%0 Journal Article
%T Sinc Collocation Method for Solving the Benjamin-Ono Equation
%A Edson Pindza
%A Eben Mar¨¦
%J Journal of Computational Methods in Physics
%D 2014
%R 10.1155/2014/392962
%X We propose a simple, though powerful, technique for numerical solutions of the Benjamin-Ono equation. This approach is based on a global collocation method using Sinc basis functions. Some properties of the Sinc collocation method required for our subsequent development are given and utilized to reduce the computation of the Benjamin-Ono equation to a system of ordinary differential equations. The propagation of one soliton and the interaction of two solitons are used to validate our numerical method. The method is easy to implement and yields accurate results. 1. Introduction It is well known that nonlinear partial differential equations (NPDEs) are widely used to describe complex phenomena in various fields of sciences, such as physics, biology, and chemistry. In this paper, we consider the initial value problem for the Benjamin-Ono (BO) equation of the form together with the following initial and the boundary conditions: Here is a real valued function and denotes the Hilbert transform defined and described by [1] (amongst others): where denotes the Cauchy principal value. This equation was derived by Benjamin [2] and later by Ono [3] as a model for one-dimensional waves in deep water and it has a close relation with the famous KdV equation which models long waves in shallow water [4]. We recall that the BO equation has an infinite sequence of invariants [5], the first three of which are The BO has been shown to admit rational analytical soliton solutions in and [6]. However, for a given arbitrary initial condition, finding analytical solutions of the BO equation becomes an intractable problem. Therefore the use of numerical methods plays an important role in the study of the dynamics of the BO equation. James and Weideman [7] used the Fourier method which implicitly assumes the periodicity of the boundary conditions and a method based on rational approximating function to compute numerical solutions of the BO. The rational method was shown to have spectacular accuracy for solutions that do not wander too far from the origin while, for long dated solutions, the Fourier method was shown to retain superior accuracy. Miloh et al. [8] proposed an efficient pseudospectral method for the numerical solution of the weakly nonlinear Benjamin-Ono equation for arbitrary initial conditions and suggested a practical new relationship for estimating the number of solitons in terms of arbitrary initial conditions. Thom¨¦e and Vsaudeva Murthy [9] used the Crank-Nicolson approximation in time and finite difference approximations in space to solve the BO equation. They
%U http://www.hindawi.com/journals/jcmp/2014/392962/