Difference Equations, Second Edition: An Introduction with Applications 2nd Edition

Book Preface

This book uses elementary analysis and linear algebra to investigate solutions to dif­ference equations. The reader likely will have encountered difference equations in one or more of the following contexts: the approximation of solutions of equations by Newton’s Method, the discretization of differential equations, the computation of special functions, the counting of elements in a defined set (combinatorics), and the discrete modeUng of economic or biological phenomena. In this book, we give ex­amples of bow difference equations arise in each of these areas, as well as examples of numerous applica.tions to other subjects.

Our goal is to present an overview of the various facets of difference equations that can be studied by elementary mathematical methods. We hope to convince the reader that the subject is a rich one, both interesting and useful. The reader will not find here a text on numerical analysis (plenty of good ones already exist). Although much of the contents of this book is closely related to the techniques of numerical analysis, we have, except in a few places, omitted discussion of computation by computer.

This book assumes no prior familiarity with difference equations. The first three chapters provide an elementary introduction to the subject. A good course in cal­culus should suffice as a preliminary to reading this material. Chapter I gives eight elementary examples, including the definition of the Gamma function, which will be important in later chapters. Chapter 2 surveys the fundamentals of the difference cal­culus: the difference operator and the computation of sums, introduces the concept of generating function, and contains a proof of the important Euler summation for­mula. In Chapter 3, the basic theory for linear difference equations is developed. and several methods are given for finding closed form solutions, including annihilators, generating functions, and z-transforms. There are also sections on applications of linear difference equations and on transforming nonlinear equations into linear ones.

Chapter 4, which is largely independent of the earlier chapters, is mainly con­cerned with stability theory for autonomous systems of equations. The Putzer al­gorithm for computing A1 , where A is an n by n matrix, is presented, leading to the solution of autonomous  linear systems with constant coefficients. The chapter covers many of the fundamental stability results for linear and nonlinear systems, using eigenvalue criteria, stairstep diagrams, Liapunov functions, and linearization. The last section is a brief introduction to chaotic behavior. The second edition con­tains two new sections: one on the behavior of solutions of systems of two linear equations (phase plane analysis) and one on the theory of systems with periodic co­efficients (Floquet Theory). Also new to this edition are discussions of the SecantMethod for finding roots of functions and of Sarkovskii’s Theorem on the existence of periodic solutions of nonlinear equations.

Approximations of solutions to difference equations for large values of the inde­pendent variable are studied in Chapter 5. This chapter is mostly independent of Chapter 4, but it uses some of the results from Chapters 2 and 3. Here, one will find the asymptotic analysis of sums, the theorems of Poincare and Perron on asymptotic behavior of solutions to linear equations, and the asymptotic behavior of solutions to nonlinear autonomous equations, with applications to Newton’s Method and to the modified Newton’s Method.

Chapters 6 through 9 develop a wide variety of distinct but related topics involv­ing second order difference equations from the theory given in Chapter 3. Chapter 6 contains a detailed study of the self-adjoint equation. This chapter includes gen­eralized zeros, interlacing of zeros of independent solutions, disconjugacy, Green’s functions, boundary value problems for linear equations, R.iccati equations, and os­cillation of solutions. Sturm-Liouville problems for difference equations are con­sidered in Chapter 7.

These problems lead to a consideration of finite Fourier series, properties of eigenpairs for self-adjoint Sturm-Liou ville probiems, nonhomogeneous problems, and a Rayleigh inequality for finding upper bounds on the smallest eigen­value. Chapter 8 treats the discrete calculus of variations for sums, including the Euler-Lagrange difference equation, transversaLity conditions, the Legendre neces­sary condition for a local extremum, and some sufficient conditions. Disconjugacy plays an important role here and, indeed. the methods in this chapter are used to sharpen some of the results from Chapter 6. In Chapter 9, several existence and uniqueness results for nonlinear boundary value problems are proved, using the con­traction mapping theorem and Brouwer fixed point theorems in Euclidean space. A final section relates these results to similar theorems for differential equations.

The last chapter takes a brief look at partial difference equations. It is shown how these arise from the discretization of partial differential equations. Computational molecules are introduced in order to determine what sort of initial and boundary conditions are needed to produce unique solutions of partial difference equations. Some special methods for finding explicit solutions are summarized.

This edition contains an appendix that illustrates how the technical computing system Mathematica can be used to assist in many of the computations that we en­counter in the study of difference equations. These examples can be easily adapted to other computer algebra systems, such as Maple and Matlab.

This book has been used as a textbook at different levels ranging from middle undergraduate to beginning graduate, depending on the choice of topics. Many new exercises and examples have been added for the second edition. Answers to selected problems can be found near the end of the book. There is also a large bibliography of books and papers on difference equations for further study.