Numerical Studies of the Fractional Korteweg-de Vries, Korteweg-de Vries-Burgers’ and Burgers’ Equations

Abstract

In this paper, we present an accurate numerical method to compute the approximate solutions of the Korteweg–de Vries, Korteweg–de Vries–Burger’s and Burger’s equations with Liouville–Caputo fractional space derivatives, respectively. We implement the spectral collocation method based on the shifted Chebyshev polynomials. The method reduces each model to a set of ODEs which is solved by using the finite difference method. The results obtained by the proposed method are compared with exact solutions, the q-homotopy analysis transform method and the variational iteration method. The efficiency and the accuracy of the results were ascertained by comparing the approximate solution with the exact solution in the case of classical models and evaluating the residual error function in the case of fractional models.