Fractional Calculus

The following sections are included:

• 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…

The following sections are included:

• Gamma function
• Mittag-Leffler functions
• Hypergeometric functions
• Miscellaneous functions

The following sections are included:

• Basics
• The fractional Leibniz product rule
• The fractional derivative in terms of finite differences - the Grünwald-Letnikov derivative
• Discussion
  • Orthogonal polynomials
  • Differential representation of the Riemann and Caputo fractional derivative

The following sections are included:

• Classical description
• Fractional friction

The following sections are included:

• The Fourier transform
• The fractional integral
  • The Liouville fractional integral
  • The Riemann fractional integral
• Correlation of fractional integration and differentiation
  • The Liouville fractional derivative
  • The Riemann fractional derivative
  • The Liouville fractional derivative with inverted operator sequence - the Liouville-Caputo fractional derivative
  • The Riemann fractional derivative with inverted operator sequence - the Caputo fractional derivative
• Fractional derivative of second order
  • The Riesz fractional derivative
  • The Feller fractional derivative
• Fractional derivatives of higher orders - the Marchaud fractional derivative
• Erdélyi-Kober operators of fractional integration
• Geometric interpretation of the fractional integral
• Low level fractionality
• Discussion
  • Semi-group property of the fractional integral

The following sections are included:

• The fractional harmonic oscillator
• The harmonic oscillator according to Fourier
• The harmonic oscillator according to Riemann
• The harmonic oscillator according to Caputo

The following sections are included:

• Fractional wave equations
• Parity and time-reversal
• Solutions of the free regularized fractional wave equation

The following sections are included:

• A short history of nonlocal concepts
• From local to nonlocal operators
• Memory effects

The following sections are included:

• The generalized fractional derivative
• Shape recovery - the local approach
• Shape recovery - the nonlocal approach

The following sections are included:

• Folded potentials in fragmentation theory
• The Riesz potential as smooth transition between Coulomb and folded Yukawa potential
• Discussion
  • Calculation of a fission yield

The following sections are included:

• Canonical quantization
• Quantization of the classical Hamilton function and free solutions
• Temperature dependence of a fission yield and determination of the corresponding fission potential
• The fractional Schrödinger equation with an infinite well potential
• Radial solutions of the fractional Schrödinger equation

The following sections are included:

• The problem - analytic part
• The solution - numerical part

The following sections are included:

• Uniqueness on a global scale - the integral representation of the Riesz derivative
• Uniqueness on a local scale - the differential representation of the Riesz derivative
• Manifest covariant differential representation of the Riesz derivative on ℝ^N
• The integral representation of the regularized Liouville derivative
• Differential representation of the regularized Liouville derivative
• Manifest covariant differential representation of the regularized Liouville derivative on ℝ^N
• Generalization of a fractional derivative

The following sections are included:

• Spin - the classical approach
• Fractional spin

The following sections are included:

• The Dirac equation
• Linearization of the collective Schrödinger equation
• The fractional Dirac equation
• The fractional Pauli equation

The following sections are included:

• Characteristics of fractional group theory
• The fractional rotation group

The following sections are included:

• The spectrum of the fractional symmetric rigid rotor
• Rotational limit
• Vibrational limit
• Davidson potential - the so called γ-unstable limit
• Linear potential limit
• The magic limit
• Comparison with experimental data

The following sections are included:

• q-deformed Lie algebras
• The fractional q-deformed harmonic oscillator
• The fractional q-deformed symmetric rotor
• Half integer representations of the fractional rotation group SO^α(3)

The following sections are included:

• The fractional quantum harmonic oscillator
• Numerical solution of the fractional quantum harmonic oscillator
• The infrared-spectrum of HCl

The following sections are included:

• Phenomenology of the baryon spectrum
• Charmonium
• Phenomenology of meson spectra
• Metaphysics: about the internal structure of quarks

The following sections are included:

• The four decompositions of the mixed fractional SO^α(9)
• Notation
• The 9-dimensional fractional Caputo-Riemann-Riemann symmetric rotor
• Magic numbers of nuclei
• Ground state properties of nuclei
• Fine structure of the single particle spectrum - the extended Caputo-Riemann-Riemann symmetric rotor
• Triaxiality
• Discussion
  • Curvature interaction in the collective Riemannian space

The following sections are included:

• The Caputo-Caputo-Riemann symmetric rotor - an analytic model for metallic clusters
• Binding energy of electronic clusters
• Metaphysics: magic numbers for clusters bound by weak and gravitational forces respectively

The following sections are included:

• Covariance for fractional tensors
• Singular fractional tensors

The following sections are included:

• Fractional Euler-Lagrange equations
• The fractional Maxwell equations
• Discussion
  • Lagrangian density for a space fractional Schrödinger equation

The following sections are included:

• Gauge invariance in first order of the coupling constant
• The fractional Riemann-Liouville-Zeeman effect

The following sections are included:

• The interplay between matter and space
• Fractional calculus with time dependent α in the adiabatic limit
• The model and possible consequences for an application in cosmology
• On the detectability of dynamic space evolution
• Meta physics: on the connection between dark matter and dark energy

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…

Elaborated solutions are provided for the exercises in this section.

The following sections are included:

• Bibliography
• Index