物理及工程中的分数维微积分: 第II卷应用 (英文版) (Fractional Derivatives for Physicists and Engine

《物理及工程中的分数维微积分: 第II卷应用(英文版)》一个运动质点位置函数的一阶导数表示速度，二阶导数表示加速度，那么分数阶导数的物理意义又是什么呢？分数阶导数是因何而产生，它对现代分析学在物理学的应用产生什么冲击，在将来又有什么发展？《物理及工程中的分数维微积分》二卷本将为你提供一个详细诠释。

《物理及工程中的分数维微积分: 第II卷应用(英文版)》由Vladimir V. Uchaikin著，本书的第Ⅰ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义清晰的分析与介绍。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，本书给读者展示一个全新的处理方式和新锐的视角。

《物理及工程中的分数维微积分: 第II卷应用(英文版)》适合于对概率和统计、数学建模和数值模拟方面感兴趣的学生、工程师、物理学家以及其他专家和学者，以及任何不想错过与这个越来越流行的数学方法接触的读者。

目录
- 7 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
- 8 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 El-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
  - References
- 9 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 rm : 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 Sch "o dinger equation
    - 12.2.2 Potential-fractional Schr "o dinger equation
    - 12.2.3 Time-fractional Schr "o dinger equation
    - 12.2.4 Fractional Heisenberg equation
    - 12.2.5 The fine structure constant
  - 12.3 Fractons
    - 12.3.1 Localized vibrational states (fractons)
    - 12.3.2 Weak fracton excitations
    - 12.3.3 Non-linear fractional Shr "o dinger equation
    - 12.3.4 Fractional Ginzburg-Landau equation
  - 12.4 Quantum dots
    - 12.4.1 Fluorescence of nanocrystals
    - 12.4.2 Binary model
    - 12.4.3 Fractional transport equations
    - 12.4.4 Quantum wires
  - 12.5 Quantum decay theory
    - 12.5.1 Krylov-Fock theorem
    - 12.5.2 Weron-Weron theorem
    - 12.5.3 Nakhushev fractional equation
  - References
- 13 Plasma Dynamics
  - 13.1 Resonance radiation transport
    - 13.1.1 A role of the dispersion profile
    - 13.1.2 Fractional Biberman-Holstein equation
    - 13.1.3 Fractional Boltzmann equation
  - 13.2 Turbulent dynamics of plasma
    - 13.2.1 Diffusion in plasma turbulence
    - 13.2.2 Stationary states and fractional dynamics
    - 13.2.3 Perturbative transport
    - 13.2.4 Electron-acoustic waves
  - 13.3 Wandering of magnetic field lines
    - 13.3.1 Normal diffusion model
    - 13.3.2 Shalchi-Kourakis equations
    - 13.3.3 Theoretical evidence of superdiffusion wandering
    - 13.3.4 Fractional Brownian motion for simulating magnetic lines
    - 13.3.5 Compound model
  - References
- 14 Cosmic Rays
  - 14.1 Unbounded anomalous diffusion
    - 14.1.1 Space-fractional equation for cosmic rays diffusion
    - 14.1.2 The ``knee''-problem
    - 14.1.3 Trapping CR by stochastic magnetic field
    - 14.1.4 Bifractional anomalous CR diffusion
  - 14.2 Bounded anomalous diffusion
    - 14.2.1 Fractal structures and finite speed
    - 14.2.2 Equations of the bounded anomalous diffusion model
    - 14.2.3 The bounded anomalous diffusion propagator
  - 14.3 Acceleration of cosmic rays
    - 14.3.1 CR reacceleration
    - 14.3.2 Fractional kinetic equations
    - 14.3.3 Fractional Fokker-Planck equations
    - 14.3.4 Integro-fractionally-differential model
  - References
- 15 Closing Chapter
  - 15.1 The problem of interpretation
  - 15.2 Geometrical interpretation
    - 15.2.1 Shadows on a fence
    - 15.2.2 Tangent vector and gradient
    - 15.2.3 Fractals and fractional derivatives
  - 15.3 Fractal and other derivatives
    - 15.3.1 Fractal derivative
    - 15.3.2 New fractal derivative
    - 15.3.3 Generalized fractional Laplaian
    - 15.3.4 Fractional derivatives in q-calculus
    - 15.3.5 Fuzzy fractional operators
  - 15.4 Probabilistic interpretation
    - 15.4.1 Probabilistic view on the G-L derivative
    - 15.4.2 Stochastic interpretation of R-L integral
    - 15.4.3 Fractional powers of operators
  - 15.5 Classical mechanic interpretation
    - 15.5.1 A car with a fractional speedometer
    - 15.5.2 Dynamical systems
    - 15.5.3 Coarse-grained-time dynamics
    - 15.5.4 Gradient systems
    - 15.5.5 Chaos kinetics
    - 15.5.6 Continuum mechanics
    - 15.5.7 Viscoelasticity
    - 15.5.8 Turbulence
    - 15.5.9 Plasma
  - 15.6 Quantum mechanic interpretations
    - 15.6.1 Feynman path integrals
    - 15.6.2 Lippmann-Schwinger equation
    - 15.6.3 Time-fractional evolution operator
    - 15.6.4 A role of environment
    - 15.6.5 Standard learning tasks
    - 15.6.6 Fractional Laplacian in a bounded domain
    - 15.6.7 Application to nuclear physics problems
  - 15.7 Concluding remarks
    - 15.7.1 Hidden variables
    - 15.7.2 Complexity
    - 15.7.3 Finishing the book
  - References
- Appendix A Some Special Functions
  - A.1 Gamma function and binomial coefficients
    - A.1.1 Gamma function
    - A.1.2 Three integrals
    - A.1.3 Binomial coefficients
  - A.2 Mittag-Leffler functions
    - A.2.1 Mittag-Leffler functions $E_ alpha (z), E_ alpha , beta (z)$
    - A.2.2 The Miller-Ross functions
    - A.2.3 Functions $C_x( nu ,a)$ and $S_x( nu ,a)$
    - A.2.4 The Wright function
    - A.2.5 The Mainardi functions
  - A.3 The Fox functions
    - A.3.1 Definition
    - A.3.2 Some properties
    - A.3.3 Some special cases
  - A.4 Fractional stable distributions
    - A.4.1 Introduction
    - A.4.2 Characteristic function
    - A.4.3 Inverse power series representation
    - A.4.4 Integral representation
    - A.4.5 Fox function representation
    - A.4.6 Multivariate fractional stable densities
  - References
- Apendix B Fractional Stable Densities
- Appendix C Fractional Operators: Symbols and Formulas
- Index
- 版权
- Front Matter

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