New Conservation Laws, Lagrangian Forms, and Exact Solutions of Modified Emden Equation

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
Gülden Gün Polat ◽  
Teoman Özer
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
First Integrals ◽  
Lie Symmetries ◽  
Jacobi Method ◽  
Emden Equation ◽  
Mathematical Connections ◽  
Integrating Factors

This study deals with the determination of Lagrangians, first integrals, and integrating factors of the modified Emden equation by using Jacobi and Prelle–Singer methods based on the Lie symmetries and λ-symmetries. It is shown that the Jacobi method enables us to obtain Jacobi last multipliers by means of the Lie symmetries of the equation. Additionally, via the Lie symmetries of modified Emden equation, we analyze some mathematical connections between λ-symmetries and Prelle–Singer method. New and nontrivial Lagrangian forms, conservation laws, and exact solutions of the equation are presented and discussed.

scholarly journals Lie symmetries, conservation laws and exact solutions of a generalized quasilinear KdV equation with degenerate dispersion

2020 ◽  
Vol 13 (10) ◽  
pp. 2691-2701
Author(s):  
María-Santos Bruzón ◽  
◽  
Elena Recio ◽  
Tamara-María Garrido ◽  
Rafael de la Rosa
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Kdv Equation ◽  
Lie Symmetries

scholarly journals On certain new and exact solutions of the Emden-Fowler equation and Emden equation via invariant variational principles and group invariance

1991 ◽  
Vol 32 (4) ◽  
pp. 457-468 ◽  
Author(s):  
O. P. Bhutani ◽  
K. Vijayakumar
Keyword(s):  
Differential Equation ◽  
Exact Solutions ◽  
Lie Group ◽  
Variational Principles ◽  
First Integrals ◽  
Noether Theorem ◽  
Repeated Application ◽  
Scale Invariant ◽  
Emden Equation ◽  
Fowler Equation

AbstractAfter formulating the alternate potential principle for the nonlinear differential equation corresponding to the generalised Emden-Fowler equation, the invariance identities of Rund [14] involving the Lagrangian and the generators of the infinitesimal Lie group are used for writing down the first integrals of the said equation via the Noether theorem. Further, for physical realisable forms of the parameters involved and through repeated application of invariance under the transformation obtained, a number of exact solutions are arrived at both for the Emden-Fowler equation and classical Emden equations. A comparative study with Bluman-Cole and scale-invariant techniques reveals quite a number of remarkable features of the techniques used here.

scholarly journals Integrating Factors and First Integrals for Liénard Type and Frequency-Damped Oscillators

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
Emrullah Yaşar
Keyword(s):  
First Integral ◽  
First Integrals ◽  
The Other ◽  
First Order ◽  
Integrating Factor ◽  
Damped Oscillators ◽  
Damped Oscillator ◽  
Integrating Factors ◽  
Oscillator Equations

We consider Liénard type and frequency-damped oscillator equations. Integrating factors and the associated first integrals are derived from the method to compute -symmetries and the associated reduction algorithm. The knowledge of a symmetry of the equation permits the determination of an integrating factor or a first integral by means of coupled first-order linear systems of partial differential equations. We will compare our results with those gained by the other methods.

scholarly journals Lie Symmetries, Conservation Laws and Exact Solutions for Jaulent-Miodek Equations

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1319
Author(s):  
Jian-Ting Pei ◽  
Yu-Shan Bai
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Lie Algebra ◽  
Adjoint Equation ◽  
Lie Symmetries ◽  
Equation Method ◽  
One Dimensional

In this paper, the Lie symmetries of the Jaulent-Miodek (JM) equations are calculated and one dimensional optimal systems of Lie algebra are obtained. Furthermore, the conservation laws are constructed by using the adjoint equation method. Finally, the exact solutions of the equations are obtained by the conservation laws.

scholarly journals On PT Symmetry Systems: Invariance, Conservation Laws, and Reductions

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
P. Masemola ◽  
A. H. Kara
Keyword(s):  
Schrödinger Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Schrodinger Equation ◽  
Lie Symmetries ◽  
Pt Symmetry ◽  
Noether Symmetries ◽  
Cubic Schrödinger Equation ◽  
The Relationship

An analysis of a PT symmetric coupler with “gain in one waveguide and loss in another” is made; a transformation in the PT system and some assumptions results in a scalar cubic Schrödinger equation. We investigate the relationship between the conservation laws and Lie symmetries and investigate a Lagrangian, corresponding Noether symmetries, conserved vectors, and exact solutions via “double reductions.”

Symmetries and conservation laws of a damped Boussinesq equation

2016 ◽  
Vol 30 (28n29) ◽  
pp. 1640012 ◽  
Author(s):  
María Luz Gandarias ◽  
María Rosa
Keyword(s):  
Differential Equations ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Ordinary Differential Equations ◽  
Source Term ◽  
Boussinesq Equation ◽  
Lie Symmetries ◽  
Multiplier Method ◽  
Independent Source ◽  
Damped Equation

In this work, we consider a damped equation with a time-independent source term. We derive the classical Lie symmetries admitted by the equation as well as the reduced ordinary differential equations. We also present some exact solutions. Conservation laws for this equation are constructed by using the multiplier method.

scholarly journals Time fractional modified KdV-type equations: Lie symmetries, exact solutions and conservation laws

Open Physics ◽  
2019 ◽  
Vol 17 (1) ◽  
pp. 480-488
Author(s):  
Fangqin He ◽  
Lianzhong Li
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Lie Symmetries ◽  
Symmetry Reductions ◽  
Kdv Type Equations

Abstract In the paper, we research a time fractional modified KdV-type equations.We give the symmetry reductions and exact solutions of the equations, and we investigate the convergence of the solutions. In addition, the conservation laws of the equations are constructed.

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