الفهرس 
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Abstract
Abstract
Many fields and some phenomena in engineering and the physical
sciences require for describing their behaviors by finding their governing
equations (ordinary or integral) equations. Some of these fields include
acoustics, aerodynamics, elasticity, electrodynamics, fluid dynamics,
geophysics (seismic wave propagation), heat transfer, meteorology,
oceanography, optics, petroleum engineering, plasma physics (ionized
liquids and gases), quantum mechanics are modeled for reducing their
equations. Giving solutions to those equations represent describing the
behavior of a phenomenon or model this requires finding computational
methods. The purpose of this thesis is to introduce the
Chebyshev-Galerkin method for solving some differential and integral
equations.
Chapter one gives an introduction of Sturm-Liouville equations and
fractional integro-differential equations. The fundamentals of fluid flow
will be discussed beside the study of the influence of mass and heat
transfer of a hydromagnetic micropolar fluid (MHD) past a stretching
surface. The contributions of researchers and their literature reviews of
the present work are presented. Chapter two gives an illustration of the
Galerkin technique and fundamental Chebyshev polynomials first kind
Tn(x) and some recurrence relations and product Chebyshev polynomials
and also gives important properties that attached to these special functions
like the orthogonality, Chebyshev-Gauss quadrature, and derivatives at
endpoints. Besides shifted Chebyshev polynomials first kind T n(x) are
presented and its properties due to their appropriate in some fractional
derivative problems. Some theorems that are used in computing the
integration of Chebyshev polynomials and their derivative. Chapter
three discuses the second, fourth, and sixth-order Sturm-Liouville
problems and showing how the Chebyshev-Galerkin method solves them
by getting Eigenvalues and Eigenfunctions and in addition to some solved
examples. The results are reformed in tabular and graphical forms and
supporting with their error functions. Chapter four shows a linear
fractional integro-differential problems and how Chebyshev-Galerkin
method is contributed in solving them. Besides giving a brief definition of
i
Abstract
Caputo’s fractional derivative and its properties. Some solved examples
are presented and their results are put in tabular and graphical forms for
validating the method. Chapter five investigate the influence of mass and
heat transfer of a magnetohydrodynamic (MHD) micropolar fluid past a
stretching surface with ohmic heating and viscous dissipation as e.g,
condensation processes in heat exchanger. The governing partial
equations are reduced to non-linear ordinary ones by using appropriate
transformations for variables and solved numerically by the
Chebyshev-Galerkin method. The effects of physical parameters on the
velocity, angular velocity, temperature, and concentration profiles are
illustrated graphically. Comparison of the numerical results made with
previously published results under the special cases. Chapter six includes
the conclusions of using the Chebyshev-Galerkin method in solving
Sturm-Liouville equations, a linear fractional integro-differential
equations and studying effect of heat and mass transfer of micropolar fluid
in the magnetic field over a stretching surface. Also, our suggestions
about the future work are included.