Linear Difference Equations  with  Discrete Transforms Method Abdul J. Jerri Clarkson University KLUWER ACADEMIC PUBLISHERS  DORDRECHT/BOSTON/LONDON 1996

Book Review

From a review of the book in the Journal of Difference Equations and Applications

“…"....This book is another addition to a growing list of books in this interesting area. The recent books are more research oriented than Jerri's book.....Chapter 4 distinguishes this book from others of its type. The author is famous for his work and his books involving transforms. Chapter 4 is concerned with the discrete Fourier Transform. The method of solving boundary value problems using the discrete Fourier transform is presented......The last chapter of the book, Chapter 7, is devoted to modeling various discrete problems with difference equations......There are a set of exercises at the end of each section and answers are given in the back of the book.
     In summary, while this book does many of the same things done in other books, one of its strengths that distinguishes it from other books is the excellent treatment of discrete transforms.”

Prof. A. Peterson
A very active researcher in the field, and the co-author of two advanced books on difference equation.
 

Preface

This book covers the basic elements of difference equations, and the tools of difference and sum calculus necessary for studying and solving, primarily, ordinary linear difference equations. It is motivated by a good number of examples from various fields, which are presented clearly in the first chapter, then discussed along with their detailed solutions in Chapters 2-7.

In addition to the familiar methods of solving difference equations that are covered in Chapter 3, this book emphasizes the use of discrete transforms. This is an attempt to introduce the methods and mechanics of discrete transforms for solving ordinary difference equations. The treatment closely parallels what many students have already learned about using the operational calculus of Laplace and Fourier transforms to solve differential equations. As in the continuous case, discrete operational methods may not solve problems that are intractable by other methods, but they can facilitate the solution of a large class of discrete initial and boundary value problems. Such operational methods, or what we shall term ``operational difference calculus'' may be extended easily to solve partial difference equations associated with initial and/or boundary value problems.

Difference equations are often used to model ``an approximation'' of differential equations, an approach which underlies the development of many numerical methods. However, there are many situations, for example, recurrence relations and the modeling of discrete processes such as traffic flow with finite number of entrances and exits, in which difference equations arise naturally, as we shall illustrate in Chapter 7. This further justifies the use of the operational difference calculus of discrete transforms. One of the primary thrusts of this book is to present various classes of difference equations and then introduce the discrete transforms which are compatible with them.

The first chapter starts with examples that illustrate how difference equations are the natural setting for problems that range from forecasting population to the electrical networks. These are in addition to the typical use of difference equations as an ``approximation'' of differential equations. In the rest of the chapter we present some of the fundamental difference operators along with their basic properties and their inverses as ``sum'' operators, which are necessary for modeling difference equations as well as developing pairs for the basic discrete transforms. Chapter 2 gives a clear introduction to difference equations, which includes their general classification, which hopefully will help establishing the structure of the remaining chapters. Chapter 2 is concluded with a brief introduction of discrete transforms as the difference analog of integral transforms and their uses in solving differential equations. Chapter 3 discusses in detail the typical methods of solving linear difference equations, along with the most basic theorems and a variety of illustrative examples. This includes constant as well as variable coefficient, and homogeneous and nonhomogeneous equations. In addition to the main topic here which is linear difference equations of one variable, or ``ordinary linear difference equation,'' a clear introduction to difference equations of several variables, or ``partial difference equations'' is also presented, which is supported by a number of interesting examples. Chapter 3 concludes with a brief presentation of the important topic of convergence and stability of the solution of difference equations with initial conditions. Chapter 4, a relatively new chapter for introductory courses in difference equations, introduces the most important transform, the discrete Fourier transform (DFT), along with its basic properties. The DFT represents the main operational difference calculus method for solving difference equations associated with boundary conditions. The efficient algorithm of computing the DFT, the fast Fourier transform (FFT) is also discussed. A number of discrete Fourier transform pairs are deduced, which are needed for illustrating the solution of boundary value problems by this discrete transform method. This method is compared with the typical direct methods of solving difference equations discussed in Chapter 3. Chapter 5 builds upon the work of the preceding chapter to derive the discrete sine and cosine transforms. The boundary value problems with which these transforms are compatible are also discussed and illustrated. Chapter 6 presents the transform used to solve difference equations associated with initial values, or ``initial value problems.'' This is in parallel to how the Laplace transform is used for solving linear differential equations associated with initial conditions, i.e., the familiar initial value problems. Chapter 7, a main chapter, proceeds by examples to sample the many kinds of practical problems which give rise, mostly in a natural way, to difference equations. This chapter is considered to be the main goal that we prepared for in the preceeding chapters with the typical methods of solution in Chapter 3 as well as the discrete transforms method of ``operational difference calculus'' in Chapters 4-6. The applications include chemical mixing, compounding interest and investments, population growth, traffic flow, coupled springs and masses system, evaluating determinants and integrals, and diffusion processes. For each section, there are good number of related exercises with answers to ``almost'' all of the exercises, which are found at the end of the book.

This book may be incorporated into either a one quarter or one semester introductory course for students of engineering, mathematics or science. We assume that students have had an elementary exposure to functions of a complex variable. While a course in differential equations or numerical analysis might increase the appreciation of the material in this book, neither course is truly a prerequisite for an understanding of this book. However, the basic sophomore course of differential equations is most helpful, as we shall often refer to the parallel with difference equations. For such an introductory one semester course we suggest Chapters 1 to 6 and problems of interest to the particular class from Chapter 7.

Of course, the book, with its added transform method flavor, represents a readable reference to students as well as researchers in the applied fields.

This book started over two decades ago with the idea of the author to use discrete Fourier transforms for solving difference equations in parallel to the use of Fourier transforms for solving differential equations, where it was termed ``Operational Difference Calculus''. The author, then, invited Professor William L. Briggs to join him in writing a short monograph. When the time came last year for writing a textbook by expanding the monograph, Prof. Briggs declined sharing the efforts due to his heavy writing commitments. Not to let this chance escapes the author, he consented that the author goes it alone, and kindly agreed to reading the initial manuscript. To my friend and former colleagues Prof. Briggs, I owe sincere thanks and appreciation, and hope that this new and complete project would meet his expectations as well as those in the field.

Table of Contents
 

Preface

Acknowledgements

Course Adoption

1 Sequences and Difference Operators

1.1  Sequences

1.2  Difference Operators

1.3  The Inverse Difference Operator as a Sum Operator

2  Sum Calculus and the Discrete Transforms Methods

2.1  Difference Equations

2.2  Summation by Parts and The Fundamental Theorm of Sum Calculus

2.3  Discrete Transforms and Review of Integral and Finite Transforms

2.4  The Discrete Fourier Transforms and Their Sum Calculus Method

3  Basic Methods of Solving Linear Difference Equations

3.1  Fundamentals of Linear Difference Equations

3.2  Solutions of Homogeneous Difference Equations with Constant Coefficients

3.3  Nonhomogeneous Difference Equations with Constant Coefficients

3.4  Linear System of Difference Equations with Constant Coefficients

3.5  Linear Partial Difference Equations with Constant Coefficients

3.6  First Order Difference Equations with Variable Coeffiicients

3.7  Linear Second Order Difference Equations with Variable Coefficients

3.8  Convergence, Equilibrium, and Stability of Linear Equations

4  Discrete Fourire Transforms

4.1  Continuous to Discrete

4.2  Properties of the Discrete Fourier Transform (DFT)

4.3  Sine - Cosine (DST & DCT) Form of the DFT

4.4  Solution of a Difference Equation by the DFT-Traffic Network

4.5  Further Properties of the DFT for its Operational Sum Calculus Method

4.6  The Fast Fourier Transform

5  The Discrete Sine (DST) and Cosine (DCT) Transforms for Boundary Value Problems

5.1  Establishing the Discrete Sine and Cosine Transforms

5.2  Solution of Difference Equations with Boundary Conditions by DST and DCT

6  The z-Transform for Initial Value Problems

6.1  Discrete Initial Value Problems for the z-Transform

6.2  The z-Transform

6.3  The Operational Sum Calculus Method of the z-Transform for Solving Initial Value Problems

7  Modeling with Difference Equations

7.1  Various Discrete Problems Modeled as Difference Equations

7.2  Solution of the Altruistic Neighborhood Modeled by DCT

References

Answers to Exercises

Index of Notations

Subject Index

March 1996, 464 p.
Hardbound, ISBN 0-7923-3940-1
NLG 285.00/USD 199.00/GBP 129.00
USD 80.00 on orders of five or more copies, for classroom use only
 

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