Fractional Calculus

The following sections are included:

• Dedication
• Preface to the Third Edition
• Preface to the Second Edition
• Preface to the First Edition
• Acknowledgments
• Contents
• List of Exercises

During recent years the interest of physicists in nonlocal field theories has been steadily increasing. The main reason for this development is the expectation that the use of these field theories will lead to a much more elegant and effective way of treating problems in particle and high-energy physics as it has been possible up to now with local field theories…

What is number?

When the Pythagoreans believed that everything is number they had in mind primarily integer numbers or ratios of integers [Burkert (1972)]. The discovery of additional number types, first ascribed to Hippasus of Metapontum, at that time must have been both, a deeply shocking as well as an extremely exciting experience which in the course of time helped to gain a deeper understanding of mathematical problems…

The historical development of fractional calculus begins with a step by step introduction of the fractional derivative for special function classes. We follow this path and present these functions and the corresponding derivatives.

In addition we will present alternative concepts to introduce a fractional derivative definition, e.g. a series expansion in terms of the standard derivative. Furthermore we introduce the fractional extension of the Leibniz product rule and we will demonstrate, that an application of this rule will extend the range of function classes, a fractional derivative may be defined for.

The strategies we present for an appropriate extension of the standard derivative definition may be applied uniquely in areas of mathematical sciences, where similar concepts are valid.

As one example we mention the important field of orthogonal polynomials, which is a vivid and actual research area.

To start with a practical application of the fractional calculus on problems in the area of classical physics we will investigate the influence of friction forces on the dynamical behavior of classical particles. Within the framework of Newtonian mechanics friction phenomena are treated mostly schematically. Therefore a description using fractional derivatives may lead to results of major importance.

There are two reasons to start with this subject. This is a nice example, where fractional calculus may be applied on a very early entry level in physics. In addition, for a thorough treatment of fractional friction forces it is not necessary to use an exceptionally difficult mathematical framework, the hitherto presented mathematical tools are sufficient. It is also the first example, that an analytic approach using the methods developed in fractional calculus leads to results, which in general reach beyond the scope of a classical treatment. More examples will be presented in the following chapters.

First we will recapitulate the classical approach and will then discuss the solutions of Newton's equations of motion including fractional friction.

It will be demonstrated, that the viewpoint of fractional calculus leads to new insights and surprising interrelations of until now unconnected classical fields of research.

Up to now we have introduced a fractional derivative definition for special simple function classes. In the following section we will present common generalizations for arbitrary analytic functions. As a starting point we will use Cauchy's formula for repeated integration which will be extended to fractional integrals. Having derived a fractional integral I^α properly we will obtain a satisfying definition of a fractional derivative, if we assume

There are only few problems which may be solved analytically in full generality. The solutions of the differential equation of the harmonic oscillator

The classical field of application for wave equations is the determination of the eigenfrequencies and eigenfunctions of vibrating systems.

Obviously there is a close analogy with the differential equation of the harmonic oscillator.

Therefore an extension of wave equations to the fractional case will benefit from solutions presented in the previous chapter.

A new aspect is the use of boundary conditions in contrast to initial conditions. Furthermore, since a wave equation besides the time dependence also contains a spatial part, we are forced to consider the behavior of solutions for negative arguments.

For free solutions, this problem may be solved by the additional requirement, that these solutions should be eigenfunctions of the parity operator, too.

Now we will embed the concept of fractional calculus into a broader theory of nonlocal operators. This strategy will lead to a successful generalization of the fractional derivative operator and will result in a promising concept for a deduction of a covariant fractional derivative calculus on the Riemann space in the future.

We will demonstrate the influence of a memory effect for a distinguished set of histories presenting a full analytic solution for a simple nonlocal model for a metal hydride battery.

Until now, we have presented only one dimensional applications of fractional calculus. This is appropriate, as long as we confine ourselves to time-dependent problems. But space is a multi-dimensional construct and therefore we have to extend the proposed definitions to the multidimensional case in order to solve fractional space dependent problems…

Convolution integrals of the type

So far we have presented several methods to extend the definition of a standard derivative from integer value n to arbitrary α values:

Wave equations play a significant role in the description of the dynamic development of particles and fields; e.g. the Maxwell-equations describe the behavior of the electro-magnetic field in terms of coupled partial differential equations. In quantum mechanics a particle may be described by the nonrelativistic Schrödinger wave equation, where the kinetic term is given by the Laplace-operator…

The concept of action-at-a-distance has dominated the interpretation of physical dynamic behavior in the early years of classical mechanics at the times of Kepler [Kepler (1609)] and Newton [Newton (1692)]. In the second half of the 19th century the introduction of a field, first successfully applied in the electromagnetic theory of Maxwell [Maxwell (1873)], has led to a change of paradigm away from the previously accepted nonlocal view to an emphasis of local aspects of a given interaction. This development found its culmination in the case of gravitational interaction with Einstein’s geometric interpretation in terms of a space deformation [Adler et al (1975)]…

Symmetries play an important role in physics. If a certain symmetry is present for a given problem, its solution will often be simplified [Greiner and Müller (1994); Frank et al (2009)]…

In the previous chapters we have presented a strategy to introduce a fractional extension for the nonrelativistic Schrödinger equation. Within the framework of canonical quantization we replaced the classical observables {x, p} by fractional derivative operators…

The complexity of a general problem may be drastically reduced if we are able to recognize inherent symmetries or if a classification according to an already known symmetry is possible…

As a first application of the fractional rotation group the ground state excitation spectra of even-even nuclei will be successfully interpreted with a fully analytic fractional symmetric rigid rotor model.

In a generalized, unique approach the fractional symmetric rigid rotor treats rotations, the γ-unstable limit, vibrations and the linear potential limit similarly as fractional rotations. They all are included in the same symmetry group, the fractional SO^α(3). This is an encouraging unifying point of view and a new powerful approach for the interpretation of nuclear ground state band spectra.

The results derived may be applied to other branches of physics as well, e.g. molecular spectroscopy.

In the previous sections we have demonstrated, that the extension of the standard rotation group SO(N), whose elements are the standard angular momentum operators to the fractional rotation group SO^α(N), whose elements are the fractional angular momentum operators leads to extended symmetries. In this case from a pure rotational symmetry to fractional rotation symmetry, which besides rotations also includes vibrational degrees of freedom…

Besides Newton’s Principia Huygens’ treatise [Huygens (1673)] on the pendulum clock may be considered as one of the most influential works in physics. Since then, the description of periodical motion in terms of the classical harmonic oscillator is a standard example for an exactly solvable problem in classical mechanics…

The calculation of the hadron spectrum is a task of central interest in quantum chromodynamics (QCD), which describes the dynamics of particles which are subject to strong interaction forces. This requires enormous efforts and computer power [Dürr et al (2008)], that is one reason, why different approximate strategies have been established, which reduce the complexity of the problem…

We have demonstrated that both fractional extensions of the standard rotation group SO(N) based on the Riemann and the Caputo fractional derivative definition respectively may successfully used to describe excitation spectra of baryons and mesons respectively. Baryons are composite particle combinations of three quarks, while mesons are built up by a quark and an anti-quark. As a consequence, baryons are fermions and obey the Pauli-exclusion principle and Fermi-Dirac statistics, while mesons are bosons, which obey the Bose-Einstein statistics…

Since 1984, an increasing amount of experimental data [Knight et al (1984); Martin et al (1991); Pellarin et al (1994)] confirms an at first unexpected shell structure in fermion systems, realized as magic numbers in metal clusters…

For a long time, the open question if there exists any geometrical or physical interpretation of the operators of the (classical) fractional calculus stayed as a challenge and was discussed at the first conferences dedicated to that topic, as at the University of New Haven - USA (1974), University of Strathclyde, Ross Priory - Scotland (1984), Nihon University, Tokyo - Japan (1989), etc. The experts’ hypotheses were pessimistic and then, the fractional calculus was still considered as an elegant theory with the idea of extending Calculus…

No specific theory of fractional tensor calculus exists up to now. In the following we propose notations, which may be of use for a formulation of such a theory. Especially we propose to interpret the fractional derivative parameter α as an extension of tensor calculus on a level similar to spin, which leads to the definition of spinors [Schmutzer (1968)].

Consequently the objects considered in fractional tensor calculus may be called fractors.

The dynamical behavior of fields may be deduced with appropriately chosen equations of motion, which in general will be a complex system of interdependent components…

Historically the first example for a quantum field theory is quantum electrodynamics (QED), which successfully describes the interaction between electrons and positrons with photons [Pauli (1941); Sakurai (1967)]. The interaction type of this quantum field theory between the electromagnetic field and the electron Dirac field is determined by the principle of minimal gauge invariant coupling of the electric charge…

When Heisenberg returned from his spring vacations at Helgoland to Göttingen in 1925, he carried in his rucksack a derivation of the quantum harmonic oscillator discrete level spectrum [Heisenberg (1925)] as a first formulation of the mechanics of quantum particles in terms of coordinate- $\widehat{x}$ and momentum- $\widehat{p}$ operators, formally given as (using Dirac’s bra-ket notation)

Fractional calculus introduces the concepts of nonlocality and memory in order to model complex dynamical behavior [Miller and Ross (1993); Samko and Ross (1993a); Podlubny (1999); Hilfer (2000)]. Literally spoken, we have learned, that the present is influenced by the past in more than one way. Within the framework of fractional calculus it really makes a difference how an actual state is achieved…

In this book we have given a concise introduction to methods and strategies used within the fractional calculus. We have demonstrated, that this specific approach may be a powerful alternative for a description of phenomena in high energy and particle physics…

This section contains the elaborated solutions of the exercises

The following sections are included:

• Bibliography
• Index