Application of meshfree collocation method to a class of nonlinear partial differential equations

In this work, an algorithm for the numerical solution of the generalized Hirota–Satsuma equations and Jaulent–Miodek equations based on meshless radial basis functions (RBFs) method using collocation points, called Kansa's method, is presented. Four model problems with six different initial conditions are considered for the computation. A fairly explicit scheme is used to approximate the solution. The comparison is made with the exact solutions of each problem of the generalized Hirota–Satsuma coupled Korteweg–de Vries equations. A system consisting highly nonlinear partial differential equations known as Jaulent–Miodek equations and generalized Hirota–Satsuma coupled modified-Korteweg–de Vries equations are considered for comparison with the work already published. The multiquadric RBF results are compared with homotopy perturbation method (HPM) and variational iteration method (VIM) to highlight the excellent performance of the method.