On the Convergence and Stability Analysis of Finite-Difference Methods for the Fractional Newell-Whitehead Equations

Abstract

In this study, standard and non-standard finite-difference methods are proposed for numerical solutions of the time-spatial fractional generalized Newell-Whitehead-Segel equations describing the dynamical behavior near the bifurcation point of the Rayleigh-Benard convection of binary fluid mixtures. The numerical solutions have been found for high values of p which shows the degree of nonlinear terms in the equations. The stability and convergence conditions of the obtained difference schemes are determined for each value of p. Errors of methods for various values of p are given in tables. The compatibility of exact solutions and numerical solutions and the effectiveness of the methods are interpreted with the help of tables and graphics. It can be said that not only standard and non-standard finite-difference methods are feasible and effective methods to solve the given equation numerically but also useful in terms of computational cost and memory.