الفهرس 
IntroductionOnly 14 pages are availabe for public view
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Abstract
Group-theoretic methods are powerful, versatile and fundamental to the
development of systematic procedures that lead to invariant solutions of
differential equations. The group-theoretic methods are applicable to both
linear and nonlinear differential equations since they are not based on linear
operators, superposition or other requirements of the linear solution
techniques.
A systematic investigation of continuous transformation group was
carried by Lie [1-3]. His original goal was the creation of a theory of
integration for ordinary differential equations analogous to the Abelian theory
for the solution of algebraic equations. He investigated the concept of the
invariance groups admitted by a given system of differential equations. These
groups have important real world applications. A number of hooks on the
application of the continuous groups of transformations relating to differential
equations have been written from a mathematical standpoint, for example
Bluman and Cole [4], Ovisannkov [5], Hill [6] Olver [7], Bluman and Kumei
[8], Hans [9] and Ibragimov [10]. In addition, the works of Hansen [11],
Ames [12,13] and Dresner [14],Na and Hensen [15] present quite extensively
the general theories involved in the similarity solution of differential
equations as applied to engineering and physics problems.
In this thesis, we applied a one-parameter Lie group method not only to
find invariants of partial differential equations but also to predict the existence
of invariants. Using this method, one can tty to obtain invariant, and partially
invariant solution to form the group in order to transform a given partial
differential equation to a less complicated or ordinary differential equation
[7,8,13].
Similarity transformations essentially reduce the number of
independent variables in partial differential equations by one. Hence, for
partial differential equations which have two independent variables, the
similarity transformations transform a partial differential equation into an
ordinary differential equation and make the determination of a class of
solutions possible for the given partial differential equation depending on
arbitrary constants. Similarity transformations have been also used to convert
moving boundary conditions to constant boundary conditions. There has been
considerable interest in symmetry reductions of partial differential equations,
mainly because the procedure reduces the number of independent variables,
and, therefore, assists in the determination of exact solutions.
The classical method [5,6,13,15] for obtaining the similarity reductions
IS using the symmetry properties of the partial differential equations. It is
possible to obtain the necessary and sufficient conditions for invariance of the
partial differential equations with respect to the one-parameter group of
transformations. Setting all the coefficients of like derivatives equal zero, one
obtains a system of determining equations for the group elements. The group
of points symmetries which leaves the partial differential equations invariant
may be determined by means of its generators. These infinitesimal generators
must form a Lie algebra determined by the structure constants. This means
that the set of generators chosen must be closed under the commutative
operator. Having defined the generators of a symmetry group, the invariance
reduction allows determination of the similarity variable and the form of
similarity solution, using the general integral of the characteristic system. The
symmetry transformation group can be generated from the corresponding Lie
algebra. In fact to find the group we only have to find its Lie algebra. We will
seek the Lie algebra and determine its corresponding transformation group.