In this paper. a new kind of generalized BBM equation is introduced and discussed Some existence theorems of periodic traveling wave solutions for this kind generalized BBM equation are given.
In this paper, the full-discrete approximation scheme of the lumped mass nonconforming finite element method for BBM equation is discussed. Without the Riesz projection used in the traditional finite element analysis,...In this paper, the full-discrete approximation scheme of the lumped mass nonconforming finite element method for BBM equation is discussed. Without the Riesz projection used in the traditional finite element analysis, the optimal error estimations are derived based on interpolation technique and special properties of element.展开更多
Based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method, the first elliptic function equation is used to get a new kind of solutions of nonlinear evolution equat...Based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method, the first elliptic function equation is used to get a new kind of solutions of nonlinear evolution equations. New exact solutions to the Jacobi elliptic function of MKdV equations and Benjamin-Bona-Mahoney （BBM） equations are obtained with the aid of computer algebraic system Maple. The method is also valid for other （l＋l）-dimensional and higher dimensional systems.展开更多
In this paper, the initial value problem for a class of BBM-BO type equation is considered. By means of priori estimates and Galerkin method, the existence and uniqueness of the solution in a suitable Sobolev space ar...In this paper, the initial value problem for a class of BBM-BO type equation is considered. By means of priori estimates and Galerkin method, the existence and uniqueness of the solution in a suitable Sobolev space are obtained.展开更多
In this paper,we approximate the solution and also discuss the periodic behavior termed as eventual periodicity of solutions of(IBVPs)for some dispersive wave equations on a bounded domain corresponding to periodic fo...In this paper,we approximate the solution and also discuss the periodic behavior termed as eventual periodicity of solutions of(IBVPs)for some dispersive wave equations on a bounded domain corresponding to periodic forcing.The constructed numerical scheme is based on radial kernels and local in nature like finite difference method.The temporal variable is executed through RK4 scheme.Due to the local nature and sparse differentiation matrices our numerical scheme efficiently recovers the solution.The results achieved are validated and examined with other methods accessible in the literature.展开更多
Boussinesq type equations have been widely studied to model the surface water wave.In this paper,we consider the abcd Boussinesq system which is a family of Boussinesq type equations including many well-known models s...Boussinesq type equations have been widely studied to model the surface water wave.In this paper,we consider the abcd Boussinesq system which is a family of Boussinesq type equations including many well-known models such as the classical Boussinesq system,the BBM-BBM system,the Bona-Smith system,etc.We propose local discontinuous Galerkin(LDG)methods,with carefully chosen numerical fluxes,to numerically solve this abcd Boussinesq system.The main focus of this paper is to rigorously establish a priori error estimate of the proposed LDG methods for a wide range of the parameters a,b,c,d.Numerical experiments are shown to test the convergence rates,and to demonstrate that the proposed methods can simulate the head-on collision of traveling wave and finite time blow-up behavior well.展开更多
This article aims to study the unconditional superconvergent behavior of nonconforming quadrilateral quasi-Wilson element for nonlinear Benjamin Bona Mahoney(BBM)equation.For the generalized rectangular meshes includi...This article aims to study the unconditional superconvergent behavior of nonconforming quadrilateral quasi-Wilson element for nonlinear Benjamin Bona Mahoney(BBM)equation.For the generalized rectangular meshes including rectangular mesh,deformed rectangular mesh and piecewise deformed rectangular mesh,by use of the special character of this element,that is,the conforming part(bilinear element)has high accuracy estimates on the generalized rectangular meshes and the consistency error can reach order O(h^(2)),one order higher than its interpolation error,the superconvergent estimates with respect to mesh size h are obtained in the broken H^(1)-norm for the semi-/fully-discrete schemes.A striking ingredient is that the restrictions between mesh size h and time stepτrequired in the previous works are removed.Finally,some numerical results are provided to confirm the theoretical analysis.展开更多
It has been observed in laboratory experiments that when nonlinear dispersive waves are forced periodically from one end of undisturbed stretch of the medium of propagation, the signal eventually becomes temporally pe...It has been observed in laboratory experiments that when nonlinear dispersive waves are forced periodically from one end of undisturbed stretch of the medium of propagation, the signal eventually becomes temporally periodic at each spatial point. The observation has been confirmed mathematically in the context of the damped Korteweg-de Vries （KdV） equation and the damped Benjamin-Bona-Mahony （BBM） equation. In this paper we intend to show the same results hold for the pure KdV equation （without the damping terms） posed on a finite domain. Consideration is given to the initial-boundary-value problem {ut＋ux＋uux＋uxxx=0, u（x,0）=φ（x）, 0〈x〈1, t〉0,u（0,t）=h（t）, u（1,t） = 0, ux（1,t） = 0, t〉0.It is shown that if the boundary forcing h is periodic with small ampitude, then the small amplitude solution u of （＊） becomes eventually time-periodic. Viewing （＊） （without the initial condition） as an infinite-dimensional dynamical system in the Hilbert space L^2（0, 1）, we also demonstrate that for a given periodic boundary forcing with small amplitude, the system （＊） admits a （locally） unique limit cycle, or forced oscillation, which is locally exponentially stable. A list of open problems are included for the interested readers to conduct further investigations.展开更多
文摘In this paper. a new kind of generalized BBM equation is introduced and discussed Some existence theorems of periodic traveling wave solutions for this kind generalized BBM equation are given.
文摘In this paper, the full-discrete approximation scheme of the lumped mass nonconforming finite element method for BBM equation is discussed. Without the Riesz projection used in the traditional finite element analysis, the optimal error estimations are derived based on interpolation technique and special properties of element.
基金Supported by the National Natural Science Foundation of China (No. 10647112)the Foundation of Donghua University
文摘Based on the homogeneous balance method, the Jacobi elliptic expansion method and the auxiliary equation method, the first elliptic function equation is used to get a new kind of solutions of nonlinear evolution equations. New exact solutions to the Jacobi elliptic function of MKdV equations and Benjamin-Bona-Mahoney （BBM） equations are obtained with the aid of computer algebraic system Maple. The method is also valid for other （l＋l）-dimensional and higher dimensional systems.
文摘In this paper, the initial value problem for a class of BBM-BO type equation is considered. By means of priori estimates and Galerkin method, the existence and uniqueness of the solution in a suitable Sobolev space are obtained.
文摘In this paper,we approximate the solution and also discuss the periodic behavior termed as eventual periodicity of solutions of(IBVPs)for some dispersive wave equations on a bounded domain corresponding to periodic forcing.The constructed numerical scheme is based on radial kernels and local in nature like finite difference method.The temporal variable is executed through RK4 scheme.Due to the local nature and sparse differentiation matrices our numerical scheme efficiently recovers the solution.The results achieved are validated and examined with other methods accessible in the literature.
基金The work of J.Sun and Y.Xing is partially sponsored by NSF grant DMS-1753581.
文摘Boussinesq type equations have been widely studied to model the surface water wave.In this paper,we consider the abcd Boussinesq system which is a family of Boussinesq type equations including many well-known models such as the classical Boussinesq system,the BBM-BBM system,the Bona-Smith system,etc.We propose local discontinuous Galerkin(LDG)methods,with carefully chosen numerical fluxes,to numerically solve this abcd Boussinesq system.The main focus of this paper is to rigorously establish a priori error estimate of the proposed LDG methods for a wide range of the parameters a,b,c,d.Numerical experiments are shown to test the convergence rates,and to demonstrate that the proposed methods can simulate the head-on collision of traveling wave and finite time blow-up behavior well.
基金supported by the National Natural Science Foundation of China(No.11671105).
文摘This article aims to study the unconditional superconvergent behavior of nonconforming quadrilateral quasi-Wilson element for nonlinear Benjamin Bona Mahoney(BBM)equation.For the generalized rectangular meshes including rectangular mesh,deformed rectangular mesh and piecewise deformed rectangular mesh,by use of the special character of this element,that is,the conforming part(bilinear element)has high accuracy estimates on the generalized rectangular meshes and the consistency error can reach order O(h^(2)),one order higher than its interpolation error,the superconvergent estimates with respect to mesh size h are obtained in the broken H^(1)-norm for the semi-/fully-discrete schemes.A striking ingredient is that the restrictions between mesh size h and time stepτrequired in the previous works are removed.Finally,some numerical results are provided to confirm the theoretical analysis.
文摘It has been observed in laboratory experiments that when nonlinear dispersive waves are forced periodically from one end of undisturbed stretch of the medium of propagation, the signal eventually becomes temporally periodic at each spatial point. The observation has been confirmed mathematically in the context of the damped Korteweg-de Vries （KdV） equation and the damped Benjamin-Bona-Mahony （BBM） equation. In this paper we intend to show the same results hold for the pure KdV equation （without the damping terms） posed on a finite domain. Consideration is given to the initial-boundary-value problem {ut＋ux＋uux＋uxxx=0, u（x,0）=φ（x）, 0〈x〈1, t〉0,u（0,t）=h（t）, u（1,t） = 0, ux（1,t） = 0, t〉0.It is shown that if the boundary forcing h is periodic with small ampitude, then the small amplitude solution u of （＊） becomes eventually time-periodic. Viewing （＊） （without the initial condition） as an infinite-dimensional dynamical system in the Hilbert space L^2（0, 1）, we also demonstrate that for a given periodic boundary forcing with small amplitude, the system （＊） admits a （locally） unique limit cycle, or forced oscillation, which is locally exponentially stable. A list of open problems are included for the interested readers to conduct further investigations.
