物理及工程中的分数维微积分（第2卷）：应用（英文版）
Mechanics
7.1 Tautochrone problem
7.1.1 Non-relativistic case
7.1.2 Relativistic case
7.2 Inverse problems
7.2.1 Finding potential from a period-energy dependence
7.2.2 Finding potential from scattering data
7.2.3 Stellar systems
7.3 Motion through a viscous fluid
7.3.1 Entrainment of fluid by a moving wall
7.3.2 Newton's equation with fractional term
7.3.3 Solution by the Laplace transform method
7.3.4 Solution by the Green functions method
7.3.5 Fractionalized fall process
7.4 Fractional oscillations
7.4.1 Fractionalized harmonic oscillator
7.4.2 Linear chain of fractional oscillators
7.4.3 Fractionalized waves
7.4.4 Fractionalized Frenkel-Kontorova model
7.4.5 Oscillations of bodies in a viscous fluid
7.5 Dynamical control problems
7.5.1 PID controller and its fractional generalization
7.5.2 Fractional transfer functions
7.5.3 Fractional optimal control problem
7.6 Analytical fractional dynamics
7.6.1 Euler-Lagrange equation
7.6.2 Discrete system Hamiltonian
7.6.3 Potentials of non-concervative forces
7.6.4 Hamilton-Jacobi mechanics
7.6.5 Hamiltonian formalism for field theory
References
Continuum Mechanics
8.1 Classical hydrodynamics
8.1.1 A simple hydraulic problem
8.1.2 Liquid drop oscillations
8.1.3 Sound radiation
8.1.4 Deep water waves
8.2 Turbulent motion
8.2.1 Kolmogorov's model of turbulence
8.2.2 From Kolmogorov's hypothesis to the space-fractional equation
8.2.3 From Boltzmann's equation to the time-fractional telegraph one
8.2.4 Turbulent diffusion in a viscous fluid
8.2.5 Navier-Stokes equation
8.2.6 Reynolds' equation
8.2.7 Diffusion in lane flows
8.2.8 Subdiffusion in a random compressible flow
8.3 Fractional models of viscoelasticity
8.3.1 Two first models of fractional viscoelasticity
8.3.2 Fractionalized Maxwell model
8.3.3 Fractionalized Kelvin-Voigt model
8.3.4 Standard model and its generalization
8.3.5 Bagley-Torvik model
8.3.6 Hysteresis loop
8.3.7 Rabotnov's model
8.3.8 Compound mechanical models
8.3.9 The Rouse model of polymers
8.3.10 Hamiltonian dynamic approach
8.4 Viscoelastic fluids motion
8.4.1 Gerasimov's results
8.4.2 E1-Shahed-Salem solutions
8.4.3 Fractional Maxwell fluid： plain flow
8.4.4 Fractional Maxwell fluid： longitudinal flow in a cylinder
8.4.5 Magnetohydrodynamic flow
8.4.6 Burgers' equation
8.5 Solid bodies
8.5.1 Viscoelastic rods
8.5.2 Local fractional approach
8.5.3 Nonlocal approach
Reference
Porous Media
9.1 Diffusion
9.1.1 Main concepts of anomalous diffusion
9.1.2 Granular porosity
9.1.3 Fiber porosity
9.1.4 Filtration
9.1.5 MHD flow in porous media
9.1.6 Advection-diffusion model
9.1.7 Reaction-diffusion equations
9.2 Fractional acoustics
9.2.1 Lokshin-Suvorova equation
9.2.2 Schneider-Wyss equation
9.2.3 Matignon et al. equation
9.2.4 Viscoelastic loss operators
9.3 Geophysical applications
9.3.1 Water transport in unsaturated soils
9.3.2 Seepage flow
9.3.3 Foam Drainage Equation
9.3.4 Seismic waves
9.3.5 Multi-degree-of-freedom system of devices
9.3.6 Spatial-temporal distribution of aftershocks
References
10 Thermodynamics
10.1 Classical heat transfer theory
10.1.1 Heat flux through boundaries
10.1.2 Flux through a spherical surface
10.1.3 Splitting inhomogeneous equations
10.1.4 Heat transfer in porous media
10.1.5 Hyperbolic heat conduction equation
10.1.6 Inverse problems
10.2 Fractional heat transfer models
10.2.1 Fractional heat conduction laws
10.2.2 Fractional equations for heat transport
10.2.3 Application to thermoelasticity
10.2.4 Some irreversible processes
10.3 Phase transitions
10.3.1 Ornstein-Zernicke equation
10.3.2 Fractional Ginzburg-Landau equation
10.3.3 Classification of phase transitions
10.4 Around equilibrium
10.4.1 Relaxation to the thermal equilibrium
10.4.2 Fractionalization of the entropy
References
11 Electrodynamics
11.1 Electromagnetic field
11.1.1 Maxwell equations
11.1.2 Fractional multipoles
11.1.3 A link between two electrostatic images
11.1.4 "Intermediate" waves
11.2 Optics
11.2.1 Fractional differentiation method
11.2.2 Wave-diffusion model of image transfer
11.2.3 Superdiffusion transfer
11.2.4 Subdiffusion and combined （bifractional） diffusion
transfer models
11.3 Laser optics
11.3.1 Laser beam equation
11.3.2 Propagation of laser beam through fractal medium
11.3.3 Free electron lasers
11.4 Dielectrics
11.4.1 Phenomenology of relaxation
11.4.2 Cole-Cole process： macroscopic view
11.4.3 Microscopic view
11.4.4 Memory phenomenon
11.4.5 Cole-Davidson process
11.4.6 Havriliak-Negami process
11.5 Semiconductors
11.5.1 Diffusion in semiconductors
11.5.2 Dispersive transport： transient current curves
11.5.3 Stability as a consequence of self-similarity
11.5.4 Fractional equations as a consequence of stability
11.6 Conductors
11.6.1 Skin-effect in a good conductor
11.6.2 Electrochemistry
11.6.3 Rough surface impedance
11.6.4 Electrical line
11.6.5 Josephson effect
References
12 Quantum Mechanics
12.1 Atom optics
12.1.1 Atoms in an optical lattice
12.1.2 Laser cooling of atoms
12.1.3 Atomic force microscopy
12.2 Quantum particles
12.2.1 Kinetic-fractional Schodinger equation
12.2.2 Potential-fractional Schrodinger equation
12.2.3 Time-fractional Schrodinger equation
……
13 Plasma Dynamics
14 Cosmic Rays
15 Closing Chapter
Appendix A Some Special Functions
Appendix B Fractional Stable Densities
Appendix C Fractional Operators： Symbols and Formulas
Index
7.1 Tautochrone problem
7.1.1 Non-relativistic case
7.1.2 Relativistic case
7.2 Inverse problems
7.2.1 Finding potential from a period-energy dependence
7.2.2 Finding potential from scattering data
7.2.3 Stellar systems
7.3 Motion through a viscous fluid
7.3.1 Entrainment of fluid by a moving wall
7.3.2 Newton's equation with fractional term
7.3.3 Solution by the Laplace transform method
7.3.4 Solution by the Green functions method
7.3.5 Fractionalized fall process
7.4 Fractional oscillations
7.4.1 Fractionalized harmonic oscillator
7.4.2 Linear chain of fractional oscillators
7.4.3 Fractionalized waves
7.4.4 Fractionalized Frenkel-Kontorova model
7.4.5 Oscillations of bodies in a viscous fluid
7.5 Dynamical control problems
7.5.1 PID controller and its fractional generalization
7.5.2 Fractional transfer functions
7.5.3 Fractional optimal control problem
7.6 Analytical fractional dynamics
7.6.1 Euler-Lagrange equation
7.6.2 Discrete system Hamiltonian
7.6.3 Potentials of non-concervative forces
7.6.4 Hamilton-Jacobi mechanics
7.6.5 Hamiltonian formalism for field theory
References
Continuum Mechanics
8.1 Classical hydrodynamics
8.1.1 A simple hydraulic problem
8.1.2 Liquid drop oscillations
8.1.3 Sound radiation
8.1.4 Deep water waves
8.2 Turbulent motion
8.2.1 Kolmogorov's model of turbulence
8.2.2 From Kolmogorov's hypothesis to the space-fractional equation
8.2.3 From Boltzmann's equation to the time-fractional telegraph one
8.2.4 Turbulent diffusion in a viscous fluid
8.2.5 Navier-Stokes equation
8.2.6 Reynolds' equation
8.2.7 Diffusion in lane flows
8.2.8 Subdiffusion in a random compressible flow
8.3 Fractional models of viscoelasticity
8.3.1 Two first models of fractional viscoelasticity
8.3.2 Fractionalized Maxwell model
8.3.3 Fractionalized Kelvin-Voigt model
8.3.4 Standard model and its generalization
8.3.5 Bagley-Torvik model
8.3.6 Hysteresis loop
8.3.7 Rabotnov's model
8.3.8 Compound mechanical models
8.3.9 The Rouse model of polymers
8.3.10 Hamiltonian dynamic approach
8.4 Viscoelastic fluids motion
8.4.1 Gerasimov's results
8.4.2 E1-Shahed-Salem solutions
8.4.3 Fractional Maxwell fluid： plain flow
8.4.4 Fractional Maxwell fluid： longitudinal flow in a cylinder
8.4.5 Magnetohydrodynamic flow
8.4.6 Burgers' equation
8.5 Solid bodies
8.5.1 Viscoelastic rods
8.5.2 Local fractional approach
8.5.3 Nonlocal approach
Reference
Porous Media
9.1 Diffusion
9.1.1 Main concepts of anomalous diffusion
9.1.2 Granular porosity
9.1.3 Fiber porosity
9.1.4 Filtration
9.1.5 MHD flow in porous media
9.1.6 Advection-diffusion model
9.1.7 Reaction-diffusion equations
9.2 Fractional acoustics
9.2.1 Lokshin-Suvorova equation
9.2.2 Schneider-Wyss equation
9.2.3 Matignon et al. equation
9.2.4 Viscoelastic loss operators
9.3 Geophysical applications
9.3.1 Water transport in unsaturated soils
9.3.2 Seepage flow
9.3.3 Foam Drainage Equation
9.3.4 Seismic waves
9.3.5 Multi-degree-of-freedom system of devices
9.3.6 Spatial-temporal distribution of aftershocks
References
10 Thermodynamics
10.1 Classical heat transfer theory
10.1.1 Heat flux through boundaries
10.1.2 Flux through a spherical surface
10.1.3 Splitting inhomogeneous equations
10.1.4 Heat transfer in porous media
10.1.5 Hyperbolic heat conduction equation
10.1.6 Inverse problems
10.2 Fractional heat transfer models
10.2.1 Fractional heat conduction laws
10.2.2 Fractional equations for heat transport
10.2.3 Application to thermoelasticity
10.2.4 Some irreversible processes
10.3 Phase transitions
10.3.1 Ornstein-Zernicke equation
10.3.2 Fractional Ginzburg-Landau equation
10.3.3 Classification of phase transitions
10.4 Around equilibrium
10.4.1 Relaxation to the thermal equilibrium
10.4.2 Fractionalization of the entropy
References
11 Electrodynamics
11.1 Electromagnetic field
11.1.1 Maxwell equations
11.1.2 Fractional multipoles
11.1.3 A link between two electrostatic images
11.1.4 "Intermediate" waves
11.2 Optics
11.2.1 Fractional differentiation method
11.2.2 Wave-diffusion model of image transfer
11.2.3 Superdiffusion transfer
11.2.4 Subdiffusion and combined （bifractional） diffusion
transfer models
11.3 Laser optics
11.3.1 Laser beam equation
11.3.2 Propagation of laser beam through fractal medium
11.3.3 Free electron lasers
11.4 Dielectrics
11.4.1 Phenomenology of relaxation
11.4.2 Cole-Cole process： macroscopic view
11.4.3 Microscopic view
11.4.4 Memory phenomenon
11.4.5 Cole-Davidson process
11.4.6 Havriliak-Negami process
11.5 Semiconductors
11.5.1 Diffusion in semiconductors
11.5.2 Dispersive transport： transient current curves
11.5.3 Stability as a consequence of self-similarity
11.5.4 Fractional equations as a consequence of stability
11.6 Conductors
11.6.1 Skin-effect in a good conductor
11.6.2 Electrochemistry
11.6.3 Rough surface impedance
11.6.4 Electrical line
11.6.5 Josephson effect
References
12 Quantum Mechanics
12.1 Atom optics
12.1.1 Atoms in an optical lattice
12.1.2 Laser cooling of atoms
12.1.3 Atomic force microscopy
12.2 Quantum particles
12.2.1 Kinetic-fractional Schodinger equation
12.2.2 Potential-fractional Schrodinger equation
12.2.3 Time-fractional Schrodinger equation
……
13 Plasma Dynamics
14 Cosmic Rays
15 Closing Chapter
Appendix A Some Special Functions
Appendix B Fractional Stable Densities
Appendix C Fractional Operators： Symbols and Formulas
Index
《物理及工程中的分数维微积分（第2卷）：应用（英文版）》的第1卷介绍分数维微积分的数学基础和相应的理论，为这个现代分析学中的重要分支提供了详细而又清晰的分析与介绍。第Ⅱ卷是应用篇，讲述了分数维微积分在物理学中的实际的应用。在湍流与半导体、等离子与热力学、力学与量子光学、纳米物理学与天体物理学等学科应用方面，《物理及工程中的分数维微积分（第2卷）：应用（英文版）》给读者展示一个全新的处理方式和新锐的视角。
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