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FRACTIONAL CALCULUS (3RD ED): An Introduction for Physicists
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FRACTIONAL CALCULUS (3RD ED): An Introduction for Physicists
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Product Details
ISBN-13: | 9789813274594 |
---|---|
Publisher: | World Scientific Publishing Company, Incorporated |
Publication date: | 07/09/2018 |
Sold by: | Barnes & Noble |
Format: | eBook |
Pages: | 636 |
File size: | 64 MB |
Note: | This product may take a few minutes to download. |
Table of Contents
Foreword v
Acknowledgments vii
1 Introduction 1
2 Functions 5
2.1 Gamma function 6
2.2 functions Mittag-Leffler 7
2.3 Hypergeometric functions 9
2.4 Miscellaneous functions 9
3 The Fractional Derivative 11
3.1 Basics 11
3.2 The fractional Leibniz product rule 16
3.3 Discussion 18
3.3.1 Orthogonal polynomials 18
3.3.2 Differential representation of the Riemann fractional derivative 20
4 Friction Forces 23
4.1 Classical description 23
4.2 Fractional friction 26
5 Fractional Calculus 33
5.1 The Fourier transform 34
5.2 The fractional integral 35
5.2.1 The Liouville fractional integral 36
5.2.2 The Riemann fractional integral 36
5.3 Correlation of fractional integration and differentiation 37
5.3.1 The Liouville fractional derivative 38
5.3.2 The Riemann fractional derivative 39
5.3.3 The Liouville fractional derivative with inverted operator sequence: the Liouville-Caputo fractional derivative 40
5.3.4 The Riemann fractional derivative with inverted operator sequence: the Caputo fractional derivative 42
5.4 Fractional derivative of second order 43
5.4.1 The Riesz fractional derivative 44
5.4.2 The Feller fractional derivative 46
5.5 Fractional derivatives of higher orders 47
5.6 Geometric interpretation of the fractional integral 51
5.7 Low level fractionality 53
5.8 Discussion 55
5.8.1 Semigroup property of the fractional integral 55
6 The Fractional Harmonic Oscillator 57
6.1 The fractional harmonic oscillator 58
6.2 The harmonic oscillator according to Fourier 58
6.3 The harmonic oscillator according to Riemann 60
6.4 The harmonic oscillator according to Caputo 62
7 Wave Equations and Parity 65
7.1 Fractional wave equations 65
7.2 Parity and time-reversal 67
7.3 Solutions of the free regularized fractional wave equation 68
8 Nonlocality and Memory Effects 75
8.1 A short history of nonlocal concepts 75
8.2 From local to nonlocal operators 77
8.3 Memory effects 88
9 Quantum Mechanics 93
9.1 Canonical quantization 95
9.2 Quantization of the classical Hamilton function and free solutions 95
9.3 Temperature dependence of a fission yield and determination of the corresponding fission potential 99
9.4 The fractional Schrödinger equation with an infinite well potential 102
9.5 Radial solutions of the fractional Schrödinger equation 107
10 Fractional Spin: a Property of Particles Described with the Fractional Schrödinger Equation 111
10.1 Spin 112
10.2 Fractional spin 113
11 Factorization 117
11.1 The Dirac equation 117
11.2 The fractional Dirac equation 118
11.3 The fractional Pauli equation 120
12 Symmetries 123
12.1 Characteristics of fractional group theory 124
12.2 The fractional rotation group SOαN 126
13 The Fractional Symmetric Rigid Rotor 133
13.1 The spectrum of the fractional symmetric rigid rotor 133
13.2 Rotational limit 136
13.3 Vibrational limit 137
13.4 Davidson potential: the so called γ-unstable limit 138
13.5 Linear potential limit 140
13.6 The magic limit 141
13.7 Comparison with experimental data 144
14 q-deformed Lie Algebras and Fractional Calculus 153
14.1 q-deformed Lie algebras 153
14.2 The fractional q-deformed harmonic oscillator 156
14.3 The fractional q-deformed symmetric rotor 160
14.4 Half-integer representations of the fractional rotation group SOα(3) 162
15 Fractional Spectroscopy of Hadrons 165
15.1 Phenomenology of the baryon spectrum 166
15.2 Charmonium 171
15.3 Phenomenology of meson spectra 176
15.4 Metaphysics: About the internal structure of quarks 184
16 Higher Dimensional Fractional Rotation Groups 187
16.1 The four decompositions of the mixed fractional SOα (9) 187
16.2 Notation 189
16.3 The nine dimensional fractional Caputo-Riemann-Riemann symmetric rotor 192
16.4 Magic numbers of nuclei 193
16.5 Ground state properties of nuclei 196
16.6 Fine structure of the single particle spectrum: the extended Caputo-Riemann-Riemann symmetric rotor 201
16.7 Magic numbers of electronic clusters: the nine dimensional fractional Caputo-Caputo-Riemann symmetric rotor 206
16.8 Binding energy of electronic clusters 210
16.9 Metaphysics: magic numbers for clusters bound by weak and gravitational forces respectively 213
17 Fractors: Fractional Tensor Calculus 219
17.1 Covariance for fractional tensors 219
17.2 Singular fractional tensors 220
18 Fractional Fields 223
18.1 Fractional Euler-Lagrange equations 224
18.2 The fractional Maxwell equations 227
19 Gauge Invariance in Fractional Field Theories 231
19.1 Gauge invariance in first order of the coupling constant g 232
19.2 The fractional Riemann-Liouville-Zeeman effect 236
20 Outlook 241
Bibliography 243
Index 257