Fractional calculus : an introduction for physicists
Richard Herrmann (Author)
"The book presents a concise introduction to the basic methods and strategies in fractional calculus which enables the reader to catch up with the state-of-the-art in this field and to participate and contribute in the development of this exciting research area. This book is devoted to the application of fractional calculus on physical problems. The fractional concept is applied to subjects in classical mechanics, image processing, folded potentials in cluster physics, infrared spectroscopy, group theory, quantum mechanics, nuclear physics, hadron spectroscopy up to quantum field theory and will surprise the reader with new intriguing insights. This new, extended edition includes additional chapters about numerical solution of the fractional Schrödinger equation, self-similarity and the geometric interpretation of non-isotropic fractional differential operators. Motivated by the positive response, new exercises with elaborated solutions are added, which significantly support a deeper understanding of the general aspects of the theory. Besides students as well as researchers in this field, this book will also be useful as a supporting medium for teachers teaching courses devoted to this subject."--Publisher's website
eBook, English, 2018
3rd edition View all formats and editions
World Scientific Publishing Co. Pte. Ltd., Singapore, 2018
1 online resource
9789813274587, 9789813274594, 9813274581, 981327459X
1047959929
Intro; Contents; Preface to the Third Edition; Preface to the Second Edition; Preface to the First Edition; Acknowledgments; List of Exercises; 1. Introduction; 2. Functions; 2.1 Gamma function; 2.2 Hypergeometric functions; 2.3 Mittag-Leffler functions; 2.4 Miscellaneous functions; 2.5 Discussion; 2.5.1 Multi-parameter Mittag-Leffler functions; 2.5.2 Fractional probability distributions; 2.5.3 Global Padé approximation of the Mittag-Leffler function; 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 Grünwald-Letnikov derivative; 3.4 Discussion; 3.4.1 Orthogonal polynomials; 3.4.2 Differential representation of the Riemann and Caputo fractional derivative; 4. Friction Forces; 4.1 Classical description; 4.2 Fractional friction; 5. Fractional Calculus; 5.1 The Fourier transform; 5.2 The fractional integral; 5.2.1 The Liouville fractional integral; 5.2.2 The Riemann fractional integral; 5.3 Correlation of fractional integration and differentiation; 5.3.1 The Liouville fractional derivative. 5.3.2 The Riemann fractional derivative; 5.3.3 The Liouville fractional derivative with inverted operator sequence
The Liouville-Caputo fractional derivative; 5.3.4 The Riemann fractional derivative with inverted operator sequence
The Caputo fractional derivative; 5.4 Fractional derivative of second order; 5.4.1 The Riesz fractional derivative; 5.4.2 The Feller fractional derivative; 5.5 Fractional derivatives of higher orders
The Marchaud fractional derivative; 5.6 Erdélyi-Kober operators of fractional integration; 5.7 Geometric interpretation of the fractional integral. 5.8 Low level fractionality; 5.9 Discussion; 5.9.1 Semi-group property of the fractional integral; 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
A Comparison of Yukawa and Coulomb Potentials with Riesz Fractional Integrals; 10.1 Folded potentials in fragmentation theory; 10.2 The Riesz potential as smooth transition between Coulomb and folded Yukawa potential; 10.3 Discussion; 10.3.1 Calculation of a fission yield. 11. Quantum Mechanics