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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

样章
作者：
Vladimir V. Uchaikin
定价：
98.00元
ISBN：
978-7-04-030734-4
版面字数：
550.000千字
开本：
16开
全书页数：
446页
装帧形式：
精装
重点项目：
暂无
出版时间：
2012-10-31
读者对象：
学术著作
一级分类：
自然科学
二级分类：
数学与统计
三级分类：
分析

《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。

  • Front Matter
  • 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

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