Fractional Calculus

An Introduction for Physicists

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.

Contents:

• Introduction
• Functions
• The Fractional Derivative
• Friction Forces
• Fractional Calculus
• The Fractional Harmonic Oscillator
• Wave Equations and Parity
• Nonlocality and Memory Effects
• Fractional Calculus in Multidimensional Space — 2D-Image Processing
• Fractional Calculus in Multidimensional Space — 3D-Folded Potentials in Cluster Physics — A Comparison of Yukawa and Coulomb Potentials with Riesz Fractional Integrals
• Quantum Mechanics
• The Fractional Schrödinger Equation with 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
• 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 One Dimensional Case
• On the Origin of Space
• Outlook

Readership: Students, researchers as well as lecturers of various physics courses.
Key Features:

• This is the only book in the market covering the full area of a physical application of fractional calculus
• This book provides a skillful insight into a vividly growing research area and guides the reader from the introductory level up to the current state-of-the-art of a physical interpretation and application in different fields
• This book enables the reader to participate and contribute in the development of this exciting research area and to apply these methods in his own research area too