Category Archives: Fractional Calculus

On Fractional Calculus

Fractional calculus provides us with a set of axioms and methods to extend the concept of a derivative operator from integer order n to arbitrary order α, where α is a real or complex value.
Despite the fact, that this concept is discussed since the days of Leibniz and since then has occupied the great mathematicians of their times, no other research area has resisted a direct application for centuries. Consequently, Abel’s treatment of the tautochrone problem from 1823 stood for a long time as a singular example for an application of fractional calculus.
Not until the works of Mandelbrot on fractal geometry in the early 1980’s the interest of physicists has been attracted by this subject and caused a first wave of publications on the area of fractional Brownian motion and anomalous diffusion processes. But these works caused only minimal impact on the progress of traditional physics, because the results obtained could also be derived using classical methods.
This situation changed drastically by the progress made in the area of fractional wave equations during the last years. Within this process, new questions in fundamental physics have been raised, which cannot be formulated adequately using traditional methods. Consequently a new research area has emerged, which allows for new insights und intriguing results using new methods and approaches.
The interest in fractional wave equations aroused in the year 2000 with a publication of Raspini. He demonstrated, that a 3-fold factorization of the Klein-Gordon equation leads to a fractional Dirac equation which contains fractional derivative operators of order α =2/3 and furthermore the resulting γ – matrices obey an extended Clifford algebra.
To state this result more precisely: the extension of Dirac’s linearization procedure which determines the correct coupling of a SU(2) symmetric charge to a 3-fold factorization of the d’Alembert-operator leads to a fractional wave equation with an inherent SU(3) symmetry. This symmetry is formally deduced by the factorization procedure. In contrast to this formal derivation a standard Yang-Mills-theory is merely a recipe, how to couple a phenomenologically deduced SU(3) symmetry.
In 2005 we calculated algebraically the Casimir operators and multiplets of the fractional extension of the standard rotation group SO(n). This may be interpreted as a first approach to investigate a fractional generalization of a standard Lie-algebra and the first nontrivial application of fractional calculus in multidimensional euclidean space. The classification scheme derived was used for a successful description of the charmonium spectrum. The derived symmetry has been used to predict the exact masses of Y(4260) and X(4664), which later have been confirmed experimentally.
In 2007 we applied the concept of local gauge invariance to fractional free fields and derived the exact interaction form in first order of the coupling constant. The fractional analogue of the normal Zeeman-effect was calculated and as a first application a mass formula was presented, which gives the masses of the baryon spectrum with an accuracy better than 1%.
It has been demonstrated, that the concept of local gauge invariance determines the exact form of this interaction, which in lowest order coincides with the derived group chain for the fractional rotation group.
Furthermore we investigated the transformation properties of the fractional Schrödinger equation under rotations with the result, that the particles described carry an additional intrinsic mixed rotational- and translational degree of freedom, which we named fractional spin. As a consequence the transformation properties of a fractional Schrödinger equation are closely related to a standard Pauli equation.
Since then the investigation of the fractional rotation group within the framework of fractional group theory has lead to a vast amount of interestening results, e.g. a theoretical foundation of magic numbers in atomic nuclei and metallic clusters
Besides group theoretical methods, the application of fractional derivatives on multi dimensional space  and the increasing importance of numerical approaches are major developments within the last years.
Furthermore, as long as the fractional derivative has been considered as the inverse of a fractional integral, which per se is nonlocal its nonlocality was a common paradigm. But recent years have seen an increasing number of alternative approaches, which are not necessarily founded on nonlocality.
Another increasing  area of research is the investigation of genetic differential equations with variable order fractional derivatives based on an idea of Samko and Ross, where the form and type of a differential operator changes with time or space respectively emphasizing evolutionary aspects of dynamic behavior.
The objective of our research program is the realization of a fractional field theory, which describes the interaction of particles in a more stringent and accurate way, than actual theories do up to now. A major contribution in this research area is the fractional group theory, which allows expressing and solving problems in a very elegant way, which cannot be sufficiently described using traditional methods.
One reason of the success of this concept is the strategy, to interpret concrete experimental data and strictly verify the theoretical results with experimental findings.
Still there are many open questions and problems to solve on the way to realize a successful universal fractional quantum field theory. The concept we follow is a promising strategy, to reach that goal.

Below follows a list of available preprints and reviewed articles.

Generalization of the fractional Poisson distribution

author: R. Herrmann

A generalization of the Poisson distribution based on the generalized Mittag-Leffler function Eα,β(λ) is proposed and the raw moments are calculated algebraically in terms of Bell polynomials. It is demonstrated, that the proposed distribution function contains the standard fractional Poisson distribution as a subset. A possible interpretation of the additional parameter β is suggested.

reference: Fract. Calc. Appl. Anal. (2016) 19(4) 832-842

Reflection symmetric Erdelyi-Kober type operators – a quasi-particle interpretation

author: R. Herrmann

Reflection symmetric Erdelyi-Kober type fractional integral operators are used to construct fractional quasi-particle generators. The eigenfunctions and eigenvalues of these operators are given analytically.
A set of fractional creation- and annihilation-operators is defined and the properties of the corresponding free Hamiltonian are investigated. Analogue to the classical approach for interacting multi-particle systems the results are interpreted as a fractional quantum model for a description of residual interactions of pairing type.

reference: Fract. Calc. Appl. Anal. (2014) 17(4) 1215-1228

A fractal approach to the dark silicon problem: a comparison of 3D computer architectures — standard slices versus fractal Menger sponge geometry

R. Herrmann
The dark silicon problem, which limits the power-growth of future computer generations, is interpreted as a heat energy transport problem when increasing the energy emitting surface area within a given volume. A comparison of two 3D-configuration models, namely a standard slicing and a fractal surface generation within the Menger sponge geometry is presented. It is shown, that for iteration orders n>3 the fractal model shows increasingly better thermal behavior. As a consequence cooling problems may be minimized by using a fractal architecture.
Therefore the Menger sponge geometry is a good example for fractal architectures applicable not only in computer science, but also e.g. in chemistry when building chemical reactors, optimizing catalytic processes or in sensor construction technology building highly effective sensors for toxic gases or water analysis.

download: arXiv: arXiv:1404.1891[cs.ET]
reference: Chaos Solitons Fractals (2015) 70,  38-41

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

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
amazon look inside

buy online:,,,,,,,,  Barnes & NobleBAM!(USA), Blackwell’s(UK), booktopia(AU), fishpond(AU),  powell’s(USA), ….


” …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

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

author: R. Herrmann


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

On the origin of space

author: R. Herrmann


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

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

R. Herrmann


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

Fractional Calculus – An Introduction to Physicists – 1st Edition


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


“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