Category Archives: Fractional Calculus

Books

Fractional Calculus – An Introduction for Physicists – 2nd revised and extended Edition

OUT OF PRINT

From the cover: 
The book presents a concise introduction to the basic methods and strategies in fractional calculus and enables the reader to catch up with the state of the art in this field as well as to participate and contribute in the development of this exciting research area.
The contents are devoted to the application of fractional calculus to physical problems. The fractional concept is applied to subjects in classical mechanics, group theory, quantum mechanics, nuclear physics, hadron spectroscopy and quantum field theory and it will surprise the reader with new intriguing insights.
This new, extended edition now also covers additional chapters about image processing, folded potentials in cluster physics, infrared spectroscopy and local aspects of fractional calculus. A new feature is exercises with elaborated solutions, which significantly supports a deeper understanding of general aspects of the theory. As a result, this book should also be useful as a supporting medium for teachers and courses devoted to this subject.

Fractional Calculus – An Introduction for Physicists (2nd revised and extended Edition)
by Richard Herrmann,
World Scientific Publishing, Singapore, March 2014, 500 pp, 6 x 9 in.
ISBN: 978-9814551076
Order information at World Scientific Publ.:
additional material:  Front matter (Contents etc.) , Chapter 1 (Introduction) , Index
preview: amazon look inside

buy online: amazon.com.au, amazon.br, amazon.ca, amazon.cn, amazon.com, amazon.co.uk, amazon.de, amazon.es, amazon.fr,  amazon.in,  amazon.it, amazon.jp, amazon.mx, amazon.nl,  Barnes & Noble, BAM!(USA), Blackwell’s(UK), booktopia(AU), fishpond(AU),  powell’s(USA), ….

reviews:

” …a popular book on fractional calculus, which has proven useful to many new researchers in the field. …A very welcome new feature in the  second edition is the inclusion of exercises at the end of every chapter, with  detailed solutions in the back of the book. This book is specifically aimed at physicists, although many of my colleagues outside physics have also found it useful. …The book takes a practical approach, which will be especially appealing to those accustomed to thinking about modeling in terms of differential equations and transforms.” 

M. M. Meerschaert, Statistics and Probability Dept. ,
Michigan State University,
.

 ” …I was pleasantly surprised not only by the amount of material the author masterly presents, but also by the timely inclusion of historical remarks that frame the discussions within aspects that the reader is familiar with…
Richard Herrmann’s Fractional Calculus is a highly recommended book…

J. Rogel-Salazar, School of Physics, Astronomy and Mathematics,
University of Hertfordshire,
For full details on this review:  Contemporary Physics,  (2015) 56(2) 240
.

“… A valuable addition to the second edition is the exercises-solutions section. … The significant change in size between the two editions within a short period indicates the importance of fractional calculus. … The reviewer strongly recommends this book for beginners as well as specialists in the fields of physics, mathematics and complex adaptive systems.” 

E. Ahmed, Zentralblatt MATH
For full details on this review: Zentralblatt MATH (2014), Zbl 06293341
Publications

Towards a geometric interpretation of generalized fractional integrals – Erdelyi-Kober type integrals on R^N as an example

author: R. Herrmann

abstract:

A family of generalized Erdelyi-Kober type fractional integrals is interpreted geometrically as a distortion of the rotationally invariant integral kernel of the Riesz fractional integral in terms of generalized Cassini ovaloids on R^N . Based on this geometric view, several extensions are discussed.

download: arXiv: arXiv:1401.6051
reference: Fract. Calc. Appl. Anal. (2014) 17(2) 361-370

Publications

On the origin of space

author: R. Herrmann

abstract:

Within the framework of fractional calculus with variable order the evolution of space in the adiabatic limit is investigated. Based on the Caputo definition of a fractional derivative using the fractional quantum harmonic oscillator a model is presented, which describes space generation as a dynamic process, where the dimension $d$ of space evolves smoothly with time in the range 0 <= d(t) <=3, where the lower and upper boundaries of dimension are derived from first principles. It is demonstrated, that a minimum threshold for the space dimension is necessary to establish an interaction with external probe particles. A possible application in cosmology is suggested.

download: arXiv: arXiv:1308.4587
reference: Cent. Eur. J. Phys. (2013) 11(10) 1212-1220

Publications

Folded potentials in cluster physics – a comparison of Yukawa and Coulomb potentials with Riesz fractional integrals

author:
R. Herrmann

abstract:

In cluster physics a single particle potential to determine the microscopic part of the total energy of a collective configuration is necessary to calculate the shell- and pairing effects. In this paper we investigate the properties of the Riesz fractional integrals and compare their properties with the standard Coulomb and Yukawa potentials commonly used. It is demonstrated, that Riesz potentials may serve as a promising extension of standard potentials and may be reckoned as a smooth transition from Coulomb to Yukawa like potentials, depending of the fractional parameter $\alpha$. For the macroscopic part of the total energy the Riesz potentials treat the Coulomb-, symmetry- and pairing-contributions from a generalized point of view, since they turn out to be similar realizations of the same fractional integral at distinct $\alpha$ values.

download: arXiv: arXiv:1305.0890
reference: J. Phys. A.: Math. Theor. (2013) 46(40) 405203

Books

Fractional Calculus – An Introduction to Physicists – 1st Edition


OUT OF PRINT

From the cover:
Fractional calculus is undergoing rapid and ongoing development. We can already recognize, that within its framework new concepts and strategies emerge, which lead to new challenging insights and surprising correlations between different branches of physics.

This book is an invitation both to the interested student and the professional researcher. It presents a thorough introduction to the basics of fractional calculus and guides the reader directly to the current state-of-the-art physical interpretation. It is also devoted to the application of fractional calculus on physical problems, in the subjects of classical mechanics, friction, damping, oscillations, group theory, quantum mechanics, nuclear physics, and hadron spectroscopy up to quantum field theory.

Fractional Calculus – An Introduction for Physicists
by Richard Herrmann,
World Scientific Publishing, Singapore, February 2011, reprinted 2012, 276 pp, 6 x 9 in.,

ISBN: 978-9814340243

reviews:

“The book is a solid introduction to fractional calculus that contains, in particular an elucidating section on geometric interpretation of fractional operators… the bulk of the book concentrates on aspects of fractional calculus related to symmetries in quantum mechanics…what is covered is presented in an authoritative, solid style and actually provides very entertaining reading…Overall, Fractional Calculus is an affordable and valuable introduction to the field that will appeal to physicists interested in scientific what-ifs…”

Ralf Metzler, Physics Today
For full details on this review, please visit: Physics Today 65(2), (2012) 55–56;
doi: 10.1063/PT.3.1443
.

“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. The reviewer strongly recommends this book for beginners as well as specialists in the fields of physics, mathematics and complex adaptive systems.”

 

E. Ahmed, Zentralblatt MATH
For full details on this review, please visit: Zentralblatt MATH (2012), Zbl 1232.26006
.

“…the first three chapters actually appear very helpful at the graduate level. Each chapter has a careful precis at the start. There are many analyses illustrating outcomes of fractional analyses…If this [fractional calculus] is the field of your research then this book is essential with numerous references…”

 

J. E. Caroll, Contemporary Physics
For full details on this review, please visit: Contemporary Physics 53(2), (2012), 187–188; doi:10.1080/00107514.2011.648957
Publications

Numerical solution of the fractional quantum mechanical harmonic oscillator based on the Riemann and Caputo derivative

author:
R. Herrmann

abstract:
Based on the Riemann- and Caputo definition of the fractional derivative we tabulate the lowest n=31 energy levels and generated graphs of the occupation probability of the fractional quantum mechanical harmonic oscillator with a precision of 32 digits for 0.50 < \alpha < 2.00, which corresponds to the transition from U(1) to SO(3).

reference: Gam. Ori. Chron. Phys. (2013) 1(1) 13-176

Publications

The fractional Schroedinger equation and the infinite potential well – numerical results using the Riesz derivative

author:
R. Herrmann

abstract:
Based on the Riesz definition of the fractional derivative  the fractional Schroedinger equation with an infinite well potential is investigated. First it is shown analytically, that the solutions of the free fractional Schroedinger equation are not eigenfunctions, but good approximations for large k and for $\alpha \approx 2$. The first lowest eigenfunctions are then calculated numerically and an approximate analytic  formula for the level spectrum is derived.

download: arxiv: arXiv:1210.4410[math-ph]
reference: Gam. Ori. Chron. Phys. (2013) 1(1) 1-12

Publications

Curvature interaction in collective space

author:
R. Herrmann

abstract:
For the Riemannian space, built from the collective coordinates used within nuclear models, an additional interaction with the metric is investigated, using the collective equivalent to Einstein’s curvature scalar. The coupling strength is determined using a fit with the AME2003 ground state masses. An extended finite-range droplet model including curvature is introduced, which generates significant improvements for light nuclei and nuclei in the trans-fermium region.

download: arxiv: arXiv:0801.0298 [nucl-th] [physics.gen-ph]
reference: International Journal of Modern Physics E (2012) 21 1250103

Publications

Infrared spectroscopy of diatomic molecules – a fractional calculus approach

author:
R. Herrmann

abstract:
The eigenvalue spectrum of the fractional quantum harmonic oscillator is calculated numerically  solving the fractional Schr\”odinger equation based on the Riemann- and Caputo definition of a fractional derivative. The fractional approach allows a smooth transition between vibrational and rotational type spectra, which is shown to be an appropriate tool to analyze IR spectra of diatomic molecules.

download: arxiv: arXiv:1209.1630 [physics.gen-ph]
reference: International Journal of Modern Physics B (2013) 27(6) 1350019