A Fourier pseudospectral method for a generalized improved Boussinesq equation

Abstract

In this article, we propose a Fourier pseudospectral method for solving the generalized improved Boussinesq equation. We prove the convergence of the semi-discrete scheme in the energy space. For various power nonlinearities, we consider three test problems concerning the propagation of a single solitary wave, the interaction of two solitary waves and a solution that blows up in finite time. We compare our numerical results with those given in the literature in terms of numerical accuracy. The numerical comparisons show that the Fourier pseudospectral method provides highly accurate results.to describe a large number of nonlinear dispersive wave phenomena, such as propagation of long waves on the surface of shallow water in both directions, propagation of long waves in one dimensional nonlinear lattices and propagation of ion-sound waves in a uniform isotropic plasma [3]. Here, the independent variables x and t denote spatial coordinate and time, respectively, and the parameter α = ±1. In the case of α = −1, (1.1) is called as the good Boussinesq equation; whereas in the case of α = 1, it is known as the bad Boussinesq equation. In [4], Bogolubsky showed that the bad Boussinesq equation is unstable under short wave perturbation and then he

Higher order dispersive effects in regularized Boussinesq equation

“…where A = 0.9 and c = 2A 3 + 1. This initial data correspond to the initial profile of the exact solitary wave solution for the IBq equation given in [11]. In Figure 2 Table III shows the dependence of the amplitude to a changing η 2 .…”

“…We therefore propose a numerical method combining a Fourier pseudo-spectral method for the space discretization and a fourthorder Runge-Kutta scheme for time discretization. To the best of our knowledge, there is no numerical study for HBq equation although there are lots of numerical studies to solve the generalized IBq equation (see [11] and references therein). In Section 2, we review some properties of the HBq equation and then we derive the solitary wave solutions.…”

Improvement of the homotopy perturbation method to nonlinear problems

“…Examples of these methods are the Adomian decomposition method (ADM) Hosseinzadeh et al, 2017), the variational iteration method (Akter and Chowdhury, 2017;Wazwaz, 2015;Glowinski, 2015;Ghorbani and Bakherad, 2017), the pseudospectral method (Bhrawy et al, 2015;Wei et al, 2017;Borluk and Muslu, 2015) and the reproducing kernel Hilbert space method (Arqub et al, 2016).…”

Numerical solution for a general class of nonlocal nonlinear wave equations arising in elasticity

“…In this case Eq. becomes the higher‐order Boussinesq equation (HBq), For these special cases there are numerous studies in the literature both numerical and analytical (see and and the references there in). The existence and stability of solitary wave solutions of the nonlocal nonlinear wave equation has been shown in a recent study .…”

“…In the recent studies a Fourier pseudo‐spectral method has been used for the numerical solutions of the IBq and HBq equations. Thus the efficiency of the method has been tested for the special cases of this class of nonlocal equations.…”