Linear Difference Equations with Discrete Transform Methods

Linear Difference Equations with Discrete Transform Methods

Mathematics and Its Applications Managing Editor: M.HAZEWINKEL Centre/or MatheTlUltics and Computer Science, Amsterdam, The Netherlands Volume 363

Linear Difference Equations with Discrete Transform Methods by Abdul J. Jerri Clarkson University SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-1-4419-4755-0 ISBN 978-1-4757-5657-9 (ebook) DOI 10.1007/978-1-4757-5657-9 Printed on acid-free paper All Rights Reserved 1996 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1996 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

To my youngest daughter Huda.

Contents PREFACE Acknowledgements COURSE ADOPTION xv xix xxi 1 SEQUENCES AND DIFFERENCE OPERATORS 1 1.1 Sequences... 1 1.2 Difference Operators.................. 13 1.3 The Inverse Difference Operator as a Sum Operator 33 2 SUM CALCULUS AND THE DISCRETE TRANS FORMS METHODS 41 2.1 Difference Equations................... 41 2.2 Summation by Parts and The Fundamental Theorem of Sum Calculus... 57 2.3 Discrete Transforms and Review of Integral and Finite Transforms "...................... 65 2.4 The Discrete Fourier Transforms and Their Sum Calculus Method............... 95 3 BASIC METHODS OF SOLVING LINEAR DIFFER ENCE EQUATIONS 139 3.1 Fundamentals of Linear Difference Equations... 139 3.2 Solutions of Homogeneous Difference Equations with Constant Coefficients..... 157 vii

Vlll 3.3 3.4 3.5 3.6 3.7 3.8 Nonhomogeneous Difference Equations with Constant Coefficients........................ 171 Linear System of Difference Equations with Constant Coefficients... 197 Linear Partial Difference Equations with Constant Coefficients.......................... 204 First Order Difference Equations with Variable Coefficients........................... 225 Linear Second Order Difference Equations with Variable Coefficients... 240 Convergence, Equilibrium, and Stability of Linear Equations... 270 4 DISCRETE FOURIER TRANSFORMS 281 4.1 Continuous to Discrete.................. 281 4.2 Properties of the Discrete Fourier Transform (DFT). 285 4.3 Sine - Cosine (DST & DCT) Form of the DFT... 297 4.4 Solution of a Difference Equation by the DFT-Traffic Network.......................... 303 4.5 Further Properties of the DFT for its Operational Sum Calculus Method....... 313 4.6 The Fast Fourier Transform... 321 5 THE DISCRETE SINE (DST) AND COSINE (DCT) TRANSFORMS FOR BOUNDARY VALUE PROB LEMS 329 5.1 Establishing the Discrete Sine and Cosine Transforms 330 5.2 Solution of Difference Equations with Boundary Conditions by DST and DCT................ 340 6 THE z-transform FOR INITIAL VALUE PROB LEMS 355 6.1 Discrete Initial Value Problems for the z-transform. 355 6.2 The z-transform.................. 359 6.3 The Operational Sum Calculus Method of the z- Transform for Solving Initial Value Problems.. 375

IX 7 MODELING WITH DIFFERENCE EQUATIONS 385 7.1 Various Discrete Problems Modeled as Difference Equations... 386 7.2 Solution of the Altruistic Neighborhood Modeled by DeT... 398 REFERENCES 405 ANSWERS TO EXERCISES 409 INDEX OF NOTATIONS 425 SUBJECT INDEX 431

List of Figures 1.1 Sampling a function.... 1.2 An electric network with N Loops..... 2 8 4.1 Equally spaced grid points at in = n'.zit 283 4.2 The partitioning of the interval (0, NT) by N points. 284 4.3 DFT of {Gn } = {n}~~o, Re{gk}' Im{gk}... 300 4.4 Sine and cosine coefficients {ad and {bd for {Gn } = {n}~~o.... 300 4.5 Spectrum Igkl =!(a%+b%)t for two 16 point sequences {Gn } = {I} and {Gn } = {cos 2iVn }, a = 2... 301 4.6 Spectrum Igkl =!(a%+bi)t for two 16 point sequences and {Gn } = {n} {Gn } = {sin 2iV n }, a = 4.. 301 4.7 A traffic network with N intersections......... 303 4.8 A traffic circle with N = 4 intersections... 308 4.9 An application of the splitting algorithm for the FFT 324 4.10 The flow of data in an N = 8 point transform... 325 5.1 The sequence {Gn } and its even extension outside its domain 0 ~ n ~ M.................... 332 5.2 The sequence {Gn } and its odd extension outside its domain 0 ~ n ~ M.................... 335 6.1 A shelfloaded with boxes (of two sizes) occupying nine (equal) spaces............. 379 7.1 A chemical concentration problem. 386 7.2 Coupled spring and mass system at equilibrium, t = 0 390 7.3 Coupled spring and mass system in motion, t > 0... 390 xi

XlI 7.4 A model for N + 1 neighbors........ 399 7.5 Single creditor - single debtor distribution. 401 7.6 Left - right distribution........... 401 7.7 Distribution of wealth in an altruistic neighborhood of 9 families (N = 8) where family 2 is a creditor and family 6 is a debtor. Gn = {2, 2, 2, 1,0, -1, -2, -2, -2}403

List of Tables 4.1 Operational and sequence pairs of discrete Fourier transform... 319 4.2 Stages of the Cooley-Tukey FFT applied to {Gn } = {cos 2"lv n }, N = 8.............. 326 6.1 Operational pairs of the z-transform.. 6.2 The z-transform of (infinite) sequences.. 371. 371 xiii

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. Examples from various fields are presented clearly in the first chapter, then discussed along with their detailed solutions in Chapters 2-7. The book is intended mainly as a text for the beginning undergraduate course in difference equations, where the "operational sum calculus" of the direct use of the discrete Fourier transforms for solving boundary value problems associated with difference equations represents an added new feature compared to other existing books on the subject at this introductory level. This means that in addition to the familiar methods of solving difference equations that are covered in Chapter 3, this book emphasizes the use of discrete transforms. It 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 (integral) 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 sum 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 situaxv

XVI tions, 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 Chapters 1 and 7. This further justifies the use of the operational sum calculus of discrete transforms. One of the goals 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, including their general classification, which hopefully will help establish the structure of the remaining chapters. Chapter 2 concludes with a very brief introduction and illustration of how the Laplace and Fourier transforms are used for simplifying the solution of differential equations, i.e., how the "operational integral calculus" method helps in algebraizing the differential operator, and at the same time involves the prescribed auxiliary conditions. This is complemented by a brief account of the discrete Fourier transform method, as the direct "sum" analog of the "integral" transforms method, for solving difference equations with boundary conditions. Thus we may coin the phrase "operational sum calculus or algebra," since the operation of the discrete Fourier transforms is no more than multiplying a column matrix by a square matrix. 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 coefficients, and homogeneous and nonhomogeneous equations. In addition to the main topic here, which is linear difference equations of one variable, or "ordinary linear difference equations," a clear introduction to difference equations of several variables, or "partial difference equations" is also presented, which is supported by a number of interesting ex-

XVll amples. Chapter 3 concludes with a brief elementary presentation of the important topics of convergence and stability of the equilibrium solution of difference equations with initial conditions. Chapter 4, a relatively new chapter for introductory courses in difference equations, presents the most important transform, the discrete Fourier transform (DFT), along with its basic properties. The DFT represents the main operational sum calculus tool for solving difference equations associated with boundary conditions in the most direct way. 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 direct discrete transform method. 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 z-transform which is used to solve difference equations associated with initial values, or "initial value problems," and which can be encompassed under the "operational sum calculus" methods. This is in parallel to how the Laplace transform is used to solve linear differential equations associated with initial conditions. The basic properties and pairs of the z-transform are established, which are very necessary for illustrating this direct transform method for solving difference equations associated with initial value problems. These solutions are compared with those obtained by the usual methods of Chapter 3. Chapter 7, a main chapter, presents examples~ that 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 culmination of the preceding chapters with the typical methods of solution in Chapter 3, as well as the discrete transforms method of "operational sum calculus" in Chapters 4-6. Of course, these problems can be distributed over the different sections of the book, where their appropriate methods of solution are covered. To accomplish such a purpose, we shall, at the end of each of these sections, make a clear reference to the corresponding examples of Chapter 7. The applications include chemical mixing, compounding interest and investments, population growth, traffic flow, coupled springs and masses system, evaluating

XVlll determinants and integrals, and diffusion processes. For each section, there are many related exercises with answers to most of the exercises, which are found at the end of the book.

Acknowledgements The project which is the subject of this book started over a decade 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 escape, he consented that the author go it alone and kindly agreed to read the initial manuscript. To my friend and former colleague Prof. Briggs, lowe sincere thanks and appreciation and hope that this new and complete project with its new flavor of the direct "operational sum method" will meet his expectations as well as those in the field. I would like to give thanks to Professor Michiel Hazewinkel, the managing editor of the Kluwer Academic Publishers series on Mathematics and its Applications, for inviting me to write this book as well as a forthcoming book on "The Gibbs Phenomenon - Analysis and Remedies." I would like to also thank Professors Saber Elaydi, Ronald E. Mickens, and Johnny L. Henderson for their careful reading of the manuscript and for their detailed constructive suggestions. Professors M. Zuhair Nashed, Allan C. Peterson, Gerry Ladas, John R. Graef, and Russel L. Herman also made many helpful suggestions; they deserve my thanks. I would like also to thank many of the students who read the manuscript and made various remarks. I have seriously attempted to accommodate almost all these valuable suggestions. Mrs. Joyce Worley was of great assistance in typesetting XIX

xx and finalizing this manuscript, and she deserves my special thanks. Thanks are also due Mr. Joseph Hruska for producing the drawings with care. Some topics and examples of the original monograph were used in the author's book "Integral and Discrete Transforms with Applications and Error Analysis," (Marcel Dekker Inc., 1992), which are used now in this book, with due thanks to Marcel Dekker Inc. Mr. John Martindale of Kluwer Academic Publishers showed a great understanding and was most helpful with both projects, and he deserves my sincere thanks. Ms. Arlene Apone, of Kluwer, who reviewed the final manuscript and made constructive editorial suggestions, deserves my thanks. My very special thanks go to my wife Suad and my daughter Huda for being so supportive and very patient during the preparation of this book.

COURSE ADOPTION This book may be incorporated into a 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. Section 2.3 is optional for those who are familiar with Laplace and Fourier integral transforms, and Fourier series. Of course, the book, with its added direct discrete transforms method flavor, represents a readable reference to students as well as to researchers in the applied fields. XXI