scholarly journals Notes on the Lie Symmetry Exact Explicit Solutions for Nonlinear Burgers' Equation

2019 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
General Solution ◽  
Nonlinear Equation ◽  
Burgers Equation ◽  
Lie Symmetry ◽  
Explicit Solutions ◽  
Numerical Example ◽  
Original Works

In light of Liu's original works, this paper revisits the solution of general Burgers's nonlinear equation. We obtain two exact and explicit solutions for group $G_4$ and $G_6$, and a most general solution as well. As applications, a numerical example is carried out.

Notes on the Lie Symmetry Exact Explicit Solutions for Nonlinear Burgers' Equation

2019 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
General Solution ◽  
Nonlinear Equation ◽  
Burgers Equation ◽  
Lie Symmetry ◽  
Numerical Calculations ◽  
Explicit Solutions ◽  
Research Article ◽  
Original Works

In light of Liu et al.'s original works, this research article revisits the solution of Burgers's nonlinear equation. The researcher found two exact and explicit solutions for groups $G_4$ and $G_6$, as well as a general solution. Their applications were conducted by using numerical calculations.

scholarly journals Notes on the Lie Symmetry Exact Explicit Solutions for Nonlinear Burgers' Equation

2021 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Numerical Simulation ◽  
General Solution ◽  
Nonlinear Equation ◽  
Burgers Equation ◽  
Lie Symmetry ◽  
Explicit Solutions ◽  
Original Works ◽  
Maple Code

In light of Liu \emph{at el.}'s original works, this paper revisits the solution of Burgers's nonlinear equation $u_t=a(u_x)^2+bu_{xx} $. The study found two exact and explicit solutions for groups $G_4$ and $G_6$, as well as a general solution. A numerical simulation is carried out. In the appendix a Maple code is provided

Nonclassical Potential Symmetries and New Explicit Solutions of the Burgers Equation

2005 ◽  
Vol 60 (1-2) ◽  
pp. 17-22 ◽  
Author(s):  
Maochang Qin ◽  
Fengxiang Mei ◽  
Xuejun Xu
Keyword(s):  
Symmetry Group ◽  
Burgers Equation ◽  
Lie Symmetry ◽  
Explicit Solutions ◽  
Potential Symmetries ◽  
Potential Symmetry ◽  
Symmetry Generators

Several new nonclassical potential symmetry generators to the Burgers equation are derived. Some explicit solutions, which cannot be derived from the Lie symmetry group of Burgers or its adjoined equation, are obtained by using these nonclassical potential symmetry generators.

A System of Matrix Equations over the Quaternion Algebra with Applications

2017 ◽  
Vol 24 (02) ◽  
pp. 233-253 ◽  
Author(s):  
Xiangrong Nie ◽  
Qingwen Wang ◽  
Yang Zhang
Keyword(s):  
General Solution ◽  
Quaternion Algebra ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Equations ◽  
Electronic Networks ◽  
Consistency Conditions ◽  
Numerical Example ◽  
System Of Matrix Equations ◽  
Necessary And Sufficient

We in this paper give necessary and sufficient conditions for the existence of the general solution to the system of matrix equations [Formula: see text] and [Formula: see text] over the quaternion algebra ℍ, and present an expression of the general solution to this system when it is solvable. Using the results, we give some necessary and sufficient conditions for the system of matrix equations [Formula: see text] over ℍ to have a reducible solution as well as the representation of such solution to the system when the consistency conditions are met. A numerical example is also given to illustrate our results. As another application, we give the necessary and sufficient conditions for two associated electronic networks to have the same branch current and branch voltage and give the expressions of the same branch current and branch voltage when the conditions are satisfied.

scholarly journals Lie Symmetry Analysis of Burgers Equation and the Euler Equation on a Time Scale

Symmetry ◽  
2019 ◽  
Vol 12 (1) ◽  
pp. 10 ◽  
Author(s):  
Mingshuo Liu ◽  
Huanhe Dong ◽  
Yong Fang ◽  
Yong Zhang
Keyword(s):  
Euler Equation ◽  
Time Scale ◽  
Flow Model ◽  
Burgers Equation ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Lie Symmetry Analysis ◽  
Group Invariant ◽  
Discrete Equations ◽  
Group Invariant Solutions

As a powerful tool that can be used to solve both continuous and discrete equations, the Lie symmetry analysis of dynamical systems on a time scale is investigated. Applying the method to the Burgers equation and Euler equation, we get the symmetry of the equation and single parameter groups on a time scale. Some group invariant solutions in explicit form for the traffic flow model simulated by a Burgers equation and Euler equation with a Coriolis force on a time scale are studied.

Explicit Solutions and Conservation Laws of a Coupled Burgers’ Equation

2017 ◽  
Vol 72 (9) ◽  
pp. 789-793
Author(s):  
Bo Xue ◽  
Fang Li ◽  
Yihao Li ◽  
Mingming Sun
Keyword(s):  
Conservation Laws ◽  
Gauge Transformation ◽  
Darboux Transformation ◽  
Burgers Equation ◽  
Integrable Equation ◽  
Explicit Solutions ◽  
Spectral Problems ◽  
Coupled Burgers Equation

AbstractBased on the gauge transformation between the corresponding 3×3 matrix spectral problems, N-fold Darboux transformation for a coupled Burgers’ equation is constructed. Considering the N=1 case of the derived Darboux transformation, explicit solutions for the coupled Burgers’ equation are given and their figures are plotted. Moreover, conservation laws of this integrable equation are deduced.

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