Discrete fractional calculus. (English) Zbl 1350.39001

Cham: Springer (ISBN 978-3-319-25560-6/hbk; 978-3-319-25562-0/ebook). xiii, 556 p. (2015).
While continuous fractional calculus has had a long history, little attention has been given to discrete fractional calculus. In the past five to seven years numerous authors, including the first author of this manuscript, have taken the challenge of developing the theory of discrete fractional calculus. This textbook is the first dedicated solely to the study of this field.
This book is written in the straight-forward, easy-to-read, and well-balanced style that can be found in many of the books written by the second author. There are seven chapters which span basic discrete calculus through the use of discrete calculus to study nonlocal boundary value problems. While the authors limit themselves to discrete cases, the notation and terminologies used will be familiar to those who study the theory of dynamical systems on time scales.
Chapter one is essentially a review of difference calculus and linear difference equations. In this chapter, the authors introduce the concept of delta discrete calculus. They define the forward difference operator \(\Delta\), review both difference and summation calculus, and introduce the discrete analogues of well-known functions (the discrete exponential function, the falling function, the discrete trigonometric functions, etc.). Additionally, they review how to solve first- and second-order linear difference equations as well as vector difference equations. They end the chapter by looking at stability of linear systems and discrete Floquet theory. This chapter can be skipped or skimmed over if one is well versed in the modern study of difference equations.
The second chapter begins with the definition of the delta Laplace transform as well as its properties. In the third section the authors define the \(\nu\)-th fractional sum based as \(a\), \(\Delta_a^{-\nu} f(t)\) as well as the Riemann-Liouville fractional difference. In the subsequent sections they state the properties of the Riemann-Liouville fraction difference operator. They end the chapter with the Laplace transform method for solving fractional initial value problems.
The authors repeat the development of discrete fractional calculus from the first two chapters in the third, fourth and fifth chapters. In Chapter 3, the authors consider the discrete nabla (backwards) difference. They show that there is an advantage of the discrete nabla fractional calculus over the discrete delta calculus. In delta fractional calculus there is a shifting of domains when one goes from the domain of the function to the domain of the delta fractional difference. This shifting is not as great in the nabla fractional difference. For both the delta and nabla fractional calculus the jumps between successive points is constant. In Chapters 4 and 5, the authors consider time scales in which the step size in the domains is not constant. In Chapter 4 they examine quantum calculus (\(q\)-calculus) while in Chapter 5 they look at fractional calculus on mixed time scales.
For the final two chapters of the book, the authors focus on fractional boundary value problems. These two chapters require more mathematical maturity than the first five chapters.
Overall this textbook is very well written and easy to follow. The topics flow smoothly both from section to section as well as from chapter to chapter. Each chapter ends with numerous exercises. This book will be very useful to those studying fractional calculus in general and discrete fractional calculus. It will also be useful as a textbook for a course in discrete fractional calculus.


39A12 Discrete version of topics in analysis
26A33 Fractional derivatives and integrals
44A10 Laplace transform
39A06 Linear difference equations
39-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to difference and functional equations
39A30 Stability theory for difference equations
39A13 Difference equations, scaling (\(q\)-differences)
34N05 Dynamic equations on time scales or measure chains
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