scholarly journals Group-invariant solutions for one dimensional, inviscid hydrodynamics

AIP Advances ◽  
2019 ◽  
Vol 9 (8) ◽  
pp. 085113 ◽  
Author(s):  
J. D. McHardy ◽  
E. J. Albright ◽  
S. D. Ramsey ◽  
J. H. Schmidt
Keyword(s):  
Invariant Solutions ◽  
One Dimensional ◽  
Group Invariant ◽  
Inviscid Hydrodynamics ◽  
Group Invariant Solutions

Optimal systems and their group-invariant solutions to geodesic equations

2019 ◽  
Vol 16 (09) ◽  
pp. 1950135
Author(s):  
Bismah Jamil ◽  
Tooba Feroze ◽  
Muhammad Safdar
Keyword(s):  
Nonlinear Systems ◽  
Separation Of Variables ◽  
Noether Symmetries ◽  
Invariant Solutions ◽  
One Dimensional ◽  
Group Invariant ◽  
First Order ◽  
Geodesic Equations ◽  
Integrating Factor ◽  
Group Invariant Solutions

We find one-dimensional optimal systems of the Lie subalgebras of Noether symmetries associated with systems of geodesic equations. Further, we find invariants corresponding to each element of the derived optimal system. The derived invariants are shown to reduce systems of geodesic equations (nonlinear systems of quadratically semi-linear second-order ordinary differential equations (ODEs)) to nonlinear systems of first-order ODEs. The resulting systems are solved via known methods (e.g. separation of variables, integrating factor, etc.). In some cases, we provide exact solutions of these systems of geodesic equations.

scholarly journals Invariant Solutions and Nonlinear Self-Adjointness of the Two-Component Chaplygin Gas Equation

2019 ◽  
Vol 2019 ◽  
pp. 1-9 ◽  
Author(s):  
Ben Gao ◽  
Yanxia Wang
Keyword(s):  
Chaplygin Gas ◽  
Group Method ◽  
Invariant Solutions ◽  
One Dimensional ◽  
Group Invariant ◽  
Lie Group Method ◽  
Two Component ◽  
Present Universe ◽  
Group Invariant Solutions ◽  
Similarity Reductions

In this paper, the Lie group method is performed on a special dark fluid, the Chaplygin gas, which describes both dark matter and dark energy in the present universe. Based on an optimal system of one-dimensional subalgebras, similarity reductions and group invariant solutions are given. Finally, by means of Ibragimov’s method, conservation laws are obtained.

Symmetry Reductions and Exact Solutions of the (2+1)-Dimensional Navier-Stokes Equations

2010 ◽  
Vol 65 (6-7) ◽  
pp. 504-510 ◽  
Author(s):  
Xiaorui Hua ◽  
Zhongzhou Dongb ◽  
Fei Huangc ◽  
Yong Chena
Keyword(s):  
Vector Fields ◽  
Stokes Equations ◽  
Stationary Wave ◽  
Navier Stokes ◽  
Invariant Solutions ◽  
Navier Stokes Equations ◽  
One Dimensional ◽  
Group Invariant ◽  
Dimensional Group ◽  
Group Invariant Solutions

By means of the classical symmetry method, we investigate the (2+1)-dimensional Navier-Stokes equations. The symmetry group of Navier-Stokes equations is studied and its corresponding group invariant solutions are constructed. Ignoring the discussion of the infinite-dimensional subalgebra, we construct an optimal system of one-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 Navier-Stokes equations are obtained. For three interesting solutions, the figures are given out to show their properties: the solution of stationary wave of fluid (real part) appears as a balance between fluid advection (nonlinear term) and friction parameterized as a horizontal harmonic diffusion of momentum.

scholarly journals On optimal system, exact solutions and conservation laws of the modified equal-width equation

2018 ◽  
Vol 3 (2) ◽  
pp. 409-418 ◽  
Author(s):  
Chaudry Masood Khalique ◽  
Oke Davies Adeyemo ◽  
Innocent Simbanefayi
Keyword(s):  
Wave Propagation ◽  
Conservation Laws ◽  
Optimal System ◽  
Lie Point Symmetries ◽  
Invariant Solutions ◽  
Nonlinear Media ◽  
One Dimensional ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
Dimensional Wave

AbstractIn this paper we study the modified equal-width equation, which is used in handling simulation of a single dimensional wave propagation in nonlinear media with dispersion processes. Lie point symmetries of this equation are computed and used to construct an optimal system of one-dimensional subalgebras. Thereafter using an optimal system of one-dimensional subalgebras, symmetry reductions and new group-invariant solutions are presented. The solutions obtained are cnoidal and snoidal waves. Furthermore, conservation laws for the modified equal-width equation are derived by employing two different methods, the multiplier method and Noether approach.

OPTIMAL SYSTEMS AND GROUP INVARIANT SOLUTIONS FOR A MODEL ARISING IN FINANCIAL MATHEMATICS

2009 ◽  
Vol 14 (4) ◽  
pp. 495-502 ◽  
Author(s):  
Bienvenue Feugang Nteumagne ◽  
Raseelo J. Moitsheki
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Financial Mathematics ◽  
Invariant Solutions ◽  
Pricing Model ◽  
One Dimensional ◽  
Group Invariant ◽  
Optimal System Of Subalgebras ◽  
Group Invariant Solutions ◽  
The One

We consider a bond‐pricing model described in terms of partial differential equations (PDEs). Classical Lie point symmetry analysis of the considered PDEs resulted in a number of point symmetries being admitted. The one‐dimensional optimal system of subalgebras is constructed. Following the symmetry reductions, we determine the group‐invariant solutions.

scholarly journals Optimal System and Group Invariant Solutions of the Whitham-Broer-Kaup System

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Xiaorui Hu ◽  
Yongyang Jin ◽  
Kai Zhou
Keyword(s):  
Lie Algebra ◽  
Nonlinear Waves ◽  
Optimal System ◽  
Invariant Solutions ◽  
One Dimensional ◽  
Nonlocal Symmetries ◽  
Group Invariant ◽  
Group Invariant Solutions

From the nonlocal symmetries of the Whitham-Broer-Kaup system, an eight-dimensional Lie algebra is found and the corresponding one-dimensional optimal system is constructed to provide an inequivalent classification. Six types of inequivalent group invariant solutions are demonstrated, some of which reflect the interactions between soliton and other nonlinear waves.

scholarly journals Some General New Einstein Walker Manifolds

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Mehdi Nadjafikhah ◽  
Mehdi Jafari
Keyword(s):  
Partial Differential Equations ◽  
Symmetry Group ◽  
Lie Symmetry ◽  
Group Method ◽  
Invariant Solutions ◽  
Point Symmetry Group ◽  
One Dimensional ◽  
Group Invariant ◽  
Group Invariant Solutions ◽  
Walker Manifold

Lie symmetry group method is applied to find the Lie point symmetry group of a system of partial differential equations that determines general form of four-dimensional Einstein Walker manifold. Also we will construct the optimal system of one-dimensional Lie subalgebras and investigate some of its group invariant 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.

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