الفهرس 
Preliminaries and Auxiliary Tools
Some Methods Used in Solving Integral EquationsOnly 14 pages are availabe for public view
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Abstract
Integral equations are encountered in various elds of science and numerous applications (in
elasticity, plasticity, heat and mass transfer, oscillation theory, uid dynamics, ltration theory,
electrostatics, electrodynamics, biomechanics, game theory, control, queuing theory, electrical
engineering, economics, medicine, etc.).
Exact (closed-form) solutions of integral equations play an important role in the proper
understanding of qualitative features of many phenomena and processes in various areas of
natural science. Lots of equations of physics, chemistry, and biology contain functions or
parameters which are obtained from experiments and hence are not strictly xed. Therefore,
it is expedient to choose the structure of these functions so that it would be easier to analyze
and solve the equation. As a possible selection criterion, one may adopt the requirement that
the model integral equation admits a solution in a closed form. Exact solutions can be used to
verify the consistency and estimate errors of various numerical, asymptotic, and approximate
methods.
Through out this thesis we come across many subjects that handle the linear integral and
nonlinear integro-di¤erential equations by using the modern mathematical methods, and some
of the powerful traditional methods.
This thesis will be powerful tool for those who are concerned by applied mathematics,
physical science, and engineering, attempts are made so that it presents both analytical and
numerical approaches in a clear and systematic fashion to make this thesis accessible to those
who work in these elds.
Also there will be a part that is devoted to thoroughly examine the nonlinear integral
equations and its applications. Mathematical physics models, such as di¤raction problems,
scattering in quantum mechanics, conformal mapping, and water waves also contributed to the
creation of nonlinear integral equations. Because it is not always possible to nd exact solutions
to problems of physical science that are posed, much work is devoted to obtaining qualitative
approximations that highlight the structure of the solution and we worked to search for new
methods to become closer or to obtain the exact solution.
Summarizing this thesis demonstrates the following ve chapters
Chapter 1: This chapter is an introduction to the basic de nitions and concepts of in-tegral and integro-di¤erential equations with some important de nitions and theorems which
are necessary for studying the properties of integral and integro-di¤erential equations. It also
contains an introduction to some methods used in solving integral equations.
Chapter 2: In this chapter, we introduce the basic de nitions and theorems of the Dif-
ferential transform method (DTM) for some linear and non-linear functions, it also contains
di¤erential transform for convolution theorem and error analysis. Also, we investigate the
di¤erential transform method for the Fresnel integrals, Singularly perturbed Volterra integral
equations, Volterra population model and a system of di¤erential equations.
Exact and approximate solutions were obtained by the mentioned method and we compared
those solutions with previous work that used other methods for the same model.
Chapter 3: In this chapter, we apply the di¤erential transform method for solving quadratic
integral equations to nd the exact solutions for some models and problems, also we study the
existence and uniqueness theorem for quadratic integral equations. It also contains applications
like singularly perturbed Volterra integral equations and Heat Radiation in a Semi-In nite Solid
and their solution using di¤erential transform method.
On the other hand, we introduce the basic idea of Homotopy Perturbation Method (HPM)
and investigate the method for Singularly perturbed Volterra integral equations, quadratic Sin-
gularly perturbed Volterra integral equations, heat radiation in a semi-in nite solid application,
and Volterra population model to nd exact and approximate solutions for these applications.
The method showed remarkable resulted and was compared to the results of another methods
used to solve the same applications.
Chapter 4: In this chapter, we study the di¤erential transform method to nd the approx-
imate solutions of nonlinear delay di¤erential equations (DDEs) and delay integro-di¤erential
equations of type u(qt ) and u(qt). Also, we prove theorems related to the di¤erential trans-
formation of the delay functions u(qt ) and u(qt). The results of some examples were tested
by applying the DTM showed remarkable performance through a comparison with the pervious
results.
Chapter 5: This chapter introduces the basic de nitions and theorems of two and three-
dimensional di¤erential transform for integral equations. By applying the di¤erential transform
method, the integral equations can be transformed to an algebraic equations and solving this equations, we nd the approximate solutions of the integral equations.
Also, we apply two and three-dimensional di¤erential transform on some Integral equations
and we compared the results with the exact solutions.
Two papers were published from the work in this chapter entitled:
1- ”Applications on Di¤erential Transform Method for Solving Singularly Perturbed
Volterra Integral Equation, Volterra Integral Equation and Integro-di¤erential Equation.” in
the International Journal of Mathematics Trends and Technology (IJMTT), (2015) [41].
2- ”Applications on Di¤erential Transform method for solving Singularly Perturbed
Volterra integral equation, Volterra integral equation and integro-di¤erential equation.” in the
Journal of Fractional Calculus and Applications, (2014) [42].
A third paper was accepted containing some work from these chapters entitled:
3- ”Exact Solutions of Quadratic Integral and Integro-di¤erential equations.” in the
Journal of Nonlinear Analysis and Optimization, (2015) [43].