A Hybrid Adomian–Runge–Kutta Method for Solving Nonlinear Reaction–Diffusion Equations

This paper presents a novel hybrid numerical framework that combines the Adomian Decomposition Method (ADM) with the classical fourth-order Runge–Kutta (RK4) scheme for solving nonlinear reaction–diffusion equations. The hybrid approach analytically handles nonlinear terms via Adomian polynomials and employs RK4 for stable and accurate time integration. This strategy addresses the limitations of both ADM and RK4 when used independently. The proposed method is applied to a range of benchmark problems, including the exponential-type, Ginzburg–Landau, Gray–Scott, and Zeldovich reaction–diffusion equations. In each case, the hybrid ADM–RK4 method demonstrates enhanced accuracy and stability compared to pure RK4. Numerical experiments confirm low absolute errors (as small as 10−4 to 10−3) and strong agreement between hybrid and standard RK4 solutions. The results are validated through detailed tabulated data and surface plots, showcasing the hybrid method’s effectiveness in capturing the dynamics of nonlinear reaction–diffusion systems over short time intervals