scholarly journals Soliton molecules and abundant interaction solutions of a general high-order Burgers equation

2021 ◽  
pp. 104052
Author(s):  
Gaizhu Qu ◽  
Xiaorui Hu ◽  
Zhengwu Miao ◽  
Shoufeng Shen ◽  
Mengmeng Wang
Keyword(s):  
Burgers Equation ◽  
High Order ◽  
Interaction Solutions

Y-shaped soliton solutions for the (2+1)-dimensional bidirectional Sawada–Kotera equation

2021 ◽  
Author(s):  
Shuxin Yang ◽  
Zhao Zhang ◽  
Biao Li
Keyword(s):  
Soliton Solutions ◽  
Complex Interaction ◽  
High Order ◽  
Hirota Bilinear Method ◽  
Bilinear Method ◽  
Dynamic Behaviors ◽  
Interaction Solutions ◽  
Localized Waves ◽  
Hirota Bilinear ◽  
Interaction Phenomenon

On the basis of the Hirota bilinear method, resonance Y-shaped soliton and its interaction with other localized waves of (2+1)-dimensional bidirectional Sawada–Kotera equation are derived by introducing the constraint conditions. These types of mixed soliton solutions exhibit complex interaction phenomenon between the resonance Y-shaped solitons and line waves, breather waves, and high-order lump waves. The dynamic behaviors of the interaction solutions are analyzed and illustrated.

A FDM for Burgers’ Equation on Unbounded Domains Using High-Order Artificial Boundary Conditions

2017 ◽  
Vol 865 ◽  
pp. 233-238
Author(s):  
Quan Zheng ◽  
Yu Feng Liu
Keyword(s):  
Boundary Conditions ◽  
Heat Equation ◽  
Unbounded Domain ◽  
Burgers Equation ◽  
High Order ◽  
Computational Domain ◽  
Artificial Boundary ◽  
Artificial Boundary Conditions ◽  
Second Order Convergence ◽  
The Stability

Burgers’ equation on an unbounded domain is an important mathematical model to treat with some external problems of fluid materials. In this paper, we study a FDM of Burgers’ equation using high-order artificial boundary conditions on the unbounded domain. First, the original problem is converted into the heat equation on an unbounded domain by Hopf-Cole transformation. Thus the difficulty of nonlinearity of Burgers’ equation is overcome. Second, high-order artificial boundary conditions are given by using Padé approximation and Laplace transformation. And the conditions confine the heat equation onto a bounded computational domain. Third, we prove the solutions of the resulting heat equation and Burgers’ equation are both stable. Fourth, we establish the FDM for Burgers’ equation on the bounded computational domain. Finally, a numerical example demonstrates the stability, the effectiveness and the second-order convergence of the proposed method.

scholarly journals Residual symmetry reductions and interaction solutions of the (2+1)-dimensional Burgers equation

2015 ◽  
Vol 24 (1) ◽  
pp. 010203
Author(s):  
Xi-Zhong Liu ◽  
Jun Yu ◽  
Bo Ren ◽  
Jian-Rong Yang
Keyword(s):  
Burgers Equation ◽  
Residual Symmetry ◽  
Interaction Solutions ◽  
Symmetry Reductions

scholarly journals Controlling the dynamics of Burgers equation with a high-order nonlinearity

2004 ◽  
Vol 2004 (62) ◽  
pp. 3321-3332 ◽  
Author(s):  
Nejib Smaoui
Keyword(s):  
Steady State ◽  
Asymptotic Stability ◽  
Global Asymptotic Stability ◽  
Nonlinear Boundary ◽  
Burgers Equation ◽  
High Order ◽  
Time Stepping ◽  
Backward Euler ◽  
High Order Nonlinearity ◽  
Steady State Solutions

We investigate analytically as well as numerically Burgers equation with a high-order nonlinearity (i.e.,ut=νuxx−unux+mu+h(x)). We show existence of an absorbing ball inL2[0,1]and uniqueness of steady state solutions for all integern≥1. Then, we use an adaptive nonlinear boundary controller to show that it guarantees global asymptotic stability in time and convergence of the solution to the trivial solution. Numerical results using Chebychev collocation method with backward Euler time stepping scheme are presented for both the controlled and the uncontrolled equations illustrating the performance of the controller and supporting the analytical results.

scholarly journals Finite Element Method of BBM-Burgers Equation with Dissipative Term Based on Adaptive Moving Mesh

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Changna Lu ◽  
Qianqian Gao ◽  
Chen Fu ◽  
Hongwei Yang
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Burgers Equation ◽  
Time Derivative ◽  
High Order ◽  
Moving Mesh ◽  
High Order Derivative ◽  
Dissipative Term ◽  
Dissipative Terms ◽  
Element Method

A finite element model is proposed for the Benjamin-Bona-Mahony-Burgers (BBM-Burgers) equation with a high-order dissipative term; the scheme is based on adaptive moving meshes. The model can be applied to the equations with spatial-time mixed derivatives and high-order derivative terms. In this scheme, new variables are needed to make the equation become a coupled system, and then the linear finite element method is used to discretize the spatial derivative and the fifth-order Radau IIA method is used to discretize the time derivative. The simulations of 1D and 2D BBM-Burgers equations with high-order dissipative terms are presented in numerical examples. The numerical results show that the method keeps a second-order convergence in space and provides a smaller error than that based on the fixed mesh, which demonstrates the effectiveness and feasibility of the finite element method based on the moving mesh. We also study the effect of the dissipative terms with different coefficients in the equation; by numerical simulations, we find that the dissipative termuxxplays a more important role thanuxxxxin dissipation.

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