Group Of Equivalence Transformations Research Articles

A complete Lie symmetry classification of a class of (1+2)-dimensional reaction-diffusion-convection equations

A class of nonlinear reaction-diffusion-convection equations describing various processes in physics, biology, chemistry etc. is under study in the case of time and two space variables. The group of equivalence transformations is constructed, which is applied for deriving a Lie symmetry classification for the class of such equations by the well-known algorithm. It is proved that the algorithm leads to 32 reaction-diffusion-convection equations admitting nontrivial Lie symmetries. Furthermore a set of form-preserving transformations for this class is constructed in order to reduce this number of the equations and obtain a complete Lie symmetry classification. As a result, the so called canonical list of all inequivalent equations admitting nontrivial Lie symmetry (up to any point transformations) and their Lie symmetries are derived. The list consists of 22 equations and it is shown that any other reaction-diffusion-convection equation admitting a nontrivial Lie symmetry is reducible to one of these 22 equations. As a nontrivial example, the symmetries derived are applied for the reduction and finding exact solutions in the case of the porous-Fisher type equation with the Burgers term.

Equivalence classes and linearization of the Riccati and Abel chain

The problem of linearization by point transformations is solved for equations in the generalized Riccati and Abel chain of order not exceeding the fourth. It is shown in particular that nonlinear third order and fourth order equations from the chain are not linearizable by any point transformations. The Lie pseudo-group of equivalence transformations for equations of arbitrary orders from the chain are then found, together with expressions for the transformed parameter functions. An important subgroup of the group of equivalence transformations found is considered and some associated equivalence classes are exhibited.

Group classification of a class of generalized nonlinear Kolmogorov equations and exact solutions

A class of generalized nonlinear Kolmogorov equations is investigated. We present the group classification of Lie symmetries of the class with respect to the group of equivalence transformations. We find a number of exact solutions of nonlinear Kolmogorov equations which have the maximal symmetry properties.

On the equivalence groups for (2+1) dimensional nonlinear diffusion equation

(2+1) dimensional diffusion equation is considered within the framework of the group of equivalence transformations. Generators for the group are obtained and admissible transformations between linear and nonlinear equations are examined. It is shown that transformations between linear and nonlinear equations are possible provided that the generators of independent variables depend on the dependent variable. Exact solutions for some nonlinear equations are obtained.

Symbolic computation of equivalence transformations and parameter reduction for nonlinear physical models

An efficient systematic procedure is provided for symbolic computation of Lie groups of equivalence transformations and generalized equivalence transformations of systems of differential equations that contain arbitrary elements (arbitrary functions and/or arbitrary constant parameters), using the software package GeM for Maple. Application of equivalence transformations to the reduction of the number of arbitrary elements in a given system of equations is discussed, and several examples are considered. The first computational example of generalized equivalence transformations where the transformation of the dependent variable involves an arbitrary constitutive function is presented.As a detailed physical example, a three-parameter family of nonlinear wave equations describing finite anti-plane shear displacements of an incompressible hyperelastic fiber-reinforced medium is considered. Equivalence transformations are computed and employed to radically simplify the model for an arbitrary fiber direction, invertibly reducing the model to a simple form that corresponds to a special fiber direction, and involves no arbitrary elements.The presented computation algorithm is applicable to wide classes of systems of differential equations containing arbitrary elements.

GROUP CLASSIFICATION FOR A GENERAL NONLINEAR MODEL OF OPTIONS PRICING

We consider a family of equations with two free functional parameters containing the classical Black–Scholes model, Schonbucher–Wilmott model, Sircar–Papanicolaou equation for option pricing as partial cases. A five-dimensional group of equivalence transformations is calculated for that family. That group is applied to a search for specifications' parameters specifications corresponding to additional symmetries of the equation. Seven pairs of specifications are found.

Coefficient characterization of linear differential equations with maximal symmetries

A characterization of the general linear equation in standard form admitting a maximal symmetry algebra is obtained in terms of a simple set of conditions relating the coefficients of the equation. As a consequence, it is shown that in its general form such an equation can be expressed in terms of only two arbitrary functions, and its connection with the Laguerre–Forsyth form is clarified. The characterizing conditions are also used to derive an infinite family of semi-invariants, each corresponding to an arbitrary order of the linear equation. Finally a simplifying ansatz is established, which allows an easier determination of the infinitesimal generators of the induced pseudo group of equivalence transformations, for all the three most common canonical forms of the equation.

Petrophysics laws as invariants of filtration model

In this paper the methods of Lie group analysis are used to investigate symmetry properties of differential equations that describe filtration of a two-phase liquid in a porousmedia. Lie algebra of operators of a group of equivalence transformations is calculated. Invariants with respect to a subgroup of the group of equivalence transformations are used for constructing the functional dependencies which define arbitrary parameters of the model. It is shown that one of the invariants of a subalgebra of operators of equivalence transformations is the Timur’s law, describing a relation between absolute permeability, porosity and residual water saturation.

Equivalence problem and integrability of the Riccati equations

We consider the class of general real Riccati equations and find its Lie group of equivalence transformations. Using the Lie algebra of this Lie group and its invariants we formulate criteria of equivalence of the Riccati equations. These criteria determine some cases of the general Riccati equations, which are integrable in quadratures.

Group classification of a family of second-order differential equations

We find the group of equivalence transformations for equations of the form y ″ = A ( x ) y ′ + F ( y ) , where A and F are arbitrary functions. We then give a complete group classification of this family of equations using a direct method of analysis, together with the equivalence transformations.

A solution to the problem of invariants for parabolic equations

The article is devoted to the solution of the invariants problem for the one-dimensional parabolic equations written in the two-coefficient canonical form used recently by N.H. Ibragimov: u t - u xx + a ( t , x ) u x + c ( t , x ) u = 0 . A simple invariant condition is obtained for determining all equations that are reducible to the heat equation by the general group of equivalence transformations. The solution to the problem of invariants is given also in the one-coefficient canonical u t - u xx + c ( t , x ) u = 0 . One of the main differences between these two canonical forms is that the equivalence group for the two-coefficient form contains the arbitrary linear transformation of the dependent variable whereas this group for the one-coefficient form contains only a special type of the linear transformations of the dependent variable.

Invariants of differential equations defined by vector fields

We determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold. We also find all invariants and differential invariants for this group up to the second order. A result on the characterization of classes of these equations by the invariant functions is also given.

Group analysis and exact solutions of a class of variable coefficient nonlinear telegraph equations

A complete group classification of a class of variable coefficient (1+1)-dimensional telegraph equations f(x)utt=(H(u)ux)x+K(u)ux, is given, by using a compatibility method and additional equivalence transformations. A number of new interesting nonlinear invariant models which have nontrivial invariance algebras are obtained. Furthermore, the possible additional equivalence transformations between equations from the class under consideration are investigated. Exact solutions of special forms of these equations are also constructed via classical Lie method and generalized conditional transformations. Local conservation laws with characteristics of order 0 of the class under consideration are classified with respect to the group of equivalence transformations.

Explicit determination of isovector fields of equivalence groups for balance equations of arbitrary order—Part II

The explicit solutions of previously given determining equations for isovector components (infinitesimal generators) associated with Lie groups of equivalence transformations for the most general family of balance equations of arbitrary order involving arbitrary but finite number of independent and dependent variables are provided. Equivalence transformations which are considered in this work map the solutions of partial differential equations containing arbitrary functions or parameters to the solutions of the equations of the same structure but with different functions or parameters. The general solutions exposed here are employed to determine the isovector fields corresponding to a sort of Korteweg–de Vries type equation involving third derivative with respect to spatial variable.

Equivalence groups for balance equations of arbitrary order––Part I

The groups of equivalence transformations for a family of balance equations of arbitrary order involving arbitrary number of independent and dependent variables are investigated. Equivalence groups are essentially much more general than symmetry groups in the sense that they map equations containing arbitrary functions or parameters onto equations of the same structure but with different functions or parameters. We attack the problem of determination of group structure of a system of arbitrary balance equations through exterior calculus. The analysis is reduced to determine isovector fields of a closed ideal of an exterior algebra over an appropriate differentiable manifold dictated by the structure of the differential equations by employing almost solely algebraic operations. The isovector fields induce point transformations, which are none other than the desired equivalence transformations, generated by their orbits which leave that particular ideal invariant. The determining equations are fully obtained. It is shown that symmetry transformations can also be deduced directly from equivalence transformations.

Equivalence transformations for first order balance equations

The groups of equivalence transformations for a family of first order partial differential equations of balance form are investigated. Equivalence transformations map a specific class of equations involving arbitrary functions or parameters into the same class. In that sense equivalence groups are much more general than symmetry groups. The method of attack to this problem is based on the exterior calculus. The analysis is reduced to determine isovector fields of a closed ideal of the exterior algebra over an appropriate differentiable manifold dictated by the structure of the family of differential equations. The determining equations in terms of the isovector field components are obtained and furthermore their explicit solutions are determined. The results are then applied to a class of Maxwell equations in rigid bodies.

EQUIVALENCE GROUPS FOR FIRST-ORDER BALANCE EQUATIONS AND APPLICATIONS TO ELECTROMAGNETISM

We investigate the groups of equivalence transformations for first-order balance equations involving an arbitrary number of dependent and independent variables. We obtain the determining equations and find their explicit solutions. The approach to this problem is based on a geometric method that depends on Cartan's exterior differential forms. The general solutions of the determining equations for equivalence transformations for first-order systems are applied to a class of the Maxwell equations of electrodynamics.

Explicit determination of isovector fields of equivalence groups for second order balance equations

The explicit solutions of previously given determining equations for isovector components (infinitesimal generators) associated with Lie groups of equivalence transformations for the most general family of second order balance equations involving finite arbitrary number of independent and dependent variables are obtained. Equivalence transformations which are considered in this work map the solutions of partial differential equations containing arbitrary functions or parameters to the solutions of the equations of the same structure but with different functions or parameters. The general solutions discovered here are employed to determine the isovector fields corresponding (i) to a one-imensional non-linear wave equation, (ii) to the field equations of non-linear homogeneous hyperelasticity. These isovector fields were found in earlier works by solving determining equations associated with the each individual case.

Equivalence groups for second order balance equations

The groups of equivalence transformations for a family of second order balance equations involving arbitrary number of independent and dependent variables are investigated. Equivalence groups are much more general than symmetry groups in the sense that they map equations containing arbitrary functions or parameters onto equations of the same structure but with different functions or parameters. Our approach to attack this problem is based on exterior calculus. The analysis is reduced to determine isovector fields of an ideal of the exterior algebra over an appropriate differentiable manifold dictated by the structure of the differential equations. The isovector fields induce point transformations, which are none other than the desired equivalence transformations, via their orbits which leave that particular ideal invariant. The general scheme is applied to a one-dimensional nonlinear wave equation and hyperelasticity. It is shown that symmetry transformations can be deduced directly from equivalence transformations.

Equivalence transformations in linear dynamical systems†

Abstract The authors analyse the equivalonce of the minimal realizations of linear dynamical systems with given properties. To this end thoy review various groups of equivalence transformations, somo of which are already known while others are introduced for the first time in this paper. More particularly, the authors study the equivalence of the minimal and periodic realizations, as well as that of the minimal, bounded and reduced realizations. For these latter they also state tho necessary and sufficient conditions governing their existence.