scholarly journals Conservation laws, classical symmetries and exact solutions of the generalized KdV-Burgers-Kuramoto equation

Open Physics ◽  
2017 ◽  
Vol 15 (1) ◽  
pp. 433-439 ◽  
Author(s):  
Maria S. Bruzón ◽  
Elena Recio ◽  
Tamara M. Garrido ◽  
Almudena P. Márquez
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Arbitrary Function ◽  
Soliton Solutions ◽  
Travelling Wave ◽  
Multiplier Method ◽  
Explicit Solutions ◽  
Lie Point Symmetries ◽  
Classical Symmetries

AbstractFor a generalized KdV-Burgers-Kuramoto equation we have studied conservation laws by using the multiplier method, and investigated its first-level and second-level potential systems. Furthermore, the Lie point symmetries of the equation and the Lie point symmetries associated with the conserved vectors are determined. We obtain travelling wave reductions depending on the form of an arbitrary function. We present some explicit solutions: soliton solutions, kinks and antikinks.

scholarly journals Exact Explicit Solutions and Conservation Laws for a Coupled Zakharov-Kuznetsov System

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Chaudry Masood Khalique
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Equation Method ◽  
Explicit Solutions ◽  
Simplest Equation Method ◽  
De Vries

We study a coupled Zakharov-Kuznetsov system, which is an extension of a coupled Korteweg-de Vries system in the sense of the Zakharov-Kuznetsov equation. Firstly, we obtain some exact solutions of the coupled Zakharov-Kuznetsov system using the simplest equation method. Secondly, the conservation laws for the coupled Zakharov-Kuznetsov system will be constructed by using the multiplier approach.

scholarly journals On the Generation of Infinitely Many Conservation Laws of the Black-Scholes Equation

Computation ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 65
Author(s):  
Winter Sinkala
Keyword(s):  
Differential Equations ◽  
Conservation Laws ◽  
Conservation Law ◽  
The Other ◽  
Multiplier Method ◽  
Lie Point Symmetries ◽  
Mathematical Study ◽  
Black Scholes

Construction of conservation laws of differential equations is an essential part of the mathematical study of differential equations. In this paper we derive, using two approaches, general formulas for finding conservation laws of the Black-Scholes equation. In one approach, we exploit nonlinear self-adjointness and Lie point symmetries of the equation, while in the other approach we use the multiplier method. We present illustrative examples and also show how every solution of the Black-Scholes equation leads to a conservation law of the same equation.

scholarly journals Derivation of Conservation Laws for the Magma Equation Using the Multiplier Method: Power Law and Exponential Law for Permeability and Viscosity

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
N. Mindu ◽  
D. P. Mason
Keyword(s):  
Conservation Laws ◽  
Power Law ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Multiplier Method ◽  
Reduction Theorem ◽  
Double Reduction ◽  
Exponential Law ◽  
Spatial Derivatives ◽  
Derivatives Of

The derivation of conservation laws for the magma equation using the multiplier method for both the power law and exponential law relating the permeability and matrix viscosity to the voidage is considered. It is found that all known conserved vectors for the magma equation and the new conserved vectors for the exponential laws can be derived using multipliers which depend on the voidage and spatial derivatives of the voidage. It is also found that the conserved vectors are associated with the Lie point symmetry of the magma equation which generates travelling wave solutions which may explain by the double reduction theorem for associated Lie point symmetries why many of the known analytical solutions are travelling waves.

scholarly journals Symmetry Analysis, Explicit Solutions, and Conservation Laws of a Sixth-Order Nonlinear Ramani Equation

Symmetry ◽  
2018 ◽  
Vol 10 (8) ◽  
pp. 341 ◽  
Author(s):  
Aliyu Aliyu ◽  
Mustafa Inc ◽  
Abdullahi Yusuf ◽  
Dumitru Baleanu
Keyword(s):  
Conservation Laws ◽  
Travelling Wave ◽  
Symmetry Analysis ◽  
Optimal System ◽  
Sixth Order ◽  
Lie Point Symmetries ◽  
Lie Symmetry Analysis ◽  
One Dimensional ◽  
Analysis Technique ◽  
Conservation Theorem

In this work, we study the completely integrable sixth-order nonlinear Ramani equation. By applying the Lie symmetry analysis technique, the Lie point symmetries and the optimal system of one-dimensional sub-algebras of the equation are derived. The optimal system is further used to derive the symmetry reductions and exact solutions. In conjunction with the Riccati Bernoulli sub-ODE (RBSO), we construct the travelling wave solutions of the equation by solving the ordinary differential equations (ODEs) obtained from the symmetry reduction. We show that the equation is nonlinearly self-adjoint and construct the conservation laws (CL) associated with the Lie symmetries by invoking the conservation theorem due to Ibragimov. Some figures are shown to show the physical interpretations of the acquired results.

scholarly journals On New Conservation Laws of Fin Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-16 ◽  
Author(s):  
Gülden Gün Polat ◽  
Özlem Orhan ◽  
Teoman Özer
Keyword(s):  
Conservation Laws ◽  
Linear Form ◽  
Mathematical Physics ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Multiplier Method ◽  
Lie Point Symmetries ◽  
Invariant Solutions ◽  
Relationship Of ◽  
The Relationship

We study the new conservation forms of the nonlinear fin equation in mathematical physics. In this study, first, Lie point symmetries of the fin equation are identified and classified. Then by using the relationship of Lie symmetry andλ-symmetry, newλ-functions are investigated. In addition, the Jacobi Last Multiplier method and the approach, which is based on the factλ-functions are assumed to be of linear form, are considered as different procedures for lambda symmetry analysis. Finally, the corresponding new conservation laws and invariant solutions of the equation are presented.

scholarly journals Nonlinear self adjointness, conservation laws and exact solutions of ill-posed Boussinesq equation

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 37-43 ◽  
Author(s):  
Emrullah Yaşar ◽  
Sait San ◽  
Yeşim Sağlam Özkan
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Shallow Water ◽  
Water Waves ◽  
Function Method ◽  
Boussinesq Equation ◽  
Lie Point Symmetries ◽  
Shallow Water Waves ◽  
Non Linear ◽  
Ill Posed

AbstractIn this work, we consider the ill-posed Boussinesq equation which arises in shallow water waves and non-linear lattices. We prove that the ill-posed Boussinesq equation is nonlinearly self-adjoint. Using this property and Lie point symmetries, we construct conservation laws for the underlying equation. In addition, the generalized solitonary, periodic and compact-like solutions are constructed by the exp-function method.

scholarly journals On the Conservation Laws and Exact Solutions of a Modified Hunter-Saxton Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Sait San ◽  
Emrullah Yaşar
Keyword(s):  
Liquid Crystals ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Nematic Liquid Crystals ◽  
Lie Point Symmetries ◽  
Local Conservation ◽  
The Relationship ◽  
Conservation Method

We study the modified Hunter-Saxton equation which arises in modelling of nematic liquid crystals. We obtain local conservation laws using the nonlocal conservation method and multiplier approach. In addition, using the relationship between conservation laws and Lie-point symmetries, some reductions and exact solutions are obtained.

scholarly journals Lagrangian Formulation, Conservation Laws, Travelling Wave Solutions: A Generalized Benney-Luke Equation

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1480
Author(s):  
Sivenathi Oscar Mbusi ◽  
Ben Muatjetjeja ◽  
Abdullahi Rashid Adem
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Lagrangian Density ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Noether Symmetries ◽  
Extended Tanh Method ◽  
Wave Solutions ◽  
Tanh Method ◽  
Derivatives Of

The aim of this paper is to find the Noether symmetries of a generalized Benney-Luke equation. Thereafter, we construct the associated conserved vectors. In addition, we search for exact solutions for the generalized Benney-Luke equation through the extended tanh method. A brief observation on equations arising from a Lagrangian density function with high order derivatives of the field variables, is also discussed.

Conservation Laws and Exact Solutions with Symmetry Reduction of Nonlinear Reaction Diffusion Equations

2015 ◽  
Vol 16 (3-4) ◽  
pp. 191-196 ◽  
Author(s):  
Filiz Tascan ◽  
Arzu Yakut
Keyword(s):  
Differential Equations ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Diffusion Equations ◽  
Lie Point Symmetries ◽  
Important Place ◽  
Physical Problems ◽  
Nonlinear Reaction

AbstractIn this work we study one of the most important applications of symmetries to physical problems, namely the construction of conservation laws. Conservation laws have important place for applications of differential equations and solutions, also in all physics applications. And so, this study deals conservation laws of first- and second-type nonlinear (NL) reaction diffusion equations. We used Ibragimov’s approach for finding conservation laws for these equations. And then, we found exact solutions of first- and second-type NL reaction diffusion equations with Lie-point symmetries.

scholarly journals Symmetry Analysis, Exact Solutions and Conservation Laws of a Benjamin–Bona–Mahony–Burgers Equation in 2+1-Dimensions

Symmetry ◽  
2021 ◽  
Vol 13 (11) ◽  
pp. 2083
Author(s):  
María S. Bruzón ◽  
Tamara M. Garrido-Letrán ◽  
Rafael de la Rosa
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Soliton Solutions ◽  
Third Order ◽  
Partial Differential ◽  
Symmetry Reductions ◽  
Multipliers Method ◽  
The Family

The Benjamin–Bona–Mahony equation describes the unidirectional propagation of small-amplitude long waves on the surface of water in a channel. In this paper, we consider a family of generalized Benjamin–Bona–Mahony–Burgers equations depending on three arbitrary constants and an arbitrary function G(u). We study this family from the standpoint of the theory of symmetry reductions of partial differential equations. Firstly, we obtain the Lie point symmetries admitted by the considered family. Moreover, taking into account the admitted point symmetries, we perform symmetry reductions. In particular, for G′(u)≠0, we construct an optimal system of one-dimensional subalgebras for each maximal Lie algebra and deduce the corresponding (1+1)-dimensional nonlinear third-order partial differential equations. Then, we apply Kudryashov’s method to look for exact solutions of the nonlinear differential equation. We also determine line soliton solutions of the family of equations in a particular case. Lastly, through the multipliers method, we have constructed low-order conservation laws admitted by the family of equations.

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