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The Classical Theory of Integral Equations is a thorough, concise, and rigorous treatment of the essential aspects of the theory of integral equations. The book provides the background and insight necessary to facilitate a complete understanding of the fundamental results in the field. With a firm foundation for the theory in their grasp, students will be well prepared and motivated for further study.

Included in the presentation are:

A section entitled Tools of the Trade at the beginning of each chapter, providing necessary background information for comprehension of the results presented in that chapter;

Thorough discussions of the analytical methods used to solve many types of integral equations;

An introduction to the numerical methods that are commonly used to produce approximate solutions to integral equations;

Over 80 illustrative examples that are explained in meticulous detail;
Nearly 300 exercises specifically constructed to enhance the understanding of both routine and challenging concepts;
Guides to Computation to assist the student with particularly complicated algorithmic procedures.

This unique textbook offers a comprehensive and balanced treatment of material needed for a general understanding of the theory of integral equations by using only the mathematical background that a typical undergraduate senior should have. The self-contained book will serve as a valuable resource for advanced undergraduate and beginning graduate-level students as well as for independent study. Scientists and engineers who are working in the field will also find this text to be user friendly and informative.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Fredholm Integral Equations of the Second Kind (Separable Kernel)

Abstract
In this chapter, our purpose is to examine the Fredholm integral equation of the second kind
$$\phi (x) = f(x) + \lambda \,\int\limits_{a}^{b}K(x,t)\,\phi (t)\,\mathrm{d}t,$$
where K(x, t) is a separable kernel.
Stephen M. Zemyan

Chapter 2. Fredholm Integral Equations of the Second Kind (General Kernel)

Abstract
In Chap. 1, we conducted a thorough examination of the Fredholm integral equation of the second kind for an arbitrary complex parameter λ, assuming that the free term f(x) is complex-valued and continuous on the interval [a, b] and that the kernel K(x, t) is complex-valued, continuous, and separable on the square Q(a, b) = { (x, t): [a, b] ×[a, b]}. We stated the four Fredholm theorems and the Fredholm Alternative Theorem which provide for the construction of all possible solutions to the equation under these assumptions.
Stephen M. Zemyan

Chapter 3. Fredholm Integral Equations of the Second Kind (Hermitian Kernel)

Abstract
A Hermitian kernel is a kernel that satisfies the property
$${K}^{{_\ast}}(x,t) = \overline{K(t,x)} = K(x,t)$$
in the square Q(a, b) = { (x, t): axb and atb}. We assume as usual that K(x, t) is continuous in Q(a, b).
Stephen M. Zemyan

Chapter 4. Volterra Integral Equations

Abstract
In this chapter, our attention is devoted to the Volterra integral equation of the second kindwhich assumes the form
$$\phi (x) = f(x) + \lambda \,{\int \nolimits }_{a}^{x}\,K(x,t)\,\phi (t)\,\mathrm{d}t.$$
(4.1)
Volterra integral equations differ from Fredholm integral equations in that the upper limit of integration is the variable x instead of the constant b.
Stephen M. Zemyan

Chapter 5. Differential and Integrodifferential Equations

Abstract
There are strong connections between the theory of integral equations and the theory of differential equations. Although there are many ways to illustrate, analyze, and interpret these connections, we can only discuss a few of them in this chapter.
Stephen M. Zemyan

Chapter 6. Nonlinear Integral Equations

Abstract
A nonlinear integral equation is an integral equation in which the unknown function appears in the equation in a nonlinear manner. The nonlinearity may occur either inside or outside of the integrand or simultaneously in both of these locations. It leads to an astonishing variety of new phenomena related to the characteristics of the solutions and to the methods of solution.
Stephen M. Zemyan

Chapter 7. Singular Integral Equations

Abstract
The theory introduced in previous chapters, especially the Fredholm Theory, was presented under the restrictive assumptions that the kernel was continuous on its domain of definition and that the interval of integration was finite. There is no guarantee that those results or similar ones will hold if the kernel has an infinite discontinuity or if the interval of integration is infinite.
Stephen M. Zemyan

Chapter 8. Linear Systems of Integral Equations

Abstract
A system of integral equationsis a set of two or more integral equations in two or more unknown functions. Usually, all of the equations belonging to a system are of the same type, but this need not be the case. Since linear systems of Fredholm, Volterra, or singular integral equations occur very commonly in practice, they are the subjects of this chapter.
Stephen M. Zemyan

Backmatter

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