scholarly journals Conservation Laws of Some Physical Models via Symbolic Package GeM

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Rehana Naz ◽  
Imran Naeem ◽  
M. Danish Khan
Keyword(s):  
Evolution Equation ◽  
Conservation Laws ◽  
Poisson Equation ◽  
Conservation Law ◽  
Direct Method ◽  
Physical Models ◽  
Lubrication Models

We study the conservation laws of evolution equation, lubrication models, sinh-Poisson equation, Kaup-Kupershmidt equation, and modified Sawada-Kotera equation. The symbolic software GeM (Cheviakov (2007) and (2010)) is used to derive the multipliers and conservation law fluxes. Software GeM is Maple-based package, and it computes conservation laws by direct method and first homotopy and second homotopy formulas.

Conserved Quantities for the General Linear Heat Equation

2016 ◽  
pp. 4437-4439
Author(s):  
Adil Jhangeer ◽  
Fahad Al-Mufadi
Keyword(s):  
Evolution Equation ◽  
Heat Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Lagrangian Approach ◽  
General Linear ◽  
Conserved Quantities ◽  
Linear Heat ◽  
The Family ◽  
Partial Lagrangian

In this paper, conserved quantities are computed for a class of evolution equation by using the partial Noether approach [2]. The partial Lagrangian approach is applied to the considered equation, infinite many conservation laws are obtained depending on the coefficients of equation for each n. These results give potential systems for the family of considered equation, which are further helpful to compute the exact solutions.

Symmetry Reduction, Exact Solutions, and Conservation Laws of the (2+1)-Dimensional Dispersive Long Wave Equations

2009 ◽  
Vol 64 (9-10) ◽  
pp. 597-603 ◽  
Author(s):  
Zhong Zhou Dong ◽  
Yong Chen
Keyword(s):  
Conservation Laws ◽  
Exact Solutions ◽  
Symmetry Group ◽  
Infinite Number ◽  
Wave Equations ◽  
Direct Method ◽  
Discrete Groups ◽  
Point Symmetry Group ◽  
Long Wave ◽  
Full Symmetry

By means of the generalized direct method, we investigate the (2+1)-dimensional dispersive long wave equations. A relationship is constructed between the new solutions and the old ones and we obtain the full symmetry group of the (2+1)-dimensional dispersive long wave equations, which includes the Lie point symmetry group S and the discrete groups D. Some new forms of solutions are obtained by selecting the form of the arbitrary functions, based on their relationship. We also find an infinite number of conservation laws of the (2+1)-dimensional dispersive long wave equations.

On conservation laws and necessary conditions in the calculus of variations

2002 ◽  
Vol 132 (6) ◽  
pp. 1361-1371 ◽  
Author(s):  
G. Francfort ◽  
J. Sivaloganathan
Keyword(s):  
Conservation Laws ◽  
Calculus Of Variations ◽  
Conservation Law ◽  
Necessary Conditions ◽  
Lagrange Equation ◽  
Integral Functional ◽  
Variational Symmetry

It is well known from the work of Noether that every variational symmetry of an integral functional gives rise to a corresponding conservation law. In this paper, we prove that each such conservation law arises directly as the Euler-Lagrange equation for the functional on taking suitable variations around a minimizer.

On the symmetry and conservation law classification of the de Sitter–Schwarzschild metric and the corresponding wave and Klein–Gordon equations

2020 ◽  
Vol 17 (11) ◽  
pp. 2050172
Author(s):  
Ashfaque H. Bokhari ◽  
A. H. Kara ◽  
F. D. Zaman ◽  
B. B. I. Gadjagboui
Keyword(s):  
Conservation Laws ◽  
Conservation Law ◽  
De Sitter ◽  
Killing Vectors ◽  
Schwarzschild Geometry ◽  
Klein Gordon ◽  
Spacetime Metric ◽  
Variational Symmetries ◽  
Symmetry And Conservation Law

The main purpose of this work is to focus on a discussion of Lie symmetries admitted by de Sitter–Schwarzschild spacetime metric, and the corresponding wave or Klein–Gordon equations constructed in the de Sitter–Schwarzschild geometry. The obtained symmetries are classified and the variational (Noether) conservation laws associated with these symmetries via the natural Lagrangians are obtained. In the case of the metric, we obtain additional variational ones when compared with the Killing vectors leading to additional conservation laws and for the wave and Klein–Gordon equations, the variational symmetries involve less tedious calculations as far as invariance studies are concerned.

scholarly journals Stability of the 1D IBVP for a non autonomous scalar conservation law

2018 ◽  
Vol 149 (03) ◽  
pp. 561-592 ◽  
Author(s):  
Rinaldo M. Colombo ◽  
Elena Rossi
Keyword(s):  
Conservation Laws ◽  
Boundary Value Problems ◽  
Conservation Law ◽  
Space Dimension ◽  
Initial Boundary ◽  
Initial Boundary Value Problems ◽  
Scalar Conservation Law ◽  
Scalar Conservation ◽  
The Stability ◽  
Initial Boundary Value

We prove the stability with respect to the flux of solutions to initial – boundary value problems for scalar non autonomous conservation laws in one space dimension. Key estimates are obtained through a careful construction of the solutions.

scholarly journals Symmetries, pseudosymmetries and conservation laws in Lagrangian and Hamiltonian k-symplectic formalisms

2016 ◽  
Vol 13 (03) ◽  
pp. 1650026
Author(s):  
Florian Munteanu
Keyword(s):  
Conservation Laws ◽  
Hamiltonian Systems ◽  
Conservation Law ◽  
Type Theorem ◽  
Natural Extension ◽  
Lagrangian Systems ◽  
Symmetry And Conservation Law

In this paper, we will present Lagrangian and Hamiltonian [Formula: see text]-symplectic formalisms, we will recall the notions of symmetry and conservation law and we will define the notion of pseudosymmetry as a natural extension of symmetry. Using symmetries and pseudosymmetries, without the help of a Noether type theorem, we will obtain new kinds of conservation laws for [Formula: see text]-symplectic Hamiltonian systems and [Formula: see text]-symplectic Lagrangian systems.

Traveling wave solutions and conservation laws of a generalized Kudryashov–Sinelshchikov equation

2019 ◽  
Vol 25 (2) ◽  
pp. 211-217 ◽  
Author(s):  
Ben Muatjetjeja ◽  
Abdullahi Rashid Adem ◽  
Sivenathi Oscar Mbusi
Keyword(s):  
Heat Transfer ◽  
Evolution Equation ◽  
Conservation Laws ◽  
Traveling Wave ◽  
Nonlinear Evolution Equation ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Pressure Waves ◽  
Wave Solutions ◽  
Kudryashov Method

Abstract Kudryashov and Sinelshchikov proposed a nonlinear evolution equation that models the pressure waves in a mixture of liquid and gas bubbles by taking into account the viscosity of the liquid and the heat transfer. Conservation laws and exact solutions are computed for this underlying equation. In the analysis of this particular equation, two approaches are employed, namely, the multiplier method and Kudryashov method.

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.

Potential Systems and Nonlocal Conservation Laws of Prandtl Boundary Layer Equations on the Surface of a Sphere

2017 ◽  
Vol 72 (4) ◽  
pp. 351-357 ◽  
Author(s):  
R. Naz
Keyword(s):  
Boundary Layer ◽  
Conservation Laws ◽  
Linear Combination ◽  
Conservation Law ◽  
Local Conservation ◽  
Boundary Layer Equations ◽  
Potential System ◽  
Nonlocal Conservation Laws ◽  
Prandtl Boundary Layer ◽  
Nonlocally Related Systems

Abstract:The potential systems and nonlocal conservation laws of Prandtl boundary layer equations on the surface of a sphere have been investigated. The multiplier approach yields two local conservation laws for the Prandtl boundary layer equations on the surface of a sphere. Two potential variables ψ and ϕ are introduced corresponding to first and second conservation law. Moreover, another potential variable p is introduced by considering the linear combination of both conservation laws. Two level one potential systems involving a single nonlocal variable ψ or ϕ are constructed. One level two potential system involving both nonlocal variables ψ and ϕ is established. The nonlocal variable p is utilised to derive a spectral potential system. The nonlocal conservation laws of Prandtl boundary layer equations on the surface of a sphere are derived by computing the local conservation laws of its potential systems. The nonlocal conservation laws are utilised to derive the further nonlocally related systems.

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