Comparison of Approximate Symmetry Methods for Differential Equations (original) (raw)

Approximate symmetries of creeping flow equations of a second grade fluid

International Journal of Non-Linear Mechanics, 2004

Creeping ow equations of a second grade uid are considered. Two current approximate symmetry methods and a modiÿed new one are applied to the equations of motion. Approximate symmetries obtained by di erent methods and the exact symmetries are contrasted. Approximate solutions corresponding to the approximate symmetries are derived for each method. Symmetries and solutions are compared and advantages and disadvantages of each method are discussed in detail. ?

14 Approximate Symmetry Analysis ofa Class of Perturbed Nonlinear Reaction-Diffusion Equations

2014

In this paper, the problem of approximate symmetries of a class of nonlinear reaction-diffusion equations called Kolmogorov-Petrovsky-Piskounov (KPP) equation is comprehensively analyzed. In order to compute the approximate symmetries, we have applied the method which was proposed by Fushchich and Shtelen [8] and fundamentally based on the expansion of the dependent variables in a perturbation series. Particularly, an optimal system of one dimensional subalgebras is constructed and some invariant solutions corresponding to the resulted symmetries are obtained.

Potential symmetry generators and associated conservation laws of perturbed nonlinear equations

Applied Mathematics and Computation, 2004

Some recent results on approximate Lie group methods and previously developed concepts on potential symmetries are extended and applied to (mainly) nonlinear systems perturbed by a small parameter. The potential (or auxilliary) form of the perturbed system necessarily requires a knowledge of an 'approximate' conservation law of the system. We then also use knowledge of 'approximate' potential symmetries to calculate new approximate potential conservation laws.

Approximate Symmetry Analysis of a Class of Perturbed Nonlinear Reaction-Diffusion Equations

Abstract and Applied Analysis, 2013

The problem of approximate symmetries of a class of nonlinear reaction-diffusion equations called Kolmogorov-Petrovsky-Piskounov (KPP) equation is comprehensively analyzed. In order to compute the approximate symmetries, we have applied the method which was proposed by Fushchich and Shtelen (1989) and fundamentally based on the expansion of the dependent variables in a perturbation series. Particularly, an optimal system of one-dimensional subalgebras is constructed and some invariant solutions corresponding to the resulted symmetries are obtained.

Determination of approximate symmetries of differential equations

Group theory and …, 2005

Centre de Recherches Mathematiques CRM Proceedings and Lecture Notes Vo1ume 39, 2005 Determination of Approximate Symmetries of Differential Equations J. Bonasia, Frangois Lemaire, GREG REID, Robin Scott, and Lihong Zhi ABSTRACT. There has been ...

Approximate Symmetries Analysis and Conservation Laws Corresponding to Perturbed Korteweg–de Vries Equation

Journal of Mathematics, 2021

The Korteweg–de Vries (KdV) equation is a weakly nonlinear third-order differential equation which models and governs the evolution of fixed wave structures. This paper presents the analysis of the approximate symmetries along with conservation laws corresponding to the perturbed KdV equation for different classes of the perturbed function. Partial Lagrange method is used to obtain the approximate symmetries and their corresponding conservation laws of the KdV equation. The purpose of this study is to find particular perturbation (function) for which the number of approximate symmetries of perturbed KdV equation is greater than the number of symmetries of KdV equation so that explore something hidden in the system.

Analytic and numerical solutions of nonlinear diffusion equations via symmetry reductions

Advances in Difference Equations, 2014

In this article, the authors study analytic and numerical solutions of nonlinear diffusion equations of Fisher's type with the help of classical Lie symmetry method. Lie symmetries are used to reduce the equations into ordinary differential equations (ODEs). Lie group classification with respect to time dependent coefficient and optimal system of one-dimensional sub-algebras is obtained. Then sub-algebras are used to construct symmetry reduction and analytic solutions. Finally, numerical solutions of nonlinear diffusion equations are obtained by using one of the differential quadrature methods.

Symmetry in Perturbation Problems

1997

The work is devoted to a new branch of application of continuous group's techniques in the investigation of nonlinear differential equations. The principal stages of the development of perturbation theory of nonlinear diffe- rential equations are considered in short. It is shown that its characteristic features make it possible a fruitful usage of continuous group's techniques in problems of per- turbation theory.

Enhanced Symmetry Analysis of Two-Dimensional Burgers System

Acta Applicandae Mathematicae

We carry out enhanced symmetry analysis of a two-dimensional Burgers system. The complete point symmetry group of this system is found using an enhanced version of the algebraic method. Lie reductions of the Burgers system are comprehensively studied in the optimal way and new Lie invariant solutions are constructed. We prove that this system admits no local conservation laws and then study hidden conservation laws, including potential ones. Various kinds of hidden symmetries (continuous, discrete and potential ones) are considered for this system as well. We exhaustively describe the solution subsets of the Burgers system that are its common solutions with its inviscid counterpart and with the two-dimensional Navier-Stokes equations. Using the method of differential constraints, which is particularly efficient for the Burgers system, we construct a number of wide families of solutions of this system that are expressed in terms of solutions of the (1+1)-dimensional linear heat equation although they are not related to the well-known linearizable solution subset of the Burgers system.

Extended symmetry analysis of generalized Burgers equations

Journal of Mathematical Physics

Using enhanced classification techniques, we carry out the extended symmetry analysis of the class of generalized Burgers equations of the form u t + uu x + f (t, x)u xx = 0. This enhances all the previous results on symmetries of these equations and includes the description of admissible transformations, Lie symmetries, Lie and nonclassical reductions, hidden symmetries, conservation laws, potential admissible transformations and potential symmetries. The study is based on the fact that the class is normalized, and its equivalence group is finite-dimensional. 1. ((f 1 , ϕ 1 , f 2)•(f 2 , ϕ 2 , f 3))•(f 3 , ϕ 3 , f 4) = (f 1 , ϕ 1 , f 2)•((f 2 , ϕ 2 , f 3)•(f 3 , ϕ 3 , f 4)), which means the associativity of the composition. 2. For each f the role of the neutral element is played by the triple (f, id, f), where id is the identical transformation,t = t,x = x,ũ = u. 3. Any admissible transformation (f, ϕ,f) is invertible, and the inverse is (f , ϕ −1 , f). Definition 3. The usual equivalence group G ∼ = G ∼ (L| S) of the class L| S is the (pseudo)group of point transformations in the extended space of (t, x, u, f),