A nondegeneracy criterion for the sn(n + 1)/2-parameter Lie group of transformations of the space ℝ sn

2010 ◽  
Vol 4 (3) ◽  
pp. 349-353
Author(s):  
V. A. Kyrov
Keyword(s):  
Lie Group ◽  
Lie Group Of Transformations

scholarly journals A comparison of the modern Lie scaling method to classical scaling techniques

2016 ◽  
Vol 20 (7) ◽  
pp. 2669-2678 ◽  
Author(s):  
James Polsinelli ◽  
M. Levent Kavvas
Keyword(s):  
Lie Group ◽  
Overland Flow ◽  
New Technology ◽  
Scaling Method ◽  
Lie Group Of Transformations ◽  
Key Points ◽  
Scaling Transformation ◽  
Scaling Models ◽  
Scaling Methods ◽  
Channel Hydraulics

Abstract. In the past 2 decades a new modern scaling technique has emerged from the highly developed theory on the Lie group of transformations. This new method has been applied by engineers to several problems in hydrology and hydraulics, including but not limited to overland flow, groundwater dynamics, sediment transport, and open channel hydraulics. This study attempts to clarify the relationship this new technology has with the classical scaling method based on dimensional analysis, non-dimensionalization, and the Vaschy–Buckingham-Π theorem. Key points of the Lie group theory, and the application of the Lie scaling transformation, are outlined and a comparison is made with two classical scaling models through two examples: unconfined groundwater flow and contaminant transport. The Lie scaling method produces an invariant scaling transformation of the prototype variables, which ensures the dynamics between the model and prototype systems will be preserved. Lie scaling can also be used to determine the conditions under which a complete model is dynamically, kinematically, and geometrically similar to the prototype phenomenon. Similarities between the Lie and classical scaling methods are explained, and the relative strengths and weaknesses of the techniques are discussed.

scholarly journals A comparison of the modern Lie scaling method to classical scaling techniques

2015 ◽  
Vol 12 (10) ◽  
pp. 10197-10219 ◽  
Author(s):  
J. Polsinelli ◽  
M. L. Kavvas
Keyword(s):  
Lie Group ◽  
New Technology ◽  
Scaling Method ◽  
Lie Group Of Transformations ◽  
Key Points ◽  
Scaling Transformation ◽  
Scaling Models ◽  
Scaling Methods ◽  
Channel Hydraulics ◽  
The Relationship

Abstract. In the past two decades a new modern scaling technique has emerged from the highly developed theory on the Lie group of transformations. This new method has been applied by engineers to several problems in hydrology and hydraulics including but not limited to groundwater dynamics, sediment transport, and open channel hydraulics. This study attempts to clarify the relationship this new technology has with the classical scaling method based on dimensional analysis, non dimensionalization, and the Buckingham Π theorem. Key points of the Lie group theory, and the application of the Lie scaling transformation, are outlined and a comparison is done with two classical scaling models through two examples: unconfined groundwater flow and contaminant transport. The Lie scaling method produces an invariant scaling transformation of the prototype variables which ensures the dynamics between the model and prototype systems will be preserved. Lie scaling can also be used to determine the conditions under which a complete model is dynamically, kinematically, and geometrically similar to the prototype phenomenon. Similarities between the Lie and classical scaling methods are explained, and the relative strengths and weaknesses of the techniques are discussed.

scholarly journals SIMILARITY SOLUTIONS FOR CYLINDRICAL SHOCK WAVE IN SELF-GRAVITATING NON-IDEAL GAS WITH AXIAL MAGNETIC FIELD: ISOTHERMAL FLOW

2021 ◽  
Author(s):  
Nandita Gupta ◽  
Kajal Sharma ◽  
Rajan Arora
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Lie Group ◽  
Axial Magnetic Field ◽  
Similarity Solutions ◽  
Ideal Gas ◽  
Isothermal Flow ◽  
Field Variation ◽  
Cylindrical Shock Wave ◽  
Lie Group Of Transformations

The purpose of this study is to obtain the solution using the Lie group of symmetry method for the problem of propagating magnetogasdynamic strong cylindrical shock wave in a self-gravitating non-ideal gas with the magnetic field which is taken to be axial. Here, isothermal flow is considered. In the undisturbed medium, varying magnetic field and density are taken. Out of four different cases, only three cases yield the similarity solutions. Numerical computations have been performed for the cases of power-law and exponential law shock paths, to find out the behavior of flow variables in the flow-field immediately behind the shock. Similarity solutions are carried out by taking arbitrary constants in the expressions of infinitesimals of the Lie group of transformations. Also, the study of the present work provides a clear picture of whether and how the variations in the non-ideal parameter of the gas, Alfven-Mach number, adiabatic exponent, ambient magnetic field variation index and gravitational parameter affect the propagation of shock and the flow behind it. Software package “MATLAB” is used for all the computations.

Symmetry properties of fractional order transport equations

2012 ◽  
Vol 9 (1) ◽  
pp. 59-64
Author(s):  
R.K. Gazizov ◽  
A.A. Kasatkin ◽  
S.Yu. Lukashchuk
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lie Group ◽  
Initial Conditions ◽  
Group Analysis ◽  
Equivalence Transformation ◽  
Point Change ◽  
Infinitesimal Operator ◽  
Symmetry Properties ◽  
Fractional Differential

In the paper some features of applying Lie group analysis methods to fractional differential equations are considered. The problem related to point change of variables in the fractional differentiation operator is discussed and some general form of transformation that conserves the form of Riemann-Liouville fractional operator is obtained. The prolongation formula for extending an infinitesimal operator of a group to fractional derivative with respect to arbitrary function is presented. Provided simple example illustrates the necessity of considering both local and non-local symmetries for fractional differential equations in particular cases including the initial conditions. The equivalence transformation forms for some fractional differential equations are discussed and results of group classification of the wave-diffusion equation are presented. Some examples of constructing particular exact solutions of fractional transport equation are given, based on the Lie group methods and the method of invariant subspaces.

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