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.

scholarly journals Analysis of the Symmetries and Conservation Laws of the Nonlinear Jaulent-Miodek Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Mehdi Nadjafikhah ◽  
Mostafa Hesamiarshad
Keyword(s):  
Conservation Laws ◽  
Symmetry Group ◽  
Symbolic Computation ◽  
Lie Symmetry ◽  
Lie Symmetry Method ◽  
Reduced Equations ◽  
Symmetry Generators ◽  
Symmetry Method

Lie symmetry method is performed for the nonlinear Jaulent-Miodek equation. We will find the symmetry group and optimal systems of Lie subalgebras. The Lie invariants associated with the symmetry generators as well as the corresponding similarity reduced equations are also pointed out. And conservation laws of the J-M equation are presented with two steps: firstly, finding multipliers for computation of conservation laws and, secondly, symbolic computation of conservation laws will be applied.

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

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.

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.

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