On New Conservation Laws of Fin Equation (original) (raw)

A note on a symmetry analysis and exact solutions of a nonlinear fin equation

Applied Mathematics Letters, 2006

A similarity analysis of a nonlinear fin equation has been carried out by M. Pakdemirli and A.Z. Sahin [Similarity analysis of a nonlinear fin equation, Appl. Math. Lett. (2005) (in press)]. Here, we consider a further group theoretic analysis that leads to an alternative set of exact solutions or reduced equations with an emphasis on travelling wave solutions, steady state type solutions and solutions not appearing elsewhere.

A Lie Symmetry Classification of a Nonlinear Fin Equation in Cylindrical Coordinates

Abstract and Applied Analysis, 2014

The nonlinear fin equation in cylindrical coordinates is considered. Assuming a radial variable heat transfer coefficient and temperature dependent thermal conductivity, a complete classification of these two functions is obtained via Lie symmetry analysis. Using these Lie symmetries, we carry out reduction of the fin equation and whenever possible exact solutions are obtained.

Conservation laws, exact solutions and nonlinear dispersion: A lie symmetry approach

Journal of AppliedMath, 2023

In this study, we investigated a set of equations that exhibit compact solutions and nonlinear dispersion. We used the classical lie symmetry approach to derive ordinary differential equations (ODEs) that are well suited for qualitative study. By examining the dynamic behavior of these ODEs, we gained insights into the intricate nature of the underlying system. We also used a powerful multiplier approach to establish nontrivial conservation laws and exact solutions for these equations. These conservation laws provide essential information regarding the underlying symmetries and invariants of the system, and shed light on its fundamental properties.

Classical Lie Point Symmetry Analysis of a Steady Nonlinear One-Dimensional Fin Problem

Journal of Applied Mathematics, 2012

We consider the one-dimensional steady fin problem with the Dirichlet boundary condition at one end and the Neumann boundary condition at the other. Both the thermal conductivity and the heat transfer coefficient are given as arbitrary functions of temperature. We perform preliminary group classification to determine forms of the arbitrary functions appearing in the considered equation for which the principal Lie algebra is extended. Some invariant solutions are constructed. The effects of thermogeometric fin parameter and the exponent on temperature are studied. Also, the fin efficiency is analyzed.

Methodical aspects about the numerical computation of the Lie symmetries and conservation laws

Romanian Journal of Physics

This paper proposes an algorithm for the Lie symmetries investigation in the case of a high order evolution equation. General Lie operators are deduced and, in the next step, the associated conservation laws are derived. Due to the large number of equations derived from the symmetries conditions, the use of mathematical software is necessary, and we employed a MAPLE application. Some models arising from physics was chosen to test the method.

Symmetries and conservation laws

Quantum Mechanics, 2009

Conservation laws play an important role in science. The aim of this thesis is to provide an overview and develop new methods for constructing conservation laws using Lie group theory. The derivation of conservation laws for invariant variational problems is based on Noether's theorem. It is shown that the use of Lie-Bäcklund transformation groups allows one to reduce the number of basic conserved quantities for differential equations obtained by Noether's theorem and construct a basis of conservation laws. Several examples on constructing a basis for some well-known equations are provided. Moreover, this approach allows one to obtain new conservation laws even for equations without Lagrangians. A formal Lagrangian can be introduced and used for computing nonlocal conservation laws. For self-adjoint or quasi-self-adjoint equations nonlocal conservation laws can be transformed into local conservation laws. One of the fields of applications of this approach is electromagnetic theory, namely, nonlocal conservation laws are obtained for the generalized Maxwell-Dirac equations. The theory is also applied to the nonlinear magma equation and its nonlocal conservation laws are computed.

On symmetry group classification of a fin equation

Journal of Inequalities and Applications, 2013

We deal with the Noether symmetry classification of a nonlinear fin equation, in which thermal conductivity and heat transfer coefficient are assumed to be functions of the temperature. In this study Noether symmetries of the fin equation are investigated using the partial Lagrangian approach. This classification includes Noether symmetries, first integrals and some invariant solutions with respect to different choices of thermal conductivity and heat transfer coefficient functions.

Lie symmetries and conservation laws of the Hirota–Ramani equation

Communications in Nonlinear Science and Numerical Simulation, 2012

In this paper, Lie symmetry method is performed for the Hirota-Ramani (H-R) equation. We will find The symmetry group and optimal systems of Lie subalgebras. Furthermore, preliminary classification of its group invariant solutions, symmetry reduction and nonclassical symmeries are investigated. Finally the conservation laws of the H-R equation are presented.

Symmetry classification and conservation laws for some nonlinear partial differential equations

2019

In this thesis we study some nonlinear partial differential equations which appear in several physical phenomena of the real world. Exact solutions and conservation laws are obtained for such equations using various methods. The equations which are studied in this work are: a hyperbolic Lane-Emden system, a generalized hyperbolic Lane-Emden system, a coupled Jaulent-Miodek system and a (2+1)-dimensional Jaulent-Miodek equation power-law nonlinearity. We carry out a complete Noether and Lie group classification of the radial form of a coupled system of hyperbolic equations. From the Noether symmetries we establish the corresponding conserved vectors. We also determine constraints that the nonlinearities should satisfy in order for the scaling symmetries to be Noetherian. This led us to a critical hyperbola for the systems under consideration. An explicit solution is also obtained for a particular choice of the parameters. We perform a complete Noether symmetry analysis of a generaliz...