الفهرس 
Chapter 1: IntroductionOnly 14 pages are availabe for public view
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Abstract
This thesis presents numerical solutions to weakly singular Volterra integral
equations of the second kind and Abel integral equations that appear in many scientific
applications in the fields of scattering theory, potential theory, radiation theory, radar,
and the effects of magnetic fields on viruses, artificial intelligence, genetic engineering,
nanotechnology, thermodynamics, virology, and epidemiology, among others.
There are a lot of methods that have been published to solve these types of
equations, but we have not found any method that uses barycentric Lagrange
interpolations to obtain interpolate solutions. from this point of view, we have
presented four methods that all depend on barycentric Lagrange interpolations in
matrix-vector formulas. In the first chapter, we introduced an overview of the desired
work in this field and our analysis of the obtained results.
In the second chapter, we study integral equations of all types and kinds. We focused
on different traditional methods to solve weakly singular equations. In the third chapter,
we illustrated in detail four methods with different techniques to obtain interpolate
solutions based on the barycentric Lagrange interpolation in matrix-vector forms. The
goal of this thesis is devoted to the interpolate solutions of Volterra and Abel integral
equations. The procedures begin by interpolating the unknown and the given data
functions based on advanced matrix-vector barycentric Lagrange interpolation. Thus,
each of the unknown and data given function is expressed in the product of four
matrices. The first matrix is the monomial basis functions; the second matrix is a known
square matrix of the coefficients of the barycentric Lagrange interpolations functions;
the third one is a diagonal matrix of the weights of the barycentric interpolation; the
fourth matrix is the unknown coefficients matrix which needed to be determined. The
kernels are interpolated twice. The first time with respect to the first variable and the
second time with respect to the second variable. So, the kernel may be expressed as the
product of five matrices, two of which are the monomial basis functions. For the
singularities of the kernels, we have created several rules based on the optimum choice
of the node distribution of the interpolation. Furthermore, the kernel of any equation
Numerical Solutions of Linear Singular Integral Equations of the Second Kind
Abstract
Page iii
never becomes infinity. That confirms that the kernel’s denominator will never become
zero or have an imaginary value.
A linear algebraic system can be obtained without using the collocation points by
substituting the interpolated unknown function on the left and right sides of the integral
equation. The obtained solution of the system yields the unknown coefficients matrix,
and thereby, we can find the interpolated solution. Moreover, we presented the fifth
method, which evaluates the improper and singular integrals to single and double
integrals. In the fourth chapter, we provided solutions for twenty-six examples, the first
method seven examples, the second method five examples, the third method six
examples, the fourth method three examples, and the fifth method five examples. The
tables and graphics provided showed that the solutions were highly accurate compared
to those provided by other methods. The proposed methods obtain exact interpolated
solutions for non-singular equations, while they yield strongly convergent interpolated
solutions for weakly singular equations. The high precision of the data obtained using
the proposed method and the efficient CPU utilization demonstrate the uniqueness and
effectiveness of our work. Finally, a list of references regarding this discipline was cited.