الفهرس 
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Abstract
Many physical phenomena can be described by differential equations
or their integrated forms. In this work, we are concentrating on, how to
find explicit approximations for functions defined by differential equations
or by their integrated forms. In particular, we consider the high
even-order elliptic differential equations of one and two space variables.
One of the main objectives of this thesis is to introduce and develop
new efficient spectral-Galerkin algorithms for solving high evenorder
elliptic differential equations in one and two dimensions by using
third and fourth kinds of Chebyshev polynomials. These algorithms are
based on appropriate construction of base functions for the Galerkin
formulation that lead to discrete systems with specially structured matrices
that can be efficiently inverted. We also aim to introduce and construct
efficient spectral-Galerkin algorithms for solving the integrated
forms of high even-order elliptic differential equations. We demonstrate
the advantage of using the integrated forms over the differentiated ones,
by noting that the algebraic systems resulting from the integrated forms
are sparse and therefore cheaper to solve than those obtained from the
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differentiated forms.
In Chapter 1, we give a brief introduction to the spectral methods
and their advantages over the standard finite-difference and finiteelement
methods. Also, we clarify the differences between the three
most commonly used spectral methods, namely, the Galerkin, collocation,
and tau methods. A brief account of orthogonal polynomials, their
properties and expansion of functions in terms of them are given. Some
general properties of third and fourth kinds of Chebyshev polynomials
are also considered.
In Chapter 2, two new analytical formulae expressing explicitly the
derivatives of Chebyshev polynomials of third and fourth kinds of any
degree and for any order in terms of Chebyshev polynomials of third and
fourth kinds themselves are given. Two other explicit formulae which
express the third and fourth kinds Chebyshev expansion coefficients
of a general-order derivative of an infinitely differentiable function in
terms of their original expansion coefficients are also given. Two new
reduction formulae for some terminating hypergeometric functions of
unit argument are deduced.
In Chapter 3, we introduce and develop in detail two new algorithms
based on Galerkin method for solving high even-order differential
equations in one and two space variables subject to homogeneous
and nonhomogeneous boundary conditions using third and fourth kinds
of Chebyshev polynomials. Other two algorithms based on choosing
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compact combinations of shifted Chebyshev polynomials of third and
fourth kinds as basis functions are also considered.
In Chapter 4, We present and implement in detail two algorithms for
solving the integrated forms of high even-order differential equations in
one and two space variables subject to homogeneous and nonhomogeneous
boundary conditions using third and fourth kinds of Chebyshev
polynomials.
The numerical results obtained in this thesis are tabulated whenever
possible. These results show that the proposed algorithms are efficient
and accurate. Comparisons with some other techniques presented by
some authors are made throughout the context whenever available.
To the best of our knowledge, this is the first work which uses the
third and fourth kinds Chebyshev-Galerkin methods for solving high
even-order differential equations in one and two space variables and we
do believe that all the theoretical results stated and proved in this thesis
are completely new. The programs used in this thesis are performed
on the PC machine, with Intel(R) Pentium(R) 4 CPU 3.00 GHz, 1.00
GB of RAM, and the symbolic computation software Mathematica 7
has been also used.