%0 Journal Article
%T Numerical Solutions of Two-Way Propagation of Nonlinear Dispersive Waves Using Radial Basis Functions
%A Pablo U. Su¨˘rez
%A J. H¨Śctor Morales
%J International Journal of Partial Differential Equations
%D 2014
%R 10.1155/2014/407387
%X We obtain the numerical solution of a Boussinesq system for two-way propagation of nonlinear dispersive waves by using the meshless method, based on collocation with radial basis functions. The system of nonlinear partial differential equation is discretized in space by approximating the solution using radial basis functions. The discretization leads to a system of coupled nonlinear ordinary differential equations. The equations are then solved by using the fourth-order Runge-Kutta method. A stability analysis is provided and then the accuracy of method is tested by comparing it with the exact solitary solutions of the Boussinesq system. In addition, the conserved quantities are calculated numerically and compared to an exact solution. The numerical results show excellent agreement with the analytical solution and the calculated conserved quantities. 1. Introduction Consider the initial and boundary value problem where , and are real constants and subscripts and denote space and time derivatives, respectively. In fluid mechanics, the functions and represent flow velocities. Solutions of this type of systems have attracted much research in the past two decades [1¨C9]. In these studies the most popular system of Boussinesq type is the one proposed by Bona and Chen in [1], to describe approximately the two-dimensional propagation of surface waves in a uniform horizontal channel of a fixed length filled with an irrotational, incompressible, and inviscid flow. The system is derived formally from EulerĄŻs equations in 2D and using small amplitude and long wave length assumptions. Further, solitary wave solutions of this system have been reported in numerous works; see for instance [1, 2, 6, 10]. This work studies the numerical solution of this system by means of radial basis functions (RBFs). The use of these types of basis functions has become very popular in recent times; see for instance the work of Buhmann [11], Franke and Schaback [12], Driscoll and Heryudono [13], and the references therein. The main inspiration for this work is the very successful application of RBFs to solve the Kawahara equation [14] and the modified regularized long wave equation (MRLW) [15]. We will examine the two cases for the Boussinesq type system. The first one we examine is the Bona and Chen system, which is important in the theory of dispersive waves, This corresponds to the case with . The second case we examine is the full system with , , , and : The paper is organized in three sections. In Section 2, we use RBFs to reduce the PDE system to a system of differential
%U http://www.hindawi.com/journals/ijpde/2014/407387/