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Abstract
The aim of this thesis is to find new solutions for some problems in
partial differential equations with boundary conditions using two
methods: (i) Symmetry group method, (ii) Discrete symmetry group
method. Both of the two methods reduce the number of the independent
variables as well as dependent variables and map the partial differential
equations to ordinary differential equations, where dealing with the
resulting ordinary differential equations is, of course, easier than the
corresponding partial differential equations. Using the symmetry
transformations analysis, we get some group of similarity transformations
and then by applying the discrete symmetry analysis we get another
group of similarity transformations, i.e., the discrete symmetry method
increases the number of the similarity transformations. So, we get a lot of
similarity representations which enable us to find more exact solution of
the resulting problem.
The thesis has been organized into four chapters;
Chapter 1:
This chapter contains a literature review, historical background and
basic concepts related to the topics of the thesis. An algorithm for the
discrete symmetry method has been proposed.
Chapter 2:
The aim of this chapter is to find new exact solutions for the
Dirichlet problem for the Burgers equation in the case of a prescribed
motion of the boundary using Lie group method and discrete symmetry.
method. Moreover, we have transformed the original equation to a forced
Burgers equation by using linear transformation to simplify the boundary
conditions. Therefore, using Lie group of transformations, the exact
solutions of the resulted forced problem are obtained and then the exact
solutions of the original problem are found. Two cases of the boundary
motion are considered; (a) linearly moving boundary, and (b) rapidly
oscillating boundary.
Chapter 3:
The purpose of this chapter is to study the Burgers equation with
time dependent flux at the origin. Symmetry reductions and similarity
solutions for the governing equation are obtained using Lie’s method of
infinitesimal transformations groups. Using the discrete symmetry
method, we have presented three groups of discrete symmetries
representing three new groups of similarity transformations.
Chapter 4:
The main objective of this chapter is to investigate the similarity and
the numerical solutions of coupled nonlinear ordinary differential
equations with boundary conditions. The selected coupled nonlinear
partial differential equations include the effect of thermal slip on a steady
two-dimensional boundary value stagnation-point flow towards a heated
stretching sheet placed in a porous medium. The symmetry analysis is
applied to reduce the coupled nonlinear partial differential equations to
coupled nonlinear ordinary differential equations. By applying discrete
symmetry analysis we have obtained general invariant relations which
enable us to create new infinitesimal generators and similarity
transformations. Numerical results are obtained and the effects of the
influences parameters are discussed. Comparisons with published results
are presented.