Fractional calculus. An introduction for physicists.

*(English)*Zbl 1232.26006
Hackensack, NJ: World Scientific (ISBN 978-981-4340-24-3/hbk; 978-981-4340-25-0/ebook). xii, 261 p. (2011).

The basic purposes of the book are:

i) To present a concise introduction to the basic methods and strategies in fractional calculus.

ii) To enable the reader to catch up with the state of the art of this field and to participate and contribute in the development of this exciting research area.

The book is devoted to the application of the fractional calculus to physical problems. The fractional concept is applied to subjects in classical mechanics, group theory, quantum mechanics, nuclear physics, hadron spectroscopy up to quantum field theory. The book realizes the interesting property of fractional order (FO) derivative, namely memory and non-locality, which possibly makes FO suitable to investigate quantum mechanics and related systems. It is also suitable to study complex adaptive systems, e.g. biological, social and economic systems. A missing concept in this book consists in the consideration of continuous non-differentiable functions (CND) [H. H. Kairies, “Functional equations of some peculiar functions”, Aequationes Math. 53, No.3, 207–241 (1997; Zbl 0876.39004)], which makes a given definition of fractional order derivative [G. Jumarie, Appl. Math. Lett. 22, No. 3, 378–385 (2009; Zbl 1171.26305)] quite interesting. CND functions are closely related to fractals. Hence, the fractional calculus offers the natural derivative on fractals which are known to exist almost everywhere. Also a relation between \(q\)-deformed algebras and strong interactions has not been mentioned.

The book has the property that derived results are directly compared with experimental findings. As a consequence, the reader is guided and encouraged to apply the fractional calculus approach in her/his research area. It is highly expected that the viewpoint of fractional calculus will lead to new insights and interesting results. The reviewer strongly recommends this book for beginners as well as specialists in the fields of physics, mathematics and complex adaptive systems.

i) To present a concise introduction to the basic methods and strategies in fractional calculus.

ii) To enable the reader to catch up with the state of the art of this field and to participate and contribute in the development of this exciting research area.

The book is devoted to the application of the fractional calculus to physical problems. The fractional concept is applied to subjects in classical mechanics, group theory, quantum mechanics, nuclear physics, hadron spectroscopy up to quantum field theory. The book realizes the interesting property of fractional order (FO) derivative, namely memory and non-locality, which possibly makes FO suitable to investigate quantum mechanics and related systems. It is also suitable to study complex adaptive systems, e.g. biological, social and economic systems. A missing concept in this book consists in the consideration of continuous non-differentiable functions (CND) [H. H. Kairies, “Functional equations of some peculiar functions”, Aequationes Math. 53, No.3, 207–241 (1997; Zbl 0876.39004)], which makes a given definition of fractional order derivative [G. Jumarie, Appl. Math. Lett. 22, No. 3, 378–385 (2009; Zbl 1171.26305)] quite interesting. CND functions are closely related to fractals. Hence, the fractional calculus offers the natural derivative on fractals which are known to exist almost everywhere. Also a relation between \(q\)-deformed algebras and strong interactions has not been mentioned.

The book has the property that derived results are directly compared with experimental findings. As a consequence, the reader is guided and encouraged to apply the fractional calculus approach in her/his research area. It is highly expected that the viewpoint of fractional calculus will lead to new insights and interesting results. The reviewer strongly recommends this book for beginners as well as specialists in the fields of physics, mathematics and complex adaptive systems.

Reviewer: E. Ahmed (Mansoura)