Discrete Fractional Calculus (2015)

Preface

The continuous fractional calculus has a long history within the broad area of mathematical analysis. Indeed, it is nearly as old as the familiar integer-order calculus. Since its inception, it can be traced back to a question L’Hôpital had asked Leibniz in 1695 regarding the meaning of a one-half derivative; it was not until the 1800s that a firm theoretical foundation for the fractional calculus was provided. Nowadays the fractional calculus is studied both for its theoretical interest as well as its use in applications.

In spite of the existence of a substantial mathematical theory of the continuous fractional calculus, there was really no substantive parallel development of a discrete fractional calculus until very recently. Within the past five to seven years however, there has been a surge of interest in developing a discrete fractional calculus. This development has demonstrated that discrete fractional calculus has a number of unexpected difficulties and technical complications.

In this text we provide the first comprehensive treatment of the discrete fractional calculus with up-to-date references. We believe that students who are interested in learning about discrete fractional calculus will find this text to be a useful starting point. Moreover, experienced researchers, who wish to have an up-to-date reference for both discrete fractional calculus and on many related topics of current interest, will find this text instrumental.

Furthermore, we present this material in a particularly novel way since we simultaneously treat the fractional- and integer-order difference calculus (on a variety of time scales, including both the usual forward and backwards difference operators). Thus, the spirit of this text is quite modern so that the reader can not only acquire a solid foundation in the classical topics of the discrete calculus, but is also introduced to the exciting recent developments that bring them to the frontiers of the subject. This dual approach should be very useful for a variety of readers with a diverse set of backgrounds and interests.

There are several ways in which this book could be used as part of a formal course, and we have designed the text to be quite flexible and accommodating in its use. For example, if one prefers, it is possible to use this text for an introductory course in difference equations with the inclusion of discrete fractional calculus. In this case coverage of the first two chapters provide a basic introduction to the delta calculus, including the fractional calculus on the time scale of integers. We also recommend Sects. 7.2 and 7.3 if time permits.

On the other hand, if students have some basic knowledge of the difference calculus or if two semesters are available, then usage of this text in a number of other courses is possible. For example, for students with a background in elementary real analysis, one can cover Chaps. 1 and 2 more quickly and then skip to Chaps. 6 and 7 which present some basic results for fractional boundary value problems (FBVPs). If one is already familiar with the basics of the fractional calculus, then Chaps. 6 and 7 together with some of the current literature indicated in the references could easily form the basis for a seminar in the current theory of FBVPs. By contrast, if one has two semesters available, one can cover Chaps. 1 –5 carefully, which will provide a very thorough introduction to both the discrete fractional calculus as well as the integer-order time scales calculus.

In short, there are a myriad of courses for which this text can serve either a primary or secondary role. And, in particular, the text has been designed so that, effectively, any chapter after Chaps. 1 and 2 can be freely omitted or included at one’s discretion.

Regarding the specific content of the book, we note that in the first chapter of this book we develop the basic delta discrete calculus using the accepted standard notation. We define the forward difference operator and develop the discrete calculus for this operator. When one applies this difference operator to the power functions, exponential function, trigonometric functions, and hyperbolic functions one often gets very complicated functions and these formulas are quite often not useful. In this book we define these functions in such a way that the formulas are very nice, and they actually resemble the formulas that we know from the continuous calculus. Many applications and interesting problems involving these functions are given.

In Chap. 2 we first introduce the discrete delta fractional calculus and then study the (delta) Laplace transform, which is a special case of the Laplace transform studied in the book by Bohner and Peterson [62]; we do not assume the reader has any knowledge of the material in that book. The delta Laplace transform is equivalent under a transformation to the well-known Z -transform, but we prefer the definition of the Laplace transform given here, which has the property that many of the Laplace transform formulas are analogous to the Laplace transform formulas in the continuous setting. We show how we can use the (delta) Laplace transform to enable us to solve certain initial value problems for difference equations and summation equations. We then develop several properties of this transform in the fractional calculus setting, giving a precise treatment of domains of convergence along the way. We then apply the Laplace transform method to solve fractional initial value problems and fractional summation equations.

In Chap. 3 we develop the calculus for the discrete nabla difference operator ∇ (backwards difference operator). Once again, the appropriate power functions, exponential function, trigonometric functions, and hyperbolic functions are defined and their properties are derived. The nabla fractional calculus is also developed and a formula relating the nabla and delta fractional calculus is proved. In Chap. 4 the quantum calculus (or q -calculus) is given. The quantum calculus has important applications to quantum theory and to combinatorics, and this chapter provides a broad introduction to the basic theory of the q -calculus. Moreover, we also provide a brief introduction to the concept of fractional q -derivatives and integrals. Finally, in Chap. 5 we present the concept of a mixed time scale, which allows us to treat in a unified way a number of individual time scales and associated operators, e.g., the q -difference operator and the forward difference operator. This chapter will provide the reader with an introduction to the basic theory of the area such as the exponential and trigonometric functions on mixed time scales, the Laplace transform, and the application of these concepts to solving initial and boundary value problems.

The final two chapters of this text, Chaps. 6 and 7, focus on the theory of FBVPs; as such, these two chapters require more mathematical maturity than the first five. In general, and, furthermore, we assume that the reader has the relevant familiarity from the first half of the book. Thus, for example, in Chap. 7 we assume that the reader is familiar with Chap. 2 regarding the fractional delta calculus. In particular, in Chap. 6 the study of Green’s functions and boundary value problems for fractional self-adjoint equations is given. Self-adjoint operators are an important classical area of differential equations and in that setting are well known to have a very pleasing mathematical theory. In Chap. 6 we present some of the known results in the discrete fractional setting, and this presentation will amply demonstrate the number of open questions that remain in this theory. Finally, in Chap. 7 the nonlocal structure of the fractional difference operator (in the delta case only) is explored in a variety of manifestations. For example, we discuss in what ways the sign of the fractional difference (for various orders) affects the behavior of the functions to which the difference is applied (e.g, monotonicity- and convexity-type results). As we show, there are some substantial and surprising differences in the case of the fractional delta operator. Furthermore, we examine how explicit nonlocal elements in discrete fractional boundary value problems may interact with the implicit nonlocal structure of the fractional difference operator, and we examine how to analyze such problems. All in all, in this final chapter of the book we aim to give the reader a sense of the tremendous complexity and mathematical richness that these nonlocal structures induce.

Finally, we should like to point out that we have included a great many exercises in this book, and the reader is encouraged to attempt as many of these as possible. To maximize the flexibility of this text as well as its potential use in independent study, we have included answers to many of the exercises.

We would like to thank Chris Ahrendt, Elvan Akin, Douglas Anderson, Ferhan Atici, Tanner Auch, Pushp Awasthi, Martin Bohner, Abigail Brackins, Paul Eloe, Lynn Erbe, Alex Estes, Scott Gensler, Julia St. Goar, Johnny Henderson, Wu Hongwu, Michael Holm, Wei Hu, Areeba Ikram, Baoguo Jia, Raziye Mert, Gordon Woodward, Rong Kun Zhuang, and the REU students Kevin Ahrendt, Lucas Castle, David Clark, Lydia DeWolf, James St. Dizier, Nicky Gaswick, Jeff Hein, Jonathan Lai, Liam Mazurowski, Sam McCarthy, Brent McKain, Kelsey Mitchell, Kaitlin Speer, Kathryn Yochman, Emily Obudzinski, Matt Olsen, Timothy Rolling, Richard Ross, Sarah Stanley, Dominic Veconi, Cory Wright and Kathryn Yochman, for their influence on this book. Finally, we would like to thank Ann Kostant and our Springer Executive Editor, Elizabeth Loew, and her assistants for the accomplished handling of our manuscript.

Christopher Goodrich

Allan C. Peterson

Omaha, NE, USALincoln, NE, USA