On the numerical solution of the nonlinear Korteweg–de Vries equation

Abstract

In this paper, we present a new method for solving a nonlinear third-order Korteweg–de Vries equation. This method is based on the multiquadric (MQ) quasi-interpolation operator LW2 and an integrated radial basis function networks scheme. In the present scheme, the second-order central divided difference of the spatial derivative is used to approximate the third-order spatial derivative, and the Taylors series expansion to discretize the temporal derivative. Then, the spatial derivative is approximated by the MQ quasi-interpolation operator LW2. This method is applied on some test experiments and the numerical results have been compared with the exact solutions and the solutions of other numerical methods. The L∞, L2 and root-mean-square errors of the solutions show the efficiency and the accuracy of the method. Furthermore, the stability analysis of the method is surveyed.