provide a more comprehensive understanding of the system's dynamics. Overall, the natural decomposition method proves to be a robust and versatile tool, offering both analytical clarity and computational efficiency in modeling complex chemical and biological systems governed by nonlinear partial differential equations.