الفهرس 
Chapter(1) Integral equationsOnly 14 pages are availabe for public view
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Abstract
In engineering and scientific applications of a natural
phenomenon we usually find ourselves in front of one equation of
three, differential equation, integral equation, or integro -
differential equation. In fact the conversion of a scientific
phenomenon to integral equation is the easy way to obtain
numerical solutions, enable us to treat the singularities of the
solution at the end - points, and enable us to prove fundamental
results on the existence and uniqueness of the solution.
In general, an integral equation is an equation where an
unknown function occurs under an integral sign. Integral
equations are classified according to three different dichotomies.
If the limits of integration are both fixed then it is called Fredholm
equations, if one limit is variable then it is called Volterra
equations. If the Placement of the unknown function only inside
the integral it is called of the first kind, and if the unknown
function is in both inside and outside the integral it is called of the
second kind. If the known function identically zero then it is called
homogeneous, if not identically zero it is called inhomogeneous.
Both Fredholm and Volterra equations are said to be linear
integral equations, due to the linear behavior of the unknown
function under the integral sign. Indeed, Integral equations are
encountered in a variety of applications: in potential theory,
geophysics, electricity and magnetism, radiation, and control
systems.
Many methods of solving Fredholm integral equations of the
second kind have been developed in recent years, such as
quadrature method, collocation method and Galerkin method,
expansion method, product-integration method, deferred
correction method, graded mesh method, and Petrov–Galerkin
method. In addition, the iterated kernel method is a Traditional
method for solving the second kind. However, this method also
requires a huge size of calculations. The objective of this thesis is
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to establish a promising algorithm that can be easily programmed
in Matlab, and therefore, computational complexity can be
considerably reduced and much computational time can be saved.
The main idea of the thesis is given in chapter 3 where a
modified Iterative Method via matrices for the solution of
Fredholm Integral Equations of the second kind has been
presented. The given method gives a very simple form for the
iterated kernels via the well - known Hilbert matrix. Thus, the
iterative solutions of an integral equation of the second kind can
be reduced to the solution of a matrix equation, whereas only one
coefficient matrix is required to be computed. Thus the
computational complexity can be considerably reduced and much
computational time can be saved. The new proposed approach
needs a small number of iterations to provide an exact result that
proofs the power of the given Method, and stimulates to find out
the relation between the integral equations and Hilbert Matrix.
The convergence of the obtained solution is studied and three
conditions for the existence of the iterative solution are given.
In the First chapter the definitions, concepts, types, and
applications of integral equations of all types and kinds are
presented. In chapter 2 all about Fredholm Integral Equations of
the second kind and their different methods of solutions is
presented (about 15 methods). In chapter 4 the matlab program
for our method with approximate examples is given. The tables
and figures of our obtained solutions are compared with the
solutions by iterated kernel method is also given, showing that
The new proposed approach needs a small number of iterations
and much computational time can be saved to provide an exact
result.