الفهرس 
Preliminary conceptsOnly 14 pages are availabe for public view
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Abstract
Integral and differential equations are the most tools used in building
mathematical models for engineering and physical problems. The theory
of integral equations has close contacts with many different areas of
mathematics. Foremost among these are ordinary differential equations
and systems of algebraic equations. In the area of differential equations
many existence and uniqueness results has its origins in integral
equations. Many Boundary Value Problems (BVP) can be formulated as
integral equations of Fredholm type while Initial Value Problems (IVP)
can be formulated as integral equations of Volterra type. The
mathematical models represented in the form of integral equations have
the advantages of considering local and global interactions, while those of
differential equations involve only local interactions. Moreover, when
solving the mathematical models numerically differential equation
models give rise to large sparse systems while integral equation models
give rise to dense coefficient matrices.
The importance of the numerical solutions of integral and differential
equations appears as a result of the difficulties with analytical solutions
when used for real problems and also as a result of the great progress in
computer tools. Also, it is interesting to remember that solutions of
second order boundary value problems are elements in and
solutions of fourth order boundary value problems are elements of
while solutions of the corresponding Fredholm equations are
elements in space.
In this work, we consider the numerical solution of integral equations and
in narrowing the scoop of the thesis we concentrate on integral equations
which have their origins in differential equations especially those of the
second and fourth order.
Now if we have a physical problem which can be formulated as BVP and
as integral equation, what is the best, from the numerical point of view, to
solve the problem as a BVP or as integral equation provided that we use
the same method and the same accuracy?
Frist case: the problem is in the standard form of the second order twopoint
boundary value problems and its integral representation.
Second case: the problem is in the standard form of the fourth order twopoint
boundary value problems and its integral representation.
ii
Third case: the problem contains some coefficients with physical
parameters like those appears in the study of steady state convectiondiffusion
problems.
Fourth case the problem is in the general form of the second order twopoint
boundary value problems with variable coefficients.
It is found that the integral representations give more acceptable results
than the differential equation representation, and the number of iteration
in the integral form is less than that of the differential form.
The thesis consists of three chapters:
Chapter one: Basic concepts of integral and differential equations are
considered and also, the relation between integral and differential
equations with high concern on second and fourth order differential
equations. Transformation of differential equation of higher order or
system of differential equations of first order into one-dimensional
integral equation using the replacement Lemma was studied. Relaxation
iterative methods for solving linear algebraic equations and the choice of
relaxation parameters to obtain the most rapid convergence are presented.
As a measure and comparison of the speed of convergence of iterative
methods we adopted an asymptotic convergence rate which depends on
the spectral radius of the corresponding iteration matrix. Four
approximate methods (the Liouville-Neumann series, the Laplace
transform, the finite difference and the cubic Spline) for solving integral
equations (Fredholm – Volterra) are discussed.
Chapter two: This chapter is dedicated to BVP of the second order in the
form:
( ) ( ) ( ) ( ) ( ) ( )
and fourth order BVP of the form
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
With the help of the replacement lemma, a single integral equation of
Fredholm type, ( ) ( ) ∫ ( ) ( )
, corresponding to the
given boundary value problem is obtained.
The case of writing the fourth order boundary value as a system of two
second order boundary value problems is considered.
Also, the general linear second order BVP
( ) ( ) ( ) ( ) ( ) ( ) ,
with its normal form ( ) ( ) ( ) ( ) is considered. The finite
difference method is used to write the differential or the integral
equations as an algebraic system. The Successive Over-Relaxation (SOR)
iii
and the K- Successive Over-Relaxation (KSOR) are used in the treatment
of the resulting algebraic system.
On the bases of the spectral radius of the iteration matrix, a suitable
value for the relaxation parameters is determined. Four numerical
examples are considered to illustrate the concepts and the asymptotic rate
of convergence is used as a comparison technique for comparing speed of
convergence for the resulting systems.
Chapter three: two practical problems one with constant coefficients
with parameter and the other with variable coefficients are considered.
We constructed the equivalent Fredholm integral representation, and
compared the numerical treatment of the differential and integral forms.
Moreover we calculated the interval of the parameter of the differential
equation and the parameter of the iterative method for which the
convergence of the corresponding algebraic system is faster.
The first problem is:
( ) ( ) ( ) ( ) ( ) ( )
The second problem is:
( ) ( ) ( ) ( )
It is worth to mention that:
All calculations are done by the use of the computer algebra system
”Mathematica 8.0”
The results of chapter two are published as (Boundary value
problems, Fredholm integral equations, SOR and KSOR methods)
see the list of references.