Table of Contents

Preface

The goal of this present second edition is still the same as that of the first edition. It is to present the subject of integral equations, their varied applications and basic methods of solutions, on a level close to that of a first (sophomore) course in ordinary differential equations. This is not such an easy task, especially when we don't assume but basic calculus and differential equaitons as prerequisites. The main thrust here is that a variety of applied problems have their natural mathematical setting as integral equations, thus they have the advantage, of the latter's, usually, simpler methods of solution. In addition, a large class of initial and boundary value problems, associated with differential equations, can be reduced to integral equations, whence enjoy the advantage of the above integral representation. Such topics also bring to light the unity of differentiation and integration, where a tendency exists nowdays for their presentation as such in the suggested revision of the basic calculus courses. It may be said that such aa basic integral equations course would complement the elementary differential euqations course, especially when the actual coverage in the latter is (most often) limited, to initial value problems, and for obvious historical reasons. This being that differential equations took of after the work of Leibnitz and Newton, with the flavor of applicaitons in dynamics, which had occurred long time before integral equations started to get the attention at the very beginning of this century.

We should point out here that for this elementary presentation of integral equations, assuming only calculus and differential equatoins preparation, the treatment in all chapters, except for the (optional) Chapter 6, is formal. This is in the sense that clear procedures and steps, for arriving at the solution or some basic results, are emphasized without, necessarily, stopping to give their complete mathematical justification. The latter, most often, requires more advanced mathematics preparation. Thus we shall be limited to give those justifications that would not require us to go beyond the level of this basic applicable undergraduate text.

In this second edition all comments, suggestions and corrections relayed by students, colleagues from around the world, and the expert reviewers of the journals of mathematics and other concerned professions, were addressed. They all deserve my sincere thanks and appreciation. Such suggestions, it is hoped, will help this edition in even more on attaining the same goal set in the first edition for an undergraduate focusing integral equations text to serve the students of science, engineering, and mathematics. To stay with this important goal, and keep the required text material to a comparable size to that of the first edition, we decided to have a new (optional) Chapter 7 for the detailed numerical methods. This includes using higher quadrature rules for the numerical approximation of the integrals. The main changes made for this second edition, in light of the suggestions received are:

1. Discussion of the basic theory and illustration of solutions to Fredholm integral equations of the first kind as done in Section 5.4.
2. Detailed discussion and illustration of numerical integration with the basic higher quadrature rules, and numerical methods of solving Fredholm and Volterra integral equations. The use of the high quadrature rules is covered in a new (optional) Chapter 7.
3. More exercises for each section including some challenging ones.
4. More emphasis on clear statements of the basic theorems for the existence and uniqueness of the solutions of integral equations. The brief introduction of basic theory in Chapter 6, can be considered optional, when seen in light of the main goal of this elementary text.
5. Conditions for the existence of integral transforms, their inverses and important operations are spelled out. A more detailed treatment is found in the author's undergraduate-graduate book on the subject (Marcel Dekker, 1992) entitled ``Integral and Discrete Transforms with Applications and Error Analysis."
6. Very clear examples of singular integral equations with general discussions of their solutions. Such discussions must be taken in light of the (undergraduate) level set for this book, where no complex analysis preparation is assumed for the reader.
7. More applications to update, replace and complement the already ample variety of applied problems as recognized by all reviews of the first edition. These now include some relevant problems in higher dimensions.
8. More emphasis are placed on the interrelation between the integral equations, and the differential equations representations of boundary and initial value problems. This is also to emphasize that differentiation and integration are inseparable, where a tendency is observed now for their presentation as such in the basic calculus courses.
9. All detected and reported typographical as well as other errors are corrected, some examples are deleted and replaced by more appropriate ones. Almost all the suggestions made by the expert reviewers of the journals of our and other concerned professions have been very seriously addressed, keeping in mind the main goal of an undergraduate book for scientists, engineers as well as mathematicians. This includes the reviews of three critical experts for the first draft of this new edition, and another three reviewers of the final draft.

For this edition we now have a "Student's Solution Manual" to accompany this book. It contains very detailed solutions to all the odd numbered problems in the text (see the end of the preface for details).

With these changes and additions, the first chapter still starts with the statements of a number of problems from different subjects, to illustrate their integral equation representation. Although the reader is warned against expecting a full understanding of some of these problems from such a brief presentation, a very detailed formulation of them is given in Chapter 2. This is followed by the usual classification of integral equations and a clear derivation and illustration of some important integral and differential identities needed for the formulations in Chapter 2 and later chapters. Such identities are essential for showing how we can go from the integral equations representation to the differential equations representation and vise versa. We have also improved upon the self-contained (short but simple) presentation of the Laplace and Fourier transforms with clear statements for the existence of the transforms. Chapter 1 is concluded by a section on simple elements of numerical integration which represents only the essentials necessary for the numerical solutions of Fredholm and Volterra integral equations that are discussed in Chapters 5 and 3, respectively. The higher quadrature numerical integration rules along with their needed tables are covered in a new (optional) Chapter 7. They are well illustrated for the numerical integration, setting up the numerical approximation of Volterra and Fredholm equations, and the numerical solution of these integral equations. Chapter 2 involves very detailed modeling of problems as integral equations with a new section on integral equations in higher dimensions illustrated with the Schrodinger equation integral representation in the momentum space. This includes population dynamics, control, mechanics, radiation transport, and boundary and initial value problems. Chapter 3 deals with methods of solving Volterra integral equations, including approximate and numerical methods, which are presented in detail. Chapter 4 is devoted to the construction and properties of Green's functions, which is very important for reducing boundary value problems to Fredholm integral equations. Chapter 5 deals with basic theory and detailed methods of solving Fredholm integral equations including the use of the Green's functions, and a detailed presentation of the familiar approximate and numerical methods of solutions. Methods of estimating the eigenvalues of homogeneous Fredholm integral equations are also presented. In this edition a new special section (Section 5.4) is added for a very elementary theory and a method of solving Fredholm integral equations of the first kind. Also, more varied numerical methods are used in the new Chapter 7 for solving the different integral equations, compared to the very basic ones in Chapters 3 and 5 as it was the case in the first edition. In Chapter 6 we have a brief and descriptive discussion of the theory regarding the convergence of the methods of solving linear as well as nonlinear integral equations. For the basic introductory undergraduate course, this chapter is clearly optional.

In each chapter we have attempted to present many clear examples in every section followed by a good number of related exercises at the end of each section with hints to (almost) all exercises, and answers to all the exercises.

To use this text for an elementary one-semester or one-quarter course in integral equations, we suggest that from Section 1.4 of Chapter 1 only the very essential elements, that are necessary for the convolution theorems, of the Laplace and the Fourier transforms be included, a selected number of mathematical modeling problems from Chapter 2 be covered, depending on the students' interest, and some selected subsections of, the relatively long, Chapter 5 be omitted. An exposure to the very basic numerical methods of solution in Chapters 3 and 5, with their exercises and detailed answers, is very beneficial. Another possibility is to present the most basic material for a one-semester course in integral equations with boundary value problems as part of the senior or first-year graduate course in methods of applied mathematics for scientists and engineers. For this purpose, we have included the added Chapter 7 on the numerical methods using higher quadrature rules.
 

1999, 464 pp.
Hardcover, ISBN 0-471-31734-9
USD 79.95

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