In this paper, we first examine the type of structure of the solutions to the modified form of a nonlinear Fisher's reaction-diffusion equation. The existence of the traveling wave solution to the equation in the long term is observed by using dynamical system theory and exhibiting a phase space analysis of its stable points. In parallel, we represent radial basis functions (RBFs)-based differential quadrature methods (DQMs) to close the solution of the equation. The stability analysis of the recommended method is demonstrated. Some initial-boundary value problems are considered test problems. The numerical results indicate extremely exact and stable initial and boundary conditions in the same domain with dissimilar time ranges.