Fractional calculus : an introduction for physicists /

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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 a...

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Detalles Bibliográficos
Clasificación:Libro Electrónico
Autor principal: Herrmann, Richard (Autor)
Formato: Electrónico eBook
Idioma:Inglés
Publicado: New Jersey : World Scientific, 2014.
Edición:Second edition.
Temas:
Mathematical physics.
Fractional calculus.
Physique mathématique.
Dérivées fractionnaires.
SCIENCE > Energy.
SCIENCE > Mechanics > General.
SCIENCE > Physics > General.
Acceso en línea:Texto completo
Tabla de Contenidos:
  • 1. Introduction
  • 2. Functions. 2.1. Gamma function. 2.2. Mittag-Leffler functions. 2.3. Hypergeometric functions. 2.4. Miscellaneous functions
  • 3. The fractional derivative. 3.1. Basics. 3.2. The fractional Leibniz product rule. 3.3. The fractional derivative in terms of finite differences
  • the Grunwald-Letnikov derivative. 3.4. Discussion
  • 4. Friction forces. 4.1. Classical description. 4.2. Fractional friction
  • 5. Fractional calculus. 5.1. The Fourier transform. 5.2. The fractional integral. 5.3. Correlation of fractional integration and differentiation. 5.4. Fractional derivative of second order. 5.5. Fractional derivatives of higher orders
  • the Marchaud fractional derivative. 5.6. Erdelyi-Kober operators of fractional integration. 5.7. Geometric interpretation of the fractional integral. 5.8. Low level fractionality. 5.9. Discussion
  • 6. The fractional harmonic oscillator. 6.1. The fractional harmonic oscillator. 6.2. The harmonic oscillator according to Fourier. 6.3. The harmonic oscillator according to Riemann. 6.4. The harmonic oscillator according to Caputo
  • 7. Wave equations and parity. 7.1. Fractional wave equations. 7.2. Parity and time-reversal. 7.3. Solutions of the free regularized fractional wave equation
  • 8. Nonlocality and memory effects. 8.1. A short history of nonlocal concepts. 8.2. From local to nonlocal operators. 8.3. Memory effects
  • 9. Fractional calculus in multidimensional space
  • 2D-image processing. 9.1. The generalized fractional derivative. 9.2. Shape recovery
  • the local approach. 9.3. Shape recovery
  • the nonlocal approach
  • 10. Fractional calculus in multidimensional space
  • 3D-folded potentials in cluster physics. 10.1. Folded potentials in fragmentation theory. 10.2. The Riesz potential as smooth transition between Coulomb and folded Yukawa potential. 10.3. Discussion
  • 11. Quantum mechanics. 11.1. Canonical quantization. 11.2. Quantization of the classical Hamilton function and free solutions. 11.3. Temperature dependence of a fission yield and determination of the corresponding fission potential. 11.4. The fractional Schrodinger equation with an infinite well potential. 11.5. Radial solutions of the fractional Schrodinger equation
  • 12. The fractional Schrodinger equation with the infinite well potential
  • Numerical results using the Riesz derivative. 12.1. The problem
  • analytic part. 12.2. The solution
  • numerical part
  • 13. Uniqueness of a fractional derivative
  • the Riesz and regularized Liouville derivative as examples. 13.1. Uniqueness on a global scale
  • the integral representation of the Riesz derivative. 13.2. Uniqueness on a local scale
  • the differential representation of the Riesz derivative. 13.3. Manifest covariant differential representation of the Riesz derivative on RN. 13.4. The integral representation of the regularized Liouville derivative. 13.5. Differential representation of the regularized Liouville derivative. 13.6. Manifest covariant differential representation of the regularized Liouville derivative on RN. 13.7. Generalization of a fractional derivative.
  • 14. Fractional spin
  • A property of particles described with the fractional Schrodinger equation. 14.1. Spin
  • the classical approach. 14.2. Fractional spin
  • 15. Factorization. 15.1. The Dirac equation. 15.2. Linearization of the collective Schrodinger equation. 15.3. The fractional Dirac equation. 15.4. The fractional Pauli equation
  • 16. Symmetries. 16.1. Characteristics of fractional group theory. 16.2. The fractional rotation group SO[symbol]
  • 17. The fractional symmetric rigid rotor. 17.1. The spectrum of the fractional symmetric rigid rotor. 17.2. Rotational limit. 17.3. Vibrational limit. 17.4. Davidson potential
  • the so called [symbol]-unstable limit. 17.5. Linear potential limit. 17.6. The magic limit. 17.7. Comparison with experimental data
  • 18. q-deformed Lie algebras and fractional calculus. 18.1. q-deformed Lie algebras. 18.2. The fractional q-deformed harmonic oscillator. 18.3. The fractional q-deformed symmetric rotor. 18.4. Half integer representations of the fractional rotation group SO[symbol](3)
  • 19. Infrared spectroscopy of diatomic molecules. 19.1. The fractional quantum harmonic oscillator. 19.2. Numerical solution of the fractional quantum harmonic oscillator. 19.3. The infrared-spectrumof HCl
  • 20. Fractional spectroscopy of hadrons. 20.1. Phenomenology of the baryon spectrum. 20.2. Charmonium. 20.3. Phenomenology ofmeson spectra. 20.4. Metaphysics: about the internal structure of quarks
  • 21. Magic numbers in atomic nuclei. 21.1. The four decompositions of the mixed fractional SO[symbol](9). 21.2. Notation. 21.3. The 9-dimensional fractional Caputo-Riemann-Riemann symmetric rotor. 21.4. Magic numbers of nuclei. 21.5. Ground state properties of nuclei. 21.6. Fine structure of the single particle spectrum
  • the extended Caputo-Riemann-Riemann symmetric rotor. 21.7. Triaxiality. 21.8. Discussion
  • 22. Magic numbers in metal clusters. 22.1. The Caputo-Caputo-Riemann symmetric rotor
  • an analytic model for metallic clusters. 22.2. Binding energy of electronic clusters. 22.3. Metaphysics: magic numbers for clusters bound by weak and gravitational forces respectively
  • 23. Fractors
  • Fractional tensor calculus. 23.1. Covariance for fractional tensors. 23.2. Singular fractional tensors
  • 24. Fractional fields. 24.1. Fractional Euler-Lagrange equations. 24.2. The fractional Maxwell equations. 24.3. Discussion
  • 25. Gauge invariance in fractional field theories. 25.1. Gauge invariance in first order of the coupling constant [symbol]. 25.2. The fractional Riemann-Liouville-Zeeman effect
  • 26. On the origin of space. 26.1. The interplay between matter and space. 26.2. Fractional calculus with time dependent [symbol] in the adiabatic limit. 26.3. The model and possible consequences for an application in cosmology. 26.4. On the detectability of dynamic space evolution. 26.5. Meta physics: on the connection between dark matter and dark energy
  • 27. Outlook.

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