Numerical solution of nonlinear reaction-diffusion systems by a novel Taylor series method versus the Galerkin finite element method

This study presents a comprehensive analysis and comparison of two numerical approaches for handling the nonlinear Lotka-Volterra interaction and diffusion system: the novel Taylor series method (NTSM) and the Galerkin finite element method (GFEM). We develop a semi-analytical NTSM method by iteratively obtaining coefficients from the governing partial differential equations and initial and boundary conditions to construct high-order Taylor series in space and time. The method is validated by comparison with an exact analytic solution, demonstrating high accuracy at the real-solution level (achieving errors below machine precision (approximately ${10}^{-16}$) in the tested scenarios) and rapid convergence, attributed to the analytic nature of the underlying solution. Furthermore, the use of linear (hierarchical) basis functions and Galerkin weighting with GFEM provides robust and physically acceptable approximations that rapidly achieve the stable equilibrium state of the system. Using Maple 18 and MATLAB, numerical tests indicate that the GFEM is robust and suitable for complex geometries and extended dynamics; however, the NTSM method significantly outperforms it in terms of accuracy and convergence speed for smooth and well-defined boundary problems in straightforward analytical problems. The results obtained show the accuracy and efficiency of NTSM in solving analytical and numerical problems, while GFEM is characterized by its strength in complex domains. Theoretical convergence analysis confirms an $O(h^2)$ order convergence for GFEM and an exponential convergence for NTSM in the smooth test cases examined in this study, thus supporting the numerical results.