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 Conservation Laws and Travelling Wave Solutions for Double Dispersion Equations in (1+1) and (2+1) Dimensions

Symmetry ◽  
2020 ◽  
Vol 12 (6) ◽  
pp. 950 ◽  
Author(s):  
María Luz Gandarias ◽  
María Rosa Durán ◽  
Chaudry Masood Khalique
Keyword(s):  
Conservation Laws ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Two Dimensions ◽  
Travelling Wave Solutions ◽  
Dispersion Equations ◽  
Different Dimensions ◽  
Double Dispersion ◽  
Long Nonlinear Wave ◽  
Line Soliton

In this article, we investigate two types of double dispersion equations in two different dimensions, which arise in several physical applications. Double dispersion equations are derived to describe long nonlinear wave evolution in a thin hyperelastic rod. Firstly, we obtain conservation laws for both these equations. To do this, we employ the multiplier method, which is an efficient method to derive conservation laws as it does not require the PDEs to admit a variational principle. Secondly, we obtain travelling waves and line travelling waves for these two equations. In this process, the conservation laws are used to obtain a triple reduction. Finally, a line soliton solution is found for the double dispersion equation in two dimensions.

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.

scholarly journals Travelling waves and conservation laws of a (2+1)-dimensional coupling system with Korteweg-de Vries equation

2018 ◽  
Vol 3 (1) ◽  
pp. 241-254 ◽  
Author(s):  
Chaudry Masood Khalique ◽  
Isaiah Elvis Mhlanga
Keyword(s):  
Conservation Laws ◽  
Vries Equation ◽  
Travelling Waves ◽  
Hamiltonian Formulation ◽  
Expansion Method ◽  
Travelling Wave ◽  
Lie Symmetry Analysis ◽  
Coupling System ◽  
De Vries ◽  
Dimensional Coupling

AbstractIn this paper we study a (2+1)-dimensional coupling system with the Korteweg-de Vries equation, which is associated with non-semisimple matrix Lie algebras. Its Lax-pair and bi-Hamiltonian formulation were obtained and presented in the literature. We utilize Lie symmetry analysis along with the (G′/G)–expansion method to obtain travelling wave solutions of this system. Furthermore, conservation laws are constructed using the multiplier method.

scholarly journals Soliton solution and conservation laws of the Zakharov equation in plasmas with power law nonlinearity

2013 ◽  
Vol 18 (2) ◽  
pp. 153-159 ◽  
Author(s):  
Richard Morris ◽  
Abdul Hamid Kara ◽  
Anjan Biswas
Keyword(s):  
Conservation Laws ◽  
Power Law ◽  
Soliton Solution ◽  
Traveling Wave ◽  
Lie Symmetries ◽  
Multiplier Method ◽  
Conserved Quantities ◽  
Zakharov Equation

This paper studies the Zakharov equation with power law nonlinearity. The traveling wave hypothesis is applied to obtain the 1-soliton solution of this equation. The multiplier method from Lie symmetries is subsequently utilized to obtain the conservation laws of the equations. Finally, using the exact 1-soliton solution, the conserved quantities are listed.

scholarly journals Some Reduction and Exact Solutions of a Higher-Dimensional Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Guangming Wang ◽  
Zhong Han
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Conservation Law ◽  
Second Order ◽  
Reduction Theorem ◽  
Double Reduction ◽  
Noether’S Theorem ◽  
Higher Dimensional ◽  
Dimensional Equation ◽  
Nonlinear Ode

The conservation laws of the(3+1)-dimensional Zakharov-Kuznetsov equation were obtained using Noether’s theorem after an interesting substitutionu=vxto the equation. Then, with the aid of an obtained conservation law, the generalized double reduction theorem was applied to this equation. It can be verified that the reduced equation is a second order nonlinear ODE. Finally, some exact solutions of the Zakharov-Kuznetsov equation were constructed after solving the reduced equation.

scholarly journals CONTRACTIVE METRICS FOR SCALAR CONSERVATION LAWS

2005 ◽  
Vol 02 (01) ◽  
pp. 91-107 ◽  
Author(s):  
FRANÇOIS BOLLEY ◽  
YANN BRENIER ◽  
GRÉGOIRE LOEPER
Keyword(s):  
Conservation Laws ◽  
Wasserstein Distance ◽  
Entropy Solutions ◽  
Scalar Conservation Laws ◽  
Contraction Property ◽  
Spatial Derivatives ◽  
Scalar Conservation ◽  
Derivatives Of

We consider non-decreasing entropy solutions to 1-d scalar conservation laws and show that the spatial derivatives of such solutions satisfy a contraction property with respect to the Wasserstein distance of any order. This result extends the L1-contraction property shown by Kružkov.

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 Reductions and New Exact Solutions of ZK, Gardner KP, and Modified KP Equations via Generalized Double Reduction Theorem

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
R. Naz ◽  
Z. Ali ◽  
I. Naeem
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
First Integral ◽  
Second Order ◽  
Lie Symmetries ◽  
Reduction Theorem ◽  
Explicit Solutions ◽  
Double Reduction ◽  
Zk Equation ◽  
New Exact Solutions

We study here the Lie symmetries, conservation laws, reductions, and new exact solutions of (2+1) dimensional Zakharov-Kuznetsov (ZK), Gardner Kadomtsev-Petviashvili (GKP), and Modified Kadomtsev-Petviashvili (MKP) equations. The multiplier approach yields three conservation laws for ZK equation. We find the Lie symmetries associated with the conserved vectors, and three different cases arise. The generalized double reduction theorem is then applied to reduce the third-order ZK equation to a second-order ordinary differential equation (ODE) and implicit solutions are established. We use the Sine-Cosine method for the reduced second-order ODE to obtain new explicit solutions of ZK equation. The Lie symmetries, conservation laws, reductions, and exact solutions via generalized double reduction theorem are computed for the GKP and MKP equations. Moreover, for the GKP equation, some new explicit solutions are constructed by applying the first integral method to the reduced equations.

scholarly journals Conservation Laws and Travelling Wave Solutions for Double Dispersion Equations in (1+1) and (2+1) Dimensions

2020 ◽  
Author(s):  
Maria Luz Gandarias ◽  
Maria Rosa Duran ◽  
Chaudry Masood khalique
Keyword(s):  
Conservation Laws ◽  
Dispersion Equation ◽  
Nonlinear Wave ◽  
Travelling Waves ◽  
Travelling Wave ◽  
Travelling Wave Solutions ◽  
Dispersion Equations ◽  
Wave Solutions ◽  
Double Dispersion ◽  
Long Nonlinear Wave

In this article, we investigate two types of double dispersion equations in two dierent dimensions. Double dispersion equation were derived to describe long nonlinear wave evolution in a thin hyperelastic rod. Conservation laws are obtained for these equations by the application of the multiplier method. Finally, travelling waves and line travelling waves are respectively considered for these two equations.

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