Lie symmetries of generalized Burgers equations: application to boundary-value problems

@article{Vaneeva2013LieSO,
  title={Lie symmetries of generalized Burgers equations: application to boundary-value problems},
  author={Olena O. Vaneeva and Christodoulos Sophocleous and Peter G. L. Leach},
  journal={Journal of Engineering Mathematics},
  year={2013},
  volume={91},
  pages={165-176}
}
There exist several approaches exploiting Lie symmetries in the reduction of boundary-value problems for partial differential equations modelling real-world phenomena to those problems for ordinary differential equations. Using an example of generalized Burgers equations appearing in non-linear acoustics we show that the direct procedure of solving boundary-value problems using Lie symmetries first described by Bluman is more general and straightforward than the method suggested by Moran and… 

Figures and Tables from this paper

Numerical solutions of boundary value problems for variable coefficient generalized KdV equations using Lie symmetries
TLDR
The found Lie symmetries are applied in order to reduce the initial and boundary value problem for the generalized KdV equations to an initialvalue problem for nonlinear third-order ODEs.
Group classification and exact solutions of variable-coefficient generalized Burgers equations with linear damping
Admissible point transformations between Burgers equations with linear damping and time-dependent coefficients are described and used in order to exhaustively classify Lie symmetries of these
Lie symmetry analysis of some conformable fractional partial differential equations
In this article, Lie symmetry analysis is used to investigate invariance properties of some nonlinear fractional partial differential equations with conformable fractional time and space derivatives.
Enhanced symmetry analysis of two-dimensional degenerate Burgers equation
We carry out enhanced symmetry analysis of a two-dimensional degenerate Burgers equation. Its complete point-symmetry group is found using the algebraic method, and its generalized symmetries are
SYMMETRIES AND EXACT SOLUTIONS OF CONFORMABLE FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS
In this paper Lie group analysis is used to investigate invariance properties of nonlinear fractional partial differential equations with conformable fractional time derivative. The analysis is
Lie Group Classification for a Class of Compound KdV–Burgers Equations with Time-Dependent Coefficients
We derive the Lie group classification for a general class of KdV–Burgers equations, where the coefficients are functions of time. We demonstrate how important is the use of equivalence
Group analysis of general Burgers–Korteweg–de Vries equations
The complete group classification problem for the class of (1+1)-dimensional rth order general variable-coefficient Burgers–Korteweg–de Vries equations is solved for arbitrary values of r greater
Group Classification and Conservation Laws of a Class of Hyperbolic Equations
  • J. Ndogmo
  • Mathematics
    Abstract and Applied Analysis
  • 2021
Abstracts. A method for the group classification of differential equations is proposed. It is based on the determination of all possible cases of linear dependence of certain indeterminates appearing
Enhanced group classification of nonlinear diffusion–reaction equations with gradient-dependent diffusivity
Abstract We carry out the enhanced group classification of a class of (1+1)-dimensional nonlinear diffusion–reaction equations with gradient-dependent diffusivity using the two-step version of the
Lie symmetries of a generalized Kuznetsov–Zabolotskaya–Khokhlov equation
Abstract We consider a class of generalized Kuznetsov–Zabolotskaya–Khokhlov (gKZK) equations and determine its equivalence group, which is then used to give a complete symmetry classification of this
...
1
2
...

References

SHOWING 1-10 OF 43 REFERENCES
Numerical solutions of boundary value problems for variable coefficient generalized KdV equations using Lie symmetries
TLDR
The found Lie symmetries are applied in order to reduce the initial and boundary value problem for the generalized KdV equations to an initialvalue problem for nonlinear third-order ODEs.
Symmetry reductions and new exact invariant solutions of the generalized Burgers equation arising in nonlinear acoustics
We perform a complete Lie symmetry classification of the generalized Burgers equation arising in nonlinear acoustics. We obtain seven functional forms of the ray tube area that allow symmetry
New results on group classification of nonlinear diffusion–convection equations
Using a new method and additional (conditional and partial) equivalence transformations, we performed group classification in a class of variable coefficient (1 + 1)-dimensional nonlinear
Group method solutions of the generalized forms of Burgers, Burgers–KdV and KdV equations with time-dependent variable coefficients
Solutions for the generalized forms of Burgers, Burgers–KdV, and KdV equations with time-dependent variable coefficients and with initial and boundary conditions are constructed. The analysis rests
Lie Symmetry Analysis of Differential Equations in Finance
Lie group theory is applied to differential equations occurring as mathematical models in financial problems. We begin with the complete symmetry analysis of the one-dimensional Black–Scholes model
Symmetry preserving parameterization schemes
Methods for the design of physical parameterization schemes that possess certain invariance properties are discussed. These methods are based on different techniques of group classification and
Reduction operators and exact solutions of generalized Burgers equations
Abstract Reduction operators of generalized Burgers equations are studied. A connection between these equations and potential fast diffusion equations with power nonlinearity of degree −1 via
Nonlinear Diffusive Waves
This monograph deals with Burgers' equation and its generalisations. Such equations describe a wide variety of nonlinear diffusive wave phenomena, for instance, in atmospheric physics, plasma
Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems
INTRODUCTION FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS Linear Partial Differential Equations of First Order Quasilinear Partial Differential Equation of First Order Reduction of ut = unux+H(x,t,u)=0
...
1
2
3
4
5
...