Abstract
The generalized regularized long wave (GRLW) equation is an important nonlinear equation for describing a large number of physical phenomena, for examples, the shallow water waves and plasma waves. In this study, numerical approximation of the GRLW using the element-free improved moving least-squares Ritz (IMLS-Ritz) method is performed. In the solution procedure, the IMLS approximation is employed to reduce the number of unknown coefficients in the trial functions. The Ritz minimization procedure is then used to derive the final algebraic equation system through discretizing the constructed energy formulation of the nonlinear GRLW equation. Time difference technique and Newton-Raphson method are adopted to solve the nonlinear equation system. Numerical experiments are conducted on the final form of the governing equation system to demonstrate the accuracy and efficiency of the element-free IMLS-Ritz method by comparing the computed IMLS-Ritz results with the existing available analytical solutions.
Highlights
The damped generalized regularized long wave (GRLW) equation is established as a model for small-amplitude long waves on the surface of water [1, 2]
Unlike the RLW and the Benjamin-Bona-Mahony equations, the stability of solutions to the GRLW equation depends on the solitary wave velocity [4]
We present an element-free computational framework to predict numerical solutions for the nonlinear GRLW equation using an improved moving least square Ritz (IMLS-Ritz) method
Summary
The damped GRLW equation is established as a model for small-amplitude long waves on the surface of water [1, 2]. The element-free or meshless method has become a popular numerical tool in recent years It has been developed and successfully applied to obtain accurate solutions for PDEs deriving from the physical and engineering fields [21,22,23,24,25,26]. The major advantage of the element-free method for solving partial differential equations (PDEs) is that it does not require domain or boundary discretization With this advantage together with its flexibility and simplicity in implementation [38,39,40], element-free methods have been employed for solving many mathematical models of wave equation [17,18,19,20, 41, 42], such as the kp-Ritz method [17], the radial basis functions method [41], and the element-free Galerkin method [20, 42]. Computational simulations for several numerical examples are presented to examine the affectivity and efficiency of the IMLS-Ritz method on the nonlinear GRLW equation
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