Fractional Calculus 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 to the development of this exciting research area. It is especially 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 the optical model in view of fractional calculus, using machine learning to detect anti-causal sound samples, remarks on covariance in non-local field theories and a completely new section on numerical algorithms for fractional calculus. Motivated by positive responses, new exercises with elaborated solutions are added, significantly supporting a deeper understanding of the general aspects of the theory.
Besides students and researchers in this field, this book will also be useful as a supporting medium for teachers teaching courses devoted to this subject.
Contents:
- Introduction
- Functions
- The Fractional Derivative
- Friction Forces
- Fractional Calculus
- The Fractional Harmonic Oscillator
- Wave Equations and Parity
- Non-locality and Memory Effects
- Causality, Anti-causality and Machine Learning
- Fractional Calculus in Multi-dimensional Space: 2D — Image Processing
- Fractional Calculus in Multi-dimensional Space: 3D — Folded Potentials in Cluster Physics — A Comparison of Yukawa and Coulomb Potentials with Riesz Fractional Integrals
- Fractional Calculus Within the Optical Model
- Quantum Mechanics
- The Fractional Schrödinger Equation With the Infinite Well Potential — Numerical Results Using the Riesz Derivative
- Uniqueness of a Fractional Derivative: the Riesz and Regularized Liouville Derivative as Examples
- Fractional Spin — A Property of Particles Described With the Fractional Schrödinger Equation
- Factorization
- Symmetries
- The Fractional Symmetric Rigid Rotor
- q-Deformed Lie Algebras and Fractional Calculus
- Infrared Spectroscopy of Diatomic Molecules
- Fractional Spectroscopy of Hadrons
- Magic Numbers in Atomic Nuclei
- Magic Numbers in Metal Clusters
- Towards a Geometric Interpretation of Generalized Fractional Integrals
- Fractors — Fractional Tensor Calculus
- Covariance in Fractional Calculus
- Fractional Fields
- Gauge Invariance in Fractional Field Theories
- Numerical Solution of the Fractional Schrödinger Equation via Diagonalization — A Plea for the Harmonic Oscillator Basis Part I: The 1D Case
- On the Origin of Space
- Numerics
- Epilogue
- Appendix A: Solutions to Exercises
- Appendix B: Numerics: Solutions to Problems
- Appendix C: Program Listings from Chapters
Readership: Students, researchers as well as lecturers of physics courses.
