scholarly journals Symmetries, Conservation Laws, and Wave Equation on the Milne Metric

2012 ◽  
Vol 2012 ◽  
pp. 1-7 ◽  
Author(s):  
Ahmad M. Ahmad ◽  
Ashfaque H. Bokhari ◽  
F. D. Zaman
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Equation ◽  
Conservation Laws ◽  
Mathematical Physics ◽  
Lie Point Symmetries ◽  
Noether Symmetries ◽  
Physical Systems ◽  
Action Integral ◽  
Lorentzian Metric

Noether symmetries provide conservation laws that are admitted by Lagrangians representing physical systems. For partial differential equation possessing Lagrangians these symmetries are obtained by the invariance of the corresponding action integral. In this paper we provide a systematic procedure for determining Noether symmetries and conserved vectors for a Lagrangian constructed from a Lorentzian metric of interest in mathematical physics. For completeness, we give Lie point symmetries and conservation laws admitted by the wave equation on this Lorentzian metric.

scholarly journals Lie Point Symmetries, Traveling Wave Solutions and Conservation Laws of a Non-linear Viscoelastic Wave Equation

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2131
Author(s):  
Almudena P. Márquez ◽  
María S. Bruzón
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Equation ◽  
Differential Equations ◽  
Conservation Laws ◽  
Lie Point Symmetries ◽  
Viscoelastic Wave ◽  
Non Linear ◽  
Linear Viscoelastic ◽  
Damping And Source Terms

This paper studies a non-linear viscoelastic wave equation, with non-linear damping and source terms, from the point of view of the Lie groups theory. Firstly, we apply Lie’s symmetries method to the partial differential equation to classify the Lie point symmetries. Afterwards, we reduce the partial differential equation to some ordinary differential equations, by using the symmetries. Therefore, new analytical solutions are found from the ordinary differential equations. Finally, we derive low-order conservation laws, depending on the form of the damping and source terms, and discuss their physical meaning.

scholarly journals Equivalent Lagrangians: Generalization, Transformation Maps, and Applications

2012 ◽  
Vol 2012 ◽  
pp. 1-19 ◽  
Author(s):  
N. Wilson ◽  
A. H. Kara
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Conservation Laws ◽  
Ordinary Differential Equations ◽  
Wave Equations ◽  
Symmetry Algebra ◽  
Lie Point Symmetries ◽  
Partial Differential

Equivalent Lagrangians are used to find, via transformations, solutions and conservation laws of a given differential equation by exploiting the possible existence of an isomorphic algebra of Lie point symmetries and, more particularly, an isomorphic Noether point symmetry algebra. Applications include ordinary differential equations such as theKummer equationand thecombined gravity-inertial-Rossbywave equationand certain classes of partial differential equations related to multidimensional wave equations.

scholarly journals Symmetry Analysis and Conservation Laws of a Generalization of the Kelvin-Voigt Viscoelasticity Equation

Symmetry ◽  
2019 ◽  
Vol 11 (7) ◽  
pp. 840 ◽  
Author(s):  
Almudena P. Márquez ◽  
María S. Bruzón
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Conservation Laws ◽  
Mechanical Behaviour ◽  
Lie Symmetry ◽  
Symmetry Analysis ◽  
Lie Point Symmetries ◽  
Lie Symmetry Analysis ◽  
Partial Differential

In this paper, we study a generalization of the well-known Kelvin-Voigt viscoelasticity equation describing the mechanical behaviour of viscoelasticity. We perform a Lie symmetry analysis. Hence, we obtain the Lie point symmetries of the equation, allowing us to transform the partial differential equation into an ordinary differential equation by using the symmetry reductions. Furthermore, we determine the conservation laws of this equation by applying the multiplier method.

Invariance, Conservation Laws, and Exact Solutions of the Nonlinear Cylindrical Fin Equation

2014 ◽  
Vol 69 (5-6) ◽  
pp. 195-198 ◽  
Author(s):  
Saeed M. Ali ◽  
Ashfaque H. Bokhari ◽  
Fiazuddin D. Zaman ◽  
Abdul H. Kara
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Exchange ◽  
Thermal Diffusivity ◽  
Conservation Laws ◽  
Cylindrical Shape ◽  
Method Of Multipliers ◽  
Double Reduction ◽  
Temperature Dependent

Fins are heat exchange surfaces which are used widely in industry. The partial differential equation arising from heat transfer in a fin of cylindrical shape with temperature dependent thermal diffusivity are studied. The method of multipliers and invariance of the differential equations is employed to obtain conservation laws and perform double reduction.

SOLVING THE ASIAN OPTION PDE USING LIE SYMMETRY METHODS

2010 ◽  
Vol 13 (08) ◽  
pp. 1265-1277 ◽  
Author(s):  
NICOLETTE C. CAISTER ◽  
JOHN G. O'HARA ◽  
KESHLAN S. GOVINDER
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Analytical Solution ◽  
Stock Price ◽  
Ad Hoc ◽  
Lie Symmetry ◽  
Asian Option ◽  
Asian Options ◽  
Lie Point Symmetries ◽  
Reduced Order

Asian options incorporate the average stock price in the terminal payoff. Examination of the Asian option partial differential equation (PDE) has resulted in many equations of reduced order that in general can be mapped into each other, although this is not always shown. In the literature these reductions and mappings are typically acquired via inspection or ad hoc methods. In this paper, we evaluate the classical Lie point symmetries of the Asian option PDE. We subsequently use these symmetries with Lie's systematic and algorithmic methods to show that one can obtain the same aforementioned results. In fact we find a familiar analytical solution in terms of a Laplace transform. Thus, when coupled with their methodic virtues, the Lie techniques reduce the amount of intuition usually required when working with differential equations in finance.

The fundamental solution for the axially symmetric wave equation II

1980 ◽  
Vol 87 (3) ◽  
pp. 515-521
Author(s):  
Albert E. Heins
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Equation ◽  
Fundamental Solution ◽  
Free Space ◽  
Axially Symmetric ◽  
Partial Differential ◽  
Symmetric Wave ◽  
Alternate Forms ◽  
Image Position

In a recent paper, hereafter referred to as I (1) we derived two alternate forms for the fundamental solution of the axially symmetric wave equation. We demonstrated that for α > 0, the fundamental solution (the so-called free space Green's function) of the partial differential equationcould be written asif b > rorif r > b.

scholarly journals On closed-form solutions of some nonlinear partial differential equations

2000 ◽  
Vol 23 (2) ◽  
pp. 81-88 ◽  
Author(s):  
S. S. Okoya
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Equation ◽  
Steady State ◽  
Closed Form ◽  
Nonlinear Wave Equation ◽  
Nonlinear Partial Differential Equations ◽  
Partial Differential ◽  
Closed Form Solutions ◽  
Thermal Explosion Theory

This paper is devoted to closed-form solutions of the partial differential equation:θxx+θyy+δexp(θ)=0, which arises in the steady state thermal explosion theory. We find simple exact solutions of the formθ(x,y)=Φ(F(x)+G(y)), andθ(x,y)=Φ(f(x+y)+g(x-y)). Also, we study the corresponding nonlinear wave equation.

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.

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