Books by the Author
 
 
 
This Student's Solution Manual is prepared to accompany and mainly supplement the second edition of the author's text "Introduction to Integral Equations with Applications - Revised and Expanded" by A.J. Jerri, Wiley & Sons, Inc. 1999".  It contains detailed solutions to all the odd numbered problems in the text plus, occasionally, one even numbered problem in a section that may be of much importance to continuity of the subject discussed.  These worked out problems are selected as representative of each section's exercise sets.  Besides the exercises in the text we have also included solutions of other related problems in most of the sections under "Additional Solved Problems" with their statements and detailed solutions, and we placed them at the end of the exercises of their related section.  These problems include some challenging or very detailed ones that may help widen the scope of the related subject in the text. 

 
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. 

 
This book represents the first attempt at a unified picture for the presence of the Gibbs (or Gibbs-Wilbraham) phenomenon in applications, its analysis and the different methods of filtering it out. The analysis and filtering cover the familiar Gibbs phenomenon in Fourier series and integral representations of functions with jump discontinuities. In addition it will include other representations, such as general orthogonal series expansions, general integral transforms, splines approximation, and continuous as well as discrete wavelet approximations. The material in this book is presented in a manner accessible to upperclassmen and graduate students in science and engineering, as well as researchers who may face the Gibbs phenomenon in the varied applications that involve the Fourier and the other approximations of functions with jump discontinuities. Those with more advanced backgrounds in analysis will find basic material, results, and motivations from which they can begin to develop deeper and more general results. We must emphasize that the aim of this book (the first on the subject): to satisfy such a diverse audience, is quite difficult. In particular, our detailed derivations and their illustrations for an introductory book may very well sound repetitive to the experts in the field who are expecting a research monograph. To answer the concern of the researchers, we can only hope that this book will prove helpful as a basic reference for their research papers. In addition, there is always the possibility of following it by a research monograph. To accommodate all those concerned with emphasis on the clarity with some intuition, we shall quote only the very basic theorems, such as those of Fourier and wavelet analysis. The basic and most likely familiar results and theorems of Fourier analysis are reviewed in Chapter 1. We will rely on a good number of the basic references that date back to Wilbraham in 1848. For completeness, we are also including most other references that deal with the Gibbs phenomenon in some way or another. They are placed separately as ``Other Related References" in an Appendix following the main bibliography of this book. To distinguish these references from the ones used in the text, we have added a prefix A to their (separate) numbers. These references also include some very recent ones, or few ``somewhat" relevant ones, that were discovered too late to be included in our general discussion. For completeness, we shall list such references at the end of their corresponding sections. 

 
Deals with the basic elements of the integral, finite, and discrete transforms, with emphasis on their use for solving boundary (and/or initial) value problems as well as facilitating the representation of signals and systems. The analysis of the discrete Fourier transforms is stressed because they represent a form for which an efficient means of computation, the fast Fourier transform algorithms, exists. The major part of the volume can serve as a text for a senior or graduate course on integral and discrete transforms for engineers and scientists, and as a reference for the researcher with interest in the applications, error analysis, and basic theory of these transforms. Annotation copyright Book News, Inc. Portland, Or. 
 

 
This book covers the basic elements of difference equations and the tools of difference and the sum calculus necessary for studying and solving, primarily, ordinary linear difference equations. It is lucidly written and carefully motivated with examples from various fields of applications. These examples are presented in the first chapter and then discussed with their detailed solutions in Chapter 2. A particular feature is the use of the discrete Fourier transforms for solving difference equations 
associated with, generally nonhomogeneous, boundary conditions. Emphasis is placed on illustrating this new method by means of applications. The primary goal of the book is to serve as a primer for a first course in linear difference equations but, with the addition of more theory and applications, the book is suitable for more advanced courses. 

  An introductory undergraduate text covering the basic elements of difference equations and the tools of difference and sum calculus necessary to study and solve ordinary linear difference equations. The volume emphasizes the use of discrete transforms, featuring applications in discrete Fourier transforms for solving boundary value problems, discrete sine and cosine transforms, Z-transfroms, and modeling with difference equations. The operational methods may be extended to solve partial difference equations associated with initial and/or boundary value problems. Annotation c. by Book News, Inc., Portland, Or