Discrete Fractional Calculus

The Fractional calculus is studied for both its theoretical interest as well as its use in applications. Within the past 5—7 years, there has been a surge of interest in developing a discrete fractional calculus to parallel the continuous theory. This text provides the first comprehensive treatment of the discrete fractional calculus and is appropriate for a myriad of potential courses. Several exercises are provided at the end of each chapter. The authors aim to maximize the flexibility of the text and its potential use in independent study. To this, select answers have been provided and most chapters may be covered or omitted, depending upon the background of the student. For example, the text may be used as primary in an introductory course in differential equations with the inclusion of discrete functional calculus. Chapters 1—2 provide a basic introduction to the delta calculus including fractional calculus on the time scale of integers. For courses where students already have background in elementary real analysis, Chapters 1—2 may be covered quickly and then one can skip to chapters 6—8 which present some basic results in fractional boundary value problems (FBVPs). Chapters 6—8 in conjunction with some of the current literature listed in the Bibliography can provide an easy basis for a seminar in the current theory of FBVPs. For a two-semester course, Chapters 1—5 may be covered carefully, providing a very thorough introduction to both the discrete fractional calculus as well as the integer-order time scale calculus. Students who are interested in learning about discrete fractional calculus will find this text to be a useful starting point and experienced researchers will find the text useful as a reference for Discrete Fractional Calculus (DFC) as well as related topics of current interest.

Allan C. Peterson is professor of mathematics at the University of Nebraska, Lincoln. Peterson, with Walter Kelley, is author of "The Theory of Differential Equations", (c) 2010, Universitext. Current research interests include dynamic equations on time scales, fractional calculus, difference equations, and differential equations.