%0 Journal Article
%T Solving the Generalized Regularized Long Wave Equation Using a Distributed Approximating Functional Method
%A Edson Pindza
%A Eben Mar¨¦
%J International Journal of Computational Mathematics
%D 2014
%R 10.1155/2014/178024
%X The generalized regularized long wave (GRLW) equation is solved numerically by using a distributed approximating functional (DAF) method realized by the regularized Hermite local spectral kernel. Test problems including propagation of single solitons, interaction of two and three solitons, and conservation properties of mass, energy, and momentum of the GRLW equation are discussed to test the efficiency and accuracy of the method. Furthermore, using the Maxwellian initial condition, we show that the number of solitons which are generated can be approximately determined. Comparisons are made between the results of the proposed method, analytical solutions, and numerical methods. It is found that the method under consideration is a viable alternative to existing numerical methods. 1. Introduction Nonlinear partial differential equations (NPDEs) are widely used to describe complex phenomena in various fields of sciences, such as physics, biology, and chemistry. Various methods have been devised to find the exact and approximate solutions of nonlinear partial differential equations [1, 2] in order to provide more information for understanding physical phenomena arising in numerous scientific and engineering fields. In this paper, we consider an important nonlinear wave equation, the generalized regularized long wave (GRLW) equation of the form where is a positive integer and and are positive constants. The boundary conditions are given by as . This equation has strong connections to the generalized Korteweg-de Vries (GKdV) equation and has solitary solutions [3]. The GRLW equation was proposed many years ago by Peregrine [4, 5] as a model for small-amplitude long waves on the surface of water in a channel. The study of the GRLW equation gives the opportunity of investigating the creation of secondary solitary waves to obtain insight into the corresponding processes of particle physics [6]. For , this equation is reduced to the regularized long wave equation as an important equation in physics media, describing phenomena with weak nonlinearity and dispersive waves [7]. The existence and uniqueness of the solution of the RLW equation are given in [8]. Its analytical solutions were obtained under restrictive initial and boundary conditions [8]; therefore, the use of numerical methods has been advocated. Several numerical methods for the solution of the RLW equation have been introduced in the literature. These include finite- difference methods [9], Fourier pseudospectral methods [10], and finite-element methods based on Galerkin and collocation principles
%U http://www.hindawi.com/journals/ijcm/2014/178024/