Abstract
The improved Boussinesq equation is solved with classical finite element method using the most basic Lagrange element k = 1, which leads us to a second order nonlinear ordinary differential equations system in time; this can be solved by any standard accurate numerical method for example Runge-Kutta-Fehlberg. The technique is validated with a typical example and a fourth order convergence in space is confirmed; the 1- and 2-soliton solutions are used to simulate wave travel, wave splitting and interaction; solution blow up is described graphically. The computer symbolic system MathLab is quite used for numerical simulation in this paper; the known results in the bibliography are confirmed.
Highlights
The improved Boussinesq equation (IBq) was proposed in Bogolyubsky’s work [1], like a correct modification to solve the bad Boussinesq equation (BBq) which describes a large group of nonlinear dispersive wave phenomena, such as propagation of long waves on the surface of shallow water in both directions, propagation of ion-sound waves in a uniform isotropic plasma, and so on [2]
Bogolyubsky has shown that the BBq equation describes an unphysical instability of short wave lengths and the Cauchy problem for this partial differential equation is incorrect
Which is the IBq and will be the principal study equation of this paper; it is convenient for computer simulation of the dynamics of different nonlinear waves with weak dispersion; in our case the IBq equation will help to formulate the finite element discretization in the spatial direction with the primal L2-Galerkin finite element formulation [4] [5]; this due to the correction in the fourth order derivative term which leads us to the integral of a discontinuous function over a set of measure zero
Summary
The improved Boussinesq equation (IBq) was proposed in Bogolyubsky’s work [1], like a correct modification to solve the bad Boussinesq equation (BBq) which describes a large group of nonlinear dispersive wave phenomena, such as propagation of long waves on the surface of shallow water in both directions, propagation of ion-sound waves in a uniform isotropic plasma, and so on [2]. Which is the IBq and will be the principal study equation of this paper; it is convenient for computer simulation of the dynamics of different nonlinear waves with weak dispersion; in our case the IBq equation will help to formulate the finite element discretization in the spatial direction with the primal L2-Galerkin finite element formulation [4] [5]; this due to the correction in the fourth order derivative term which leads us to the integral of a discontinuous function over a set of measure zero (for the Lagrange finite element k = 1 ). Wave break up, inelastic and elastic head-on collision, and blow-up solution are modeled and graphics representations are done [14]
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