The convergence and stability analysis of a numerical method for solving a mathematical model of language competition

Abstract

In this study, a combination of spectral and fixed point methods is applied to solve a mathematical model of language competition. The considered model has formulated the competition between two languages in which the population density of speakers of each language is modeled by a nonlinear parabolic equation. Due to the fact that the problem is nonlinear, the fixed point method is employed to change the problem to a linear one in each step of iterations. After that, the coupled parabolic equations are solved using the spectral method in each step. It is proven that the constructed sequence converges to the exact solution of the problem, which results in the convergence of the method. Then, by applying the method for solving the perturbed problem, the stability of method is proven. By considering some examples the efficiency of the method is demonstrated. Finally, by the use of an example, the influences of various factors on the evolution of the population densities are illustrated step by step.