物理及工程中的分数维微积分 第2卷 应用PDF电子书下载

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