LIE SYMMETRIES OF DIFFERENTIAL EQUATIONS BY COMPUTER ALGEBRA

In this paper we restrict ourselves to Lie point symmetries an applications to the fourth order generalized Burgers equation GBE4. Using computer programs under the computer algebra package MATHEMATICA we find a three dimensional solvable Lie algebra of point symmetries of the GBE4 equation. The similarity reductions due to these symmetries have also been obtained.


Introduction
The idea of applying Lie group theory to solve differential equations is as old as Lie group theory itself [1].An extensive literature exist on this subject [2][3][4][5][6][7][8] but until quite recently, group theory has in this respect been unused.The reason may be several misconceptions in the minds of potential users of group theory, such as i) It is as difficult to find the symmetry group of an equation as to solve it; ii) Group theory only provides randomly occurring particular solutions; iii) Group theory is only useful for linear equations [9].The symmetry group of a system of differential equations is, roughly speaking, a group of transformations of independent and dependent variables leaving the set of all solutions invariant.Once the symmetry group of a system of equations is known, it can be used to generate new solutions from the old ones, often interesting ones from trivial ones.It can be used to classify solutions into conjugacy classes and to classify and simplify differential equations.An important application is the symmetry reduction: the reduction of an ordinary differential equation (ODE) to a lower order one, the reduction of a partial differential equation (PDE) to one with fewer independent variables.
In this paper we restrict ourselves to Lie point symmetries and applications to three nonlinear system of partial differential equations.We don't enter into the territory of contact, higher and generalized symmetries [9][10][11][12].

Classical Lie Symmetries of Differential Equations
Now we introduce the classical Lie symmetries or Lie point symmetries of the ordinary and partial differential equations which can be obtained through the Lie group method of infinitesimal transformations, originally developed by S. Lie [1].Though the method is entirely algorithmic, it often involves a large amount of tedious algebra and auxiliary calculations which are virtually unmanageable manually.
During the last two decades a change has occurred in applied mathematics that is even more severe than the reduction of computers for performing numerical calculations about forty years ago.It means that large computers have rendered it feasible to perform analytical calculations automatically as well.Although the idea of mechanizing analytical calculations is already more than 100 years old.
The most important general-purpose computer algebra systems available today are Macsyma by Mathlab Group at MIT, Reduce by A. C. Hearn at the Rand Corporation, Maple by B. Char at Waterloo, mu−Math by D. R.Stoutemyer of "Software House" in Honolulu, SMP by S. Wolfram and Scratchpad II by R. D. Jenks and D. Yun at IBM. Typically a computer algebra system provides modules for performing basic operations like simplification, differentiation, integration, factorization, etc.These algorithms are the building blocks for any other packages which may be developed by the user for special applications.
The availability of these computer algebra systems has a particular strong influence on those areas of applied mathematics where large analytical manipulations are necessary for obtaining a certain result.Applying a computer algebra system means to become accustomed to a completely new working style.The purpose of this subsection is to demonstrate this new working style for the symmetry analysis of differential equations.
Although the concept of the symmetry of a differential equation was introduced by Sophus Lie at the end of the last century while he was searching for a general theory of solving differential equations, it did not receive the proper attention for a long time.The reason is quickly recognized if we try to apply it to specific problems.To find the symmetry group of a differential equation almost always requires tremendous algebraic calculations.Especially in partial differential equations, they often assume such proportions that they can not be performed in the conventional way.The largest number of algebraic calculations is required for solving the so called determining system of linear partial differential equations which may have a simple structure; however, as it can be observed in the example of GBE, it may comprise several dozens or even hundreds of equations.The solution of algorithm for this determining system is the heart of the symmetry packages.In addition to providing the symmetry generators of the full symmetry group, its structure is determined automatically and communicated to the user in the terms of its commutator table.
In order to facilitate the determination of the classical Lie symmetries we use the package Lie[] of G. Baumann [13] under the computer algebra software MATHEMATICA and find a three dimensional solvable Lie algebra of infinitesimal transformations.When we use SPDE package in the computer algebra software REDUCE [14], it gave the same result with a misplaced sign.Then we tried Symman package of Vorob'ev [15] under the computer algebra software MATHEMATICA and did give only a two dimensional subalgebra.

Invariance of a PDE
In this subsection we apply infinitesimal transformations to the construction of solutions of partial differential equations.We will consider systems of PDE's as well as scalar PDE's.
As for ordinary differential equations we will show that infinites-imal criterion for invariance of partial differential equations leads directly to an algorithm to determine infinitesimal generators X admitted by given partial differential equations.Invariant surfaces of the corresponding Lie group of point transformations lead to invariant solutions (similarity solutions).These solutions are obtained by solving partial differential equations with fewer independent variables than the given PDE's.
First we consider a scalar partial differential equation.We represent a k th order partial differential equation by where x = (x 1 , x 2 , • • • , x n ) denotes n independent variables, u denotes the coordinate corresponding to the dependent variable, and u j denotes the set of coordinates corresponding to all j th order partial derivatives of u with respect to x; the coordinate u j corresponding to 71) becomes an algebraic equation which defines a hyper surface in We assume that the partial differential equation ( 1) can be written in solved form in terms of some l th order partial derivative of u: Now we are going to give a criterion for the invariance of a partial differential equation [16].
be the infinitesimal generator of the one-parameter Lie group of transformations Let be the corresponding k th extended infinitesimal generator of (3) where η and η (j) Then one parameter Lie group of transformations (4-6) is admitted by the partial differential equation (2) iff

Classical Lie Symmetries of the GBE4 Equation
We consider the one parameter Lie point transformations of (t, x, u) given by The GBE4 equation is invariant under this transformation if it is invariant under the operator Hence ξ 1 , ξ 2 , η 1 , η 2 are found from the determining equations derived from this invariancy condition [17].
To determine the classical Lie symmetries of the GBE4 equation we first start mathematica.Then we call the package Lie The determining equation as given in the output consists of 31 linear partial differential equations.
Rewritten in the usual style, the infinitesimal generators of the finite subgroup of the symmetry group of the GBE4 equation are given by The nonvanishing commutators are Hence a commutator table can be formed as follows: The linear space {X 1 , X 2 , X 3 } spanned by X 1 , X 2 , X 3 is a Lie algebra with the skew symmetric operator in the table.The subspaces L 2 = {X 1 , X 2 } and L 1 = {X 1 } are 2 and 1 dimensional subalgebras of L 3 = {X 1 , X 2 , X 3 } accordingly.Furthermore these subalgebras have the inclusion property and hence L 3 is a solvable Lie algebra.

Similarity Reductions of the GBE4 Equation
Similarity reduction corresponding to the symmetry generator X 2 + (20)