物理及工程中的分数维微积分 第2卷 应用2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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

注意：本站所有压缩包均有解压码： 点击下载压缩包解压工具

7 Mechanics1

7.1 Tautochrone problem1

7.1.1 Non-relativistic case1

7.1.2 Relativistic case2

7.2 Inverse problems4

7.2.1 Finding potential from a period-energy dependence4

7.2.2 Finding potential from scattering data5

7.2.3 Stellar systems6

7.3 Motion through a viscous fluid7

7.3.1 Entrainment of fluid by a moving wall7

7.3.2 Newton's equation with fractional term12

7.3.3 Solution by the Laplace transform method13

7.3.4 Solution by the Green functions method14

7.3.5 Fractionalized fall process15

7.4 Fractional oscillations18

7.4.1 Fractionalized harmonic oscillator18

7.4.2 Linear chain of fractional oscillators24

7.4.3 Fractionalized waves25

7.4.4 Fractionalized Frenkel-Kontorova model27

7.4.5 Oscillations of bodies in a viscous fluid30

7.5 Dynamical control problems32

7.5.1 PID controller and its fractional generalization32

7.5.2 Fractional transfer functions35

7.5.3 Fractional optimal control problem36

7.6 Analytical fractional dynamics38

7.6.1 Euler-Lagrange equation38

7.6.2 Discrete system Hamiltonian40

7.6.3 Potentials of non-concervative forces41

7.6.4 Hamilton-Jacobi mechanics42

7.6.5 Hamiltonian formalism for field theory43

References44

8 Continuum Mechanics49

8.1 Classical hydrodynamics49

8.1.1 A simple hydraulic problem49

8.1.2 Liquid drop oscillations50

8.1.3 Sound radiation52

8.1.4 Deep water waves52

8.2 Turbulent motion54

8.2.1 Kolmogorov's model of turbulence54

8.2.2 From Kolmogorov's hypothesis to the space-fractional equation55

8.2.3 From Boltzmann's equation to the time-fractional telegraph one58

8.2.4 Turbulent diffusion in a viscous fluid60

8.2.5 Navier-Stokes equation62

8.2.6 Reynolds'equation64

8.2.7 Diffusion in lane flows66

8.2.8 Subdiffusion in a random compressible flow69

8.3 Fractional models of viscoelasticity70

8.3.1 Two first models of fractional viscoelasticity70

8.3.2 Fractionalized Maxwell model73

8.3.3 Fractionalized Kelvin-Voigt model74

8.3.4 Standard model and its generalization75

8.3.5 Bagley-Torvik model76

8.3.6 Hysteresis loop78

8.3.7 Rabotnov's model79

8.3.8 Compound mechanical models81

8.3.9 The Rouse model of polymers83

8.3.10 Hamiltonian dynamic approach85

8.4 Viscoelastic fluids motion87

8.4.1 Gerasimov's results88

8.4.2 El-Shahed-Salem solutions93

8.4.3 Fractional Maxwell fluid:plain flow96

8.4.4 Fractional Maxwell fluid:longitudinal flow in a cylinder98

8.4.5 Magnetohydrodynamic flow99

8.4.6 Burgers'equation101

8.5 Solid bodies104

8.5.1 Viscoelastic rods104

8.5.2 Local fractional approach106

8.5.3 Nonlocal approach107

References108

9 Porous Media115

9.1 Diffusion115

9.1.1 Main concepts of anomalous diffusion115

9.1.2 Granular porosity117

9.1.3 Fiber porosity121

9.1.4 Filtration123

9.1.5 MHD flow in porous media125

9.1.6 Advection-diffusion model126

9.1.7 Reaction-diffusion equations128

9.2 Fractional acoustics130

9.2.1 Lokshin-Suvorova equation130

9.2.2 Schneider-Wyss equation132

9.2.3 Matignon et al.equation133

9.2.4 Viscoelastic loss operators136

9.3 Geophysical applications138

9.3.1 Water transport in unsaturated soils138

9.3.2 Seepage flow139

9.3.3 Foam Drainage Equation139

9.3.4 Seismic waves141

9.3.5 Multi-degree-of-freedom system of devices144

9.3.6 Spatial-temporal distribution of aftershocks146

References147

10 Thermodynamics153

10.1 Classical heat transfer theory153

10.1.1 Heat flux through boundaries153

10.1.2 Flux through a spherical surface156

10.1.3 Splitting inhomogeneous equations157

10.1.4 Heat transfer in porous media158

10.1.5 Hyperbolic heat conduction equation160

10.1.6 Inverse problems161

10.2 Fractional heat transfer models163

10.2.1 Fractional heat conduction laws163

10.2.2 Fractional equations for heat transport165

10.2.3 Application to thermoelasticity166

10.2.4 Some irreversible processes169

10.3 Phase transitions175

10.3.1 Ornstein-Zernicke equation175

10.3.2 Fractional Ginzburg-Landau equation178

10.3.3 Classification of phase transitions180

10.4 Around equilibrium182

10.4.1 Relaxation to the thermal equilibrium182

10.4.2 Fractionalization of the entropy183

References186

11 Electrodynamics191

11.1 Electromagnetic field191

11.1.1 Maxwell equations191

11.1.2 Fractional multipoles197

11.1.3 A link between two electrostatic images199

11.1.4 "Intermediate"waves200

11.2 Optics201

11.2.1 Fractional differentiation method201

11.2.2 Wave-diffusion model of image transfer202

11.2.3 Superdiffusion transfer205

11.2.4 Subdiffusion and combined(bifractional)diffusion transfer models207

11.3 Laser optics207

11.3.1 Laser beam equation207

11.3.2 Propagation of laser beam through fractal medium208

11.3.3 Free electron lasers209

11.4 Dielectrics211

11.4.1 Phenomenology of relaxation211

11.4.2 Cole-Cole process:macroscopic view213

11.4.3 Microscopic view214

11.4.4 Memory phenomenon216

11.4.5 Cole-Davidson process220

11.4.6 Havriliak-Negami process222

11.5 Semiconductors226

11.5.1 Diffusion in semiconductors226

11.5.2 Dispersive transport:transient current curves227

11.5.3 Stability as a consequence of self-similarity228

11.5.4 Fractional equations as a consequence of stability230

11.6 Conductors231

11.6.1 Skin-effect in a good conductor231

11.6.2 Electrochemistry233

11.6.3 Rough surface impedance233

11.6.4 Electrical line235

11.6.5 Josephson effect237

References238

12 Quantum Mechanics245

12.1 Atom optics245

12.1.1 Atoms in an optical lattice245

12.1.2 Laser cooling of atoms247

12.1.3 Atomic force microscopy248

12.2 Quantum particles250

12.2.1 Kinetic-fractional Sch？dinger equation250

12.2.2 Potential-fractional Schr？dinger equation254

12.2.3 Time-fractional Schr？dinger equation256

12.2.4 Fractional Heisenberg equation259

12.2.5 The fine structure constant260

12.3 Fractons262

12.3.1 Localized vibrational states(fractons)262

12.3.2 Weak fracton excitations264

12.3.3 Non-linear fractional Shr？dinger equation265

12.3.4 Fractional Ginzburg-Landau equation265

12.4 Quantum dots266

12.4.1 Fluorescence of nanocrystals266

12.4.2 Binary model267

12.4.3 Fractional transport equations269

12.4.4 Quantum wires271

12.5 Quantum decay theory272

12.5.1 Krylov-Fock theorem272

12.5.2 Weron-Weron theorem274

12.5.3 Nakhushev fractional equation275

References276

13 Plasma Dynamics281

13.1 Resonance radiation transport281

13.1.1 A role of the dispersion profile281

13.1.2 Fractional Biberman-Holstein equation284

13.1.3 Fractional Boltzmann equation286

13.2 Turbulent dynamics of plasma293

13.2.1 Diffusion in plasma turbulence293

13.2.2 Stationary states and fractional dynamics295

13.2.3 Perturbative transport297

13.2.4 Electron-acoustic waves299

13.3 Wandering of magnetic field lines300

13.3.1 Normal diffusion model300

13.3.2 Shalchi-Kourakis equations302

13.3.3 Theoretical evidence of superdiffusion wandering303

13.3.4 Fractional Brownian motion for simulating magnetic lines304

13.3.5 Compound model305

References307

14 Cosmic Rays311

14.1 Unbounded anomalous diffusion311

14.1.1 Space-fractional equation for cosmic rays diffusion311

14.1.2 The"knee"-problem312

14.1.3 Trapping CR by stochastic magnetic field316

14.1.4 Bifractional anomalous CR diffusion320

14.2 Bounded anomalous diffusion323

14.2.1 Fractal structures and finite speed323

14.2.2 Equations of the bounded anomalous diffusion model324

14.2.3 The bounded anomalous diffusion propagator327

14.3 Acceleration of cosmic rays329

14.3.1 CR reacceleration329

14.3.2 Fractional kinetic equations331

14.3.3 Fractional Fokker-Planck equations333

14.3.4 Integro-fractionally-differential model336

References338

15 Closing Chapter343

15.1 The problem of interpretation343

15.2 Geometrical interpretation345

15.2.1 Shadows on a fence345

15.2.2 Tangent vector and gradient347

15.2.3 Fractals and fractional derivatives348

15.3 Fractal and other derivatives355

15.3.1 Fractal derivative355

15.3.2 New fractal derivative356

15.3.3 Generalized fractional Laplaian356

15.3.4 Fractional derivatives in q-calculus357

15.3.5 Fuzzy fractional operators358

15.4 Probabilistic interpretation358

15.4.1 Probabilistic view on the G-L derivative358

15.4.2 Stochastic interpretation of R-L integral359

15.4.3 Fractional powers of operators359

15.5 Classical mechanic interpretation361

15.5.1 A car with a fractional speedometer361

15.5.2 Dynamical systems362

15.5.3 Coarse-grained-time dynamics364

15.5.4 Gradient systems364

15.5.5 Chaos kinetics366

15.5.6 Continuum mechanics367

15.5.7 Viscoelasticity369

15.5.8 Turbulence370

15.5.9 Plasma371

15.6 Quantum mechanic interpretations373

15.6.1 Feynman path integrals373

15.6.2 Lippmann-Schwinger equation374

15.6.3 Time-fractional evolution operator374

15.6.4 A role of environment375

15.6.5 Standard learning tasks377

15.6.6 Fractional Laplacian in a bounded domain378

15.6.7 Application to nuclear physics problems381

15.7 Concluding remarks382

15.7.1 Hidden variables382

15.7.2 Complexity384

15.7.3 Finishing the book385

References386

Appendix A Some Special Functions393

A. 1 Gamma function and binomial coefficients393

A.1.1 Gamma function393

A.1.2 Three integrals394

A.1.3 Binomial coefficients395

A.2 Mittag-Leffler functions395

A.2.1 Mittag-Leffler functions Eα(z),Eα,β(z)395

A.2.2 The Miller-Ross functions398

A.2.3 Functions Cx(ν,α)and Sx(ν,α)400

A.2.4 The Wright function402

A.2.5 The Mainardi functions403

A.3 The Fox functions404

A.3.1 Definition404

A.3.2 Some properties405

A.3.3 Some special cases408

A.4 Fractional stable distributions409

A.4.1 Introduction409

A.4.2 Characteristic function410

A.4.3 Inverse power series representation411

A.4.4 Integral representation411

A.4.5 Fox function representation414

A.4.6 Multivariate fractional stable densities417

References426

Appendix B Fractional Stable Densities429

Appendix C Fractional Operators:Symbols and Formulas435

Index445