Fractional calculus for scientists and engineers.

*(English)*Zbl 1251.26005
Lecture Notes in Electrical Engineering 84. Dordrecht: Springer (ISBN 978-94-007-0746-7/hbk; 978-94-007-0747-4/ebook). xiv, 152 p. (2011).

The subject of fractional calculus (FC) is gaining increasing popularity nowadays due to many applications, e.g. electromagnetism, control engineering, signal processing, etc. The increase in the number of physical and engineering processes that are best described by fractional differential equations has motivated its study. This small and well-written book (written in a cursive way like a divulgation text) is an addition to the growing literature in the subject e.g. [S. G. Samko; A. A. Kilbas and O. I. Marichev, Fractional integrals and derivatives: theory and applications. New York, NY: Gordon and Breach (1993; Zbl 0818.26003)] and [K. Nishimoto, Fractional calculus. (Calculus in the 21st century). Vol. V. Integrations and differentiations of arbitrary order. Koriyama: Descartes Press Co. (1996; Zbl 0911.26004)] . One of the main difference is the strong emphasis in applications of FC giving a practical overview of FC as it relates to Signal Processing. Thus this book will be valuable for a vast audience, ranging from pure mathematicians to engineers. It is also noteworthy to note that the author has many results in this topic.

\indent The book is divided in seven chapters. Each chapter is followed by a suitable bibliography. The inclusion of a subject index would be beneficial to the reader.

Chapter 1 gives an abbreviated history about the author incursion into the world of fractional calculus as well as a small abstract of the book content. Chapter 2 gives a brief historical account of FC as well as a panorama of the different types of fractional integrals and fractional derivatives. Using a signal processing point of view, the author chooses the approach of Grünwald-Letnikov stressing the fact that this choice is better since : 1) it does not need superfluous derivative computations, 2) it does not insert unwanted initial conditions, 3) it is more flexible, 4) it allows sequential computations, and 5) it leads to the other definitions. The rest of the chapter is dedicated to a detailed study of the Grünwald-Letnikov approach. Chapter 3 is devoted to derive the integral representations of Grünwald-Letnikov derivatives. Using this representation, the general Cauchy derivative is obtained. This is used to obtain the Riemann-Liouville and the Caputo derivatives in the complex plane. Interesting examples are also given. The objective of chapter 4, as stated in the introduction, is to treat the fractional continuous-time linear shift-invariant systems. After a brief description, the transfer function and the impulse response are studied in detail. The initial condition problem was treated with generality, and a brief study of the stability of the fractional continous-time linear is given. Chapter 5 introduces a general framework for defining the fractional central differences, obtaining derivative integrals similar to Cauchy, and this is in turn used to obtain generalizations of the Riesz potentials (e.g. summation formulas for the Riesz potentials). Chapter 6 discusses Quantum fractional derivative, in the integral and summation formulation. In Chapter 7, it is briefly described some interesting applications, e.g. biomedical applications, fractional dynamics model, fractional impedance model, fractional brownian motion, and also some open problems.

Contents. Chapter 1 - A Travel Through the World of Fractional Calculus. Chapter 2 - The Causal Fractional Derivatives. Chapter 3 - Integral Representations. Chapter 4 - Fractional Linear Shift Invariant Systems. Chapter 5 - TwoSided Fractional Derivatives. Chapter 6 - The Fractional Quantum Derivative and the Fractional Linear Scale Invariant Systems. Chapter 7 -Where are We Going To?. Bibliography.

\indent The book is divided in seven chapters. Each chapter is followed by a suitable bibliography. The inclusion of a subject index would be beneficial to the reader.

Chapter 1 gives an abbreviated history about the author incursion into the world of fractional calculus as well as a small abstract of the book content. Chapter 2 gives a brief historical account of FC as well as a panorama of the different types of fractional integrals and fractional derivatives. Using a signal processing point of view, the author chooses the approach of Grünwald-Letnikov stressing the fact that this choice is better since : 1) it does not need superfluous derivative computations, 2) it does not insert unwanted initial conditions, 3) it is more flexible, 4) it allows sequential computations, and 5) it leads to the other definitions. The rest of the chapter is dedicated to a detailed study of the Grünwald-Letnikov approach. Chapter 3 is devoted to derive the integral representations of Grünwald-Letnikov derivatives. Using this representation, the general Cauchy derivative is obtained. This is used to obtain the Riemann-Liouville and the Caputo derivatives in the complex plane. Interesting examples are also given. The objective of chapter 4, as stated in the introduction, is to treat the fractional continuous-time linear shift-invariant systems. After a brief description, the transfer function and the impulse response are studied in detail. The initial condition problem was treated with generality, and a brief study of the stability of the fractional continous-time linear is given. Chapter 5 introduces a general framework for defining the fractional central differences, obtaining derivative integrals similar to Cauchy, and this is in turn used to obtain generalizations of the Riesz potentials (e.g. summation formulas for the Riesz potentials). Chapter 6 discusses Quantum fractional derivative, in the integral and summation formulation. In Chapter 7, it is briefly described some interesting applications, e.g. biomedical applications, fractional dynamics model, fractional impedance model, fractional brownian motion, and also some open problems.

Contents. Chapter 1 - A Travel Through the World of Fractional Calculus. Chapter 2 - The Causal Fractional Derivatives. Chapter 3 - Integral Representations. Chapter 4 - Fractional Linear Shift Invariant Systems. Chapter 5 - TwoSided Fractional Derivatives. Chapter 6 - The Fractional Quantum Derivative and the Fractional Linear Scale Invariant Systems. Chapter 7 -Where are We Going To?. Bibliography.

Reviewer: Humberto Rafeiro (Bogotá)

##### MSC:

26A33 | Fractional derivatives and integrals |

26-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions |

92C55 | Biomedical imaging and signal processing |