Numerical and Analytical Solutions for a Nonlinear System

This paper investigates two nonlinear reaction–diffusion systems: (i) a spatially extended competitive species model and (ii) the FitzHugh–Nagumo system, which serves as a canonical model of excitable media. For the first system, we derive exact closed-form solutions using the exponential function method; these solutions exhibit spatially periodic structures and reflect fundamental features of competitive and diffusion-driven pattern formation. To handle both systems numerically, particularly where analytical solutions are intractable, we propose an enhanced variant of the modified Adomian decomposition method (E-MADM), incorporating adaptive decomposition and convergence-accelerating modifications. The accuracy and robustness of E-MADM are rigorously assessed by benchmarking its results against (a) the exact analytical solution (where available) and (b) high-resolution numerical approximations obtained via a second-order finite difference method (FDM). This comparative study confirms that E-MADM not only preserves structural properties of the solution but also achieves significant gains in computational efficiency and stability over classical approaches. A genetic algorithm is employed to determine the optimal diffusion coefficient (μ = 0.2281) for the competitive model. The results demonstrate that MADM achieves higher accuracy than FDM while maintaining comparable computational efficiency, confirming its effectiveness for solving nonlinear reaction–diffusion systems, as supported by the comparative analysis with established methods presented in Table 1.