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物理及工程中的分数维微积分 第2卷 应用pdf电子书版本下载
- （俄罗斯）尤查金著 著
- 出版社： 北京：高等教育出版社
- ISBN：7040307344
- 出版时间：2013
- 标注页数：446页
- 文件大小：77MB
- 文件页数：457页
- 主题词：
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物理及工程中的分数维微积分 第2卷 应用PDF格式电子书版下载
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图书目录
7 Mechanics 1
7.1 Tautochrone problem 1
7.1.1 Non-relativistic case 1
7.1.2 Relativistic case 2
7.2 Inverse problems 4
7.2.1 Finding potential from a period-energy dependence 4
7.2.2 Finding potential from scattering data 5
7.2.3 Stellar systems 6
7.3 Motion through a viscous fluid 7
7.3.1 Entrainment of fluid by a moving wall 7
7.3.2 Newton's equation with fractional term 12
7.3.3 Solution by the Laplace transform method 13
7.3.4 Solution by the Green functions method 14
7.3.5 Fractionalized fall process 15
7.4 Fractional oscillations 18
7.4.1 Fractionalized harmonic oscillator 18
7.4.2 Linear chain of fractional oscillators 24
7.4.3 Fractionalized waves 25
7.4.4 Fractionalized Frenkel-Kontorova model 27
7.4.5 Oscillations of bodies in a viscous fluid 30
7.5 Dynamical control problems 32
7.5.1 PID controller and its fractional generalization 32
7.5.2 Fractional transfer functions 35
7.5.3 Fractional optimal control problem 36
7.6 Analytical fractional dynamics 38
7.6.1 Euler-Lagrange equation 38
7.6.2 Discrete system Hamiltonian 40
7.6.3 Potentials of non-concervative forces 41
7.6.4 Hamilton-Jacobi mechanics 42
7.6.5 Hamiltonian formalism for field theory 43
References 44
8 Continuum Mechanics 49
8.1 Classical hydrodynamics 49
8.1.1 A simple hydraulic problem 49
8.1.2 Liquid drop oscillations 50
8.1.3 Sound radiation 52
8.1.4 Deep water waves 52
8.2 Turbulent motion 54
8.2.1 Kolmogorov's model of turbulence 54
8.2.2 From Kolmogorov's hypothesis to the space-fractional equation 55
8.2.3 From Boltzmann's equation to the time-fractional telegraph one 58
8.2.4 Turbulent diffusion in a viscous fluid 60
8.2.5 Navier-Stokes equation 62
8.2.6 Reynolds'equation 64
8.2.7 Diffusion in lane flows 66
8.2.8 Subdiffusion in a random compressible flow 69
8.3 Fractional models of viscoelasticity 70
8.3.1 Two first models of fractional viscoelasticity 70
8.3.2 Fractionalized Maxwell model 73
8.3.3 Fractionalized Kelvin-Voigt model 74
8.3.4 Standard model and its generalization 75
8.3.5 Bagley-Torvik model 76
8.3.6 Hysteresis loop 78
8.3.7 Rabotnov's model 79
8.3.8 Compound mechanical models 81
8.3.9 The Rouse model of polymers 83
8.3.10 Hamiltonian dynamic approach 85
8.4 Viscoelastic fluids motion 87
8.4.1 Gerasimov's results 88
8.4.2 El-Shahed-Salem solutions 93
8.4.3 Fractional Maxwell fluid:plain flow 96
8.4.4 Fractional Maxwell fluid:longitudinal flow in a cylinder 98
8.4.5 Magnetohydrodynamic flow 99
8.4.6 Burgers'equation 101
8.5 Solid bodies 104
8.5.1 Viscoelastic rods 104
8.5.2 Local fractional approach 106
8.5.3 Nonlocal approach 107
References 108
9 Porous Media 115
9.1 Diffusion 115
9.1.1 Main concepts of anomalous diffusion 115
9.1.2 Granular porosity 117
9.1.3 Fiber porosity 121
9.1.4 Filtration 123
9.1.5 MHD flow in porous media 125
9.1.6 Advection-diffusion model 126
9.1.7 Reaction-diffusion equations 128
9.2 Fractional acoustics 130
9.2.1 Lokshin-Suvorova equation 130
9.2.2 Schneider-Wyss equation 132
9.2.3 Matignon et al.equation 133
9.2.4 Viscoelastic loss operators 136
9.3 Geophysical applications 138
9.3.1 Water transport in unsaturated soils 138
9.3.2 Seepage flow 139
9.3.3 Foam Drainage Equation 139
9.3.4 Seismic waves 141
9.3.5 Multi-degree-of-freedom system of devices 144
9.3.6 Spatial-temporal distribution of aftershocks 146
References 147
10 Thermodynamics 153
10.1 Classical heat transfer theory 153
10.1.1 Heat flux through boundaries 153
10.1.2 Flux through a spherical surface 156
10.1.3 Splitting inhomogeneous equations 157
10.1.4 Heat transfer in porous media 158
10.1.5 Hyperbolic heat conduction equation 160
10.1.6 Inverse problems 161
10.2 Fractional heat transfer models 163
10.2.1 Fractional heat conduction laws 163
10.2.2 Fractional equations for heat transport 165
10.2.3 Application to thermoelasticity 166
10.2.4 Some irreversible processes 169
10.3 Phase transitions 175
10.3.1 Ornstein-Zernicke equation 175
10.3.2 Fractional Ginzburg-Landau equation 178
10.3.3 Classification of phase transitions 180
10.4 Around equilibrium 182
10.4.1 Relaxation to the thermal equilibrium 182
10.4.2 Fractionalization of the entropy 183
References 186
11 Electrodynamics 191
11.1 Electromagnetic field 191
11.1.1 Maxwell equations 191
11.1.2 Fractional multipoles 197
11.1.3 A link between two electrostatic images 199
11.1.4 "Intermediate"waves 200
11.2 Optics 201
11.2.1 Fractional differentiation method 201
11.2.2 Wave-diffusion model of image transfer 202
11.2.3 Superdiffusion transfer 205
11.2.4 Subdiffusion and combined(bifractional)diffusion transfer models 207
11.3 Laser optics 207
11.3.1 Laser beam equation 207
11.3.2 Propagation of laser beam through fractal medium 208
11.3.3 Free electron lasers 209
11.4 Dielectrics 211
11.4.1 Phenomenology of relaxation 211
11.4.2 Cole-Cole process:macroscopic view 213
11.4.3 Microscopic view 214
11.4.4 Memory phenomenon 216
11.4.5 Cole-Davidson process 220
11.4.6 Havriliak-Negami process 222
11.5 Semiconductors 226
11.5.1 Diffusion in semiconductors 226
11.5.2 Dispersive transport:transient current curves 227
11.5.3 Stability as a consequence of self-similarity 228
11.5.4 Fractional equations as a consequence of stability 230
11.6 Conductors 231
11.6.1 Skin-effect in a good conductor 231
11.6.2 Electrochemistry 233
11.6.3 Rough surface impedance 233
11.6.4 Electrical line 235
11.6.5 Josephson effect 237
References 238
12 Quantum Mechanics 245
12.1 Atom optics 245
12.1.1 Atoms in an optical lattice 245
12.1.2 Laser cooling of atoms 247
12.1.3 Atomic force microscopy 248
12.2 Quantum particles 250
12.2.1 Kinetic-fractional Sch？dinger equation 250
12.2.2 Potential-fractional Schr？dinger equation 254
12.2.3 Time-fractional Schr？dinger equation 256
12.2.4 Fractional Heisenberg equation 259
12.2.5 The fine structure constant 260
12.3 Fractons 262
12.3.1 Localized vibrational states(fractons) 262
12.3.2 Weak fracton excitations 264
12.3.3 Non-linear fractional Shr？dinger equation 265
12.3.4 Fractional Ginzburg-Landau equation 265
12.4 Quantum dots 266
12.4.1 Fluorescence of nanocrystals 266
12.4.2 Binary model 267
12.4.3 Fractional transport equations 269
12.4.4 Quantum wires 271
12.5 Quantum decay theory 272
12.5.1 Krylov-Fock theorem 272
12.5.2 Weron-Weron theorem 274
12.5.3 Nakhushev fractional equation 275
References 276
13 Plasma Dynamics 281
13.1 Resonance radiation transport 281
13.1.1 A role of the dispersion profile 281
13.1.2 Fractional Biberman-Holstein equation 284
13.1.3 Fractional Boltzmann equation 286
13.2 Turbulent dynamics of plasma 293
13.2.1 Diffusion in plasma turbulence 293
13.2.2 Stationary states and fractional dynamics 295
13.2.3 Perturbative transport 297
13.2.4 Electron-acoustic waves 299
13.3 Wandering of magnetic field lines 300
13.3.1 Normal diffusion model 300
13.3.2 Shalchi-Kourakis equations 302
13.3.3 Theoretical evidence of superdiffusion wandering 303
13.3.4 Fractional Brownian motion for simulating magnetic lines 304
13.3.5 Compound model 305
References 307
14 Cosmic Rays 311
14.1 Unbounded anomalous diffusion 311
14.1.1 Space-fractional equation for cosmic rays diffusion 311
14.1.2 The"knee"-problem 312
14.1.3 Trapping CR by stochastic magnetic field 316
14.1.4 Bifractional anomalous CR diffusion 320
14.2 Bounded anomalous diffusion 323
14.2.1 Fractal structures and finite speed 323
14.2.2 Equations of the bounded anomalous diffusion model 324
14.2.3 The bounded anomalous diffusion propagator 327
14.3 Acceleration of cosmic rays 329
14.3.1 CR reacceleration 329
14.3.2 Fractional kinetic equations 331
14.3.3 Fractional Fokker-Planck equations 333
14.3.4 Integro-fractionally-differential model 336
References 338
15 Closing Chapter 343
15.1 The problem of interpretation 343
15.2 Geometrical interpretation 345
15.2.1 Shadows on a fence 345
15.2.2 Tangent vector and gradient 347
15.2.3 Fractals and fractional derivatives 348
15.3 Fractal and other derivatives 355
15.3.1 Fractal derivative 355
15.3.2 New fractal derivative 356
15.3.3 Generalized fractional Laplaian 356
15.3.4 Fractional derivatives in q-calculus 357
15.3.5 Fuzzy fractional operators 358
15.4 Probabilistic interpretation 358
15.4.1 Probabilistic view on the G-L derivative 358
15.4.2 Stochastic interpretation of R-L integral 359
15.4.3 Fractional powers of operators 359
15.5 Classical mechanic interpretation 361
15.5.1 A car with a fractional speedometer 361
15.5.2 Dynamical systems 362
15.5.3 Coarse-grained-time dynamics 364
15.5.4 Gradient systems 364
15.5.5 Chaos kinetics 366
15.5.6 Continuum mechanics 367
15.5.7 Viscoelasticity 369
15.5.8 Turbulence 370
15.5.9 Plasma 371
15.6 Quantum mechanic interpretations 373
15.6.1 Feynman path integrals 373
15.6.2 Lippmann-Schwinger equation 374
15.6.3 Time-fractional evolution operator 374
15.6.4 A role of environment 375
15.6.5 Standard learning tasks 377
15.6.6 Fractional Laplacian in a bounded domain 378
15.6.7 Application to nuclear physics problems 381
15.7 Concluding remarks 382
15.7.1 Hidden variables 382
15.7.2 Complexity 384
15.7.3 Finishing the book 385
References 386
Appendix A Some Special Functions 393
A. 1 Gamma function and binomial coefficients 393
A.1.1 Gamma function 393
A.1.2 Three integrals 394
A.1.3 Binomial coefficients 395
A.2 Mittag-Leffler functions 395
A.2.1 Mittag-Leffler functions Eα(z),Eα,β(z) 395
A.2.2 The Miller-Ross functions 398
A.2.3 Functions Cx(ν,α)and Sx(ν,α) 400
A.2.4 The Wright function 402
A.2.5 The Mainardi functions 403
A.3 The Fox functions 404
A.3.1 Definition 404
A.3.2 Some properties 405
A.3.3 Some special cases 408
A.4 Fractional stable distributions 409
A.4.1 Introduction 409
A.4.2 Characteristic function 410
A.4.3 Inverse power series representation 411
A.4.4 Integral representation 411
A.4.5 Fox function representation 414
A.4.6 Multivariate fractional stable densities 417
References 426
Appendix B Fractional Stable Densities 429
Appendix C Fractional Operators:Symbols and Formulas 435
Index 445