scholarly journals Group-Invariant Solutions for Two-Dimensional Free, Wall, and Liquid Jets Having Finite Fluid Velocity at Orifice

2011 ◽  
Vol 2011 ◽  
pp. 1-12
Author(s):  
R. Naz
Keyword(s):  
Fluid Velocity ◽  
Invariant Solution ◽  
Jet Flows ◽  
Order Partial Differential Equation ◽  
Free Wall ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
Third Order ◽  
Group Invariant ◽  
Group Invariant Solutions

The group-invariant solutions for nonlinear third-order partial differential equation (PDE) governing flow in two-dimensional jets (free, wall, and liquid) having finite fluid velocity at orifice are constructed. The symmetry associated with the conserved vector that was used to derive the conserved quantity for the jets (free, wall, and liquid) generated the group invariant solution for the nonlinear third-order PDE for the stream function. The comparison between results for two-dimensional jet flows having finite and infinite fluid velocity at orifice is presented. The general form of the group invariant solution for two-dimensional jets is given explicitly.

Group Invariant Solutions of the Full Plastic Torsion of Rod with Arbitrary Shaped Cross Sections

2012 ◽  
Vol 4 (03) ◽  
pp. 382-388 ◽  
Author(s):  
Kefu Huang ◽  
Houguo Li
Keyword(s):  
Symmetry Group ◽  
Cross Sections ◽  
Lie Group ◽  
Equilibrium Equation ◽  
Yield Criterion ◽  
Invariant Solution ◽  
Invariant Solutions ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
Full Symmetry

AbstractBased on the theory of Lie group analysis, the full plastic torsion of rod with arbitrary shaped cross sections that consists in the equilibrium equation and the non-linear Saint Venant-Mises yield criterion is studied. Full symmetry group admitted by the equilibrium equation and the yield criterion is a finitely generated Lie group with ten parameters. Several subgroups of the full symmetry group are used to generate invariants and group invariant solutions. Moreover, physical explanations of each group invariant solution are discussed by all appropriate transformations. The methodology and solution techniques used belong to the analytical realm.

scholarly journals λ-Symmetry and μ-Symmetry Reductions and Invariant Solutions of Four Nonlinear Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1138
Author(s):  
Yu-Shan Bai ◽  
Jian-Ting Pei ◽  
Wen-Xiu Ma
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Order Partial Differential Equation ◽  
Invariant Solutions ◽  
Specific Group ◽  
Partial Differential ◽  
Group Invariant ◽  
Dimensional Diffusion ◽  
Group Invariant Solutions ◽  
Integrating Factors

On one hand, we construct λ-symmetries and their corresponding integrating factors and invariant solutions for two kinds of ordinary differential equations. On the other hand, we present μ-symmetries for a (2+1)-dimensional diffusion equation and derive group-reductions of a first-order partial differential equation. A few specific group invariant solutions of those two partial differential equations are constructed.

The reducibility of partially invariant solutions of systems of partial differential equations

1995 ◽  
Vol 6 (4) ◽  
pp. 329-354 ◽  
Author(s):  
Jeffrey Ondich
Keyword(s):  
Differential Equations ◽  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Invariant Solution ◽  
Elliptic Systems ◽  
Invariant Solutions ◽  
Hyperbolic Conservation ◽  
Group Invariant ◽  
Partially Invariant Solution ◽  
Group Invariant Solutions

Ovsiannikov's partially invariant solutions of differential equations generalize Lie's group invariant solutions. A partially invariant solution is only interesting if it cannot be discovered more readily as an invariant solution. Roughly, a partially invariant solution that can be discovered more directly by Lie's method is said to be reducible. In this paper, I develop conditions under which a partially invariant solution or a class of such solutions must be reducible, and use these conditions both to obtain non-reducible solutions to a system of hyperbolic conservation laws, and to demonstrate that some systems have no non-reducible solutions. I also demonstrate that certain elliptic systems have no non-reducible solutions.

Symmetry Reductions and Exact Solutions of the Two-Layer Model in Atmosphere

2011 ◽  
Vol 66 (1-2) ◽  
pp. 75-86 ◽  
Author(s):  
Zhongzhou Dong ◽  
Fei Huang ◽  
Yong Chen
Keyword(s):  
Vector Fields ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
Layer Model ◽  
Horizontal Structure ◽  
One Dimensional ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
Two Layer Model ◽  
Model Equations

By means of the classical symmetry method, we investigate the two-layer model in atmosphere. The symmetry group of two-layer model equations is studied and its corresponding group invariant solutions are constructed. Ignoring the discussion of the infinite-dimensional subalgebra, we construct the optimal system of one-dimensional and two-dimensional group invariant solutions. Furthermore, using the associated vector fields of the obtained symmetry, we give out the reductions by one-dimensional and two-dimensional subalgebras, and some explicit solutions of two-layer model equations are obtained. For some interesting solutions, the figures are given out to show their properties. Some solutions can describe the horizontal structure of tropical cyclones (TC). Especially, a new solution of double-eyewall structure of TCs is firstly found in this two-layer model.

scholarly journals Symmetry Reduction of the Two-Dimensional Ricci Flow Equation

Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Mehdi Nadjafikhah ◽  
Mehdi Jafari
Keyword(s):  
Ricci Flow ◽  
Adjoint Representation ◽  
Flow Equation ◽  
Two Dimensional ◽  
Invariant Solutions ◽  
Similarity Reduction ◽  
One Dimensional ◽  
Group Invariant ◽  
Dimensional Group ◽  
Group Invariant Solutions

This paper is devoted to obtain the one-dimensional group invariant solutions of the two-dimensional Ricci flow ((2D) Rf) equation. By classifying the orbits of the adjoint representation of the symmetry group on its Lie algebra, the optimal system of one-dimensional subalgebras of the ((2D) Rf) equation is obtained. For each class, we will find the reduced equation by the method of similarity reduction. By solving these reduced equations, we will obtain new sets of group invariant solutions for the ((2D) Rf) equation.

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