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物理及工程中的分数维微积分: 第I卷 数学基础及其理论(英文版) (Fractional Derivatives for Physicists and Engi

样章
作者：
Vladimir V. Uchaikin
定价：
98.00元
ISBN：
978-7-04-032235-4
版面字数：
490.000千字
开本：
16开
全书页数：
385页
装帧形式：
精装
重点项目：
暂无
出版时间：
2012-11-06
读者对象：
学术著作
一级分类：
自然科学
二级分类：
交叉学科

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1卷数学基础及其理论)(精)》(作者尤查金)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，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1卷数学基础及其理论)(精)》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
  • Part I Background
  • 1 Heredity and Nonlocality
    • 1.1 Heredity
      • 1.1.1 Concept of heredity
      • 1.1.2 A short excursus in history
    • 1.2 Volterra’s heredity theory
      • 1.2.1 Volterra’s heredity laws
      • 1.2.2 Hereditary string
      • 1.2.3 Hereditary oscillator
      • 1.2.4 Energy principle
      • 1.2.5 Hereditary electrodynamics
    • 1.3 Hereditary kinetics
      • 1.3.1 Mechanical origin of heredity
      • 1.3.2 Hereditary Boltzmann equation
      • 1.3.3 Fokker-Planck equation
      • 1.3.4 Pauli and Van Hove equations
      • 1.3.5 Hybrid kinetic equations
    • 1.4 Hereditary hydrodynamics
      • 1.4.1 Physical motivation
      • 1.4.2 Polymeric liquids
      • 1.4.3 Turbulent diffusion
      • 1.4.4 Coarse-grained diffusion models
    • 1.5 Hereditary viscoelasticity
      • 1.5.1 Boltzmann’s viscoelasticity model
      • 1.5.2 Elastic solid: a mesoscopic approach
      • 1.5.3 One-dimensional harmonic lattice
      • 1.5.4 Axiomatic approach to continuum mechanics
    • 1.6 Hereditary thermodynamics
      • 1.6.1 Mechanical approach
      • 1.6.2 Hereditary heat-transfer
      • 1.6.3 Extended irreversible thermodynamics
      • 1.6.4 Axiomatic approach
      • 1.6.5 Ecology and climatology
    • 1.7 Nonlocal models
      • 1.7.1 Many-electron atoms
      • 1.7.2 Electron correlation in metals
      • 1.7.3 Plasma
      • 1.7.4 Vlasov’s nonlocal statistical mechanics
      • 1.7.5 Turbulence
      • 1.7.6 Aggregation equations
      • 1.7.7 Nonlocal models in nano-plasticity
      • 1.7.8 Nonlocal wave equations
    • References
  • 2 Selfsimilarity
    • 2.1 Power functions
      • 2.1.1 Standard power function
      • 2.1.2 Properties of power functions
      • 2.1.3 Memory
      • 2.1.4 Fractals
    • 2.2 Hydrodynamics
      • 2.2.1 Newtonian fluids
      • 2.2.2 Turbulence
      • 2.2.3 Microscopic fluctuations
      • 2.2.4 Non-Newtonian fluids
    • 2.3 Polymers
      • 2.3.1 The Nutting law
      • 2.3.2 Relaxation of polymer chains
      • 2.3.3 Interpenetrating polymer networks
    • 2.4 Reaction-diffusion
      • 2.4.1 Diffusion
      • 2.4.2 Polymerization
      • 2.4.3 Coagulation and fragmentation
    • 2.5 Solids
      • 2.5.1 Dielectrics
      • 2.5.2 Semiconductors
      • 2.5.3 Spinglasses
      • 2.5.4 Jonscher’s universal relaxation law
    • 2.6 Optics
      • 2.6.1 Luminescence decay
      • 2.6.2 Anomalous exciton kinetics
      • 2.6.3 Blinking fluorescence of quantum dots
    • 2.7 Geophysics
      • 2.7.1 Atmosphere and ocean turbulence
      • 2.7.2 Groundwater
      • 2.7.3 Earthquakes
      • 2.7.4 Tsunami
      • 2.7.5 Fractal approach
    • 2.8 Astrophysics and cosmology
      • 2.8.1 Solar wind
      • 2.8.2 Interstellar magnetic fields
      • 2.8.3 Scintillation statistics
      • 2.8.4 Velocity and density statistics from spectral lines
      • 2.8.5 Large-scale structure
      • 2.8.6 Stochastic selfsimilarity
    • 2.9 Some statistical mechanisms
      • 2.9.1 Three simple examples
      • 2.9.2 Activation mechanism
      • 2.9.3 Tunneling
      • 2.9.4 Multiple trapping
      • 2.9.5 Averaging over a parameter
      • 2.9.6 Fermi acceleration
    • References
  • 3 Stochasticity
    • 3.1 Brownian motion
      • 3.1.1 Two kinds of motion
      • 3.1.2 Dynamic selfsimilarity
      • 3.1.3 Stochastic selfsimilarity
      • 3.1.4 Selfsimilarity and stationarity
      • 3.1.5 Brownian motion
      • 3.1.6 Bm in a nonstationary nonhomogeneous environment
    • 3.2 One-dimensional L´evy motion
      • 3.2.1 Stable random variables
      • 3.2.2 Stable characteristic functions
      • 3.2.3 Stable probability densities
      • 3.2.4 Discrete time L´evy motion
      • 3.2.5 Generalized limit theorem
      • 3.2.6 Continuous time L´evy motion
    • 3.3 Multidimensional L´evy motion
      • 3.3.1 Multivariate symmetric stable vectors
      • 3.3.2 Sub-Gaussian random vectors
      • 3.3.3 Isotropic stable distributions as limit distributions
      • 3.3.4 Isotropic stable densities
      • 3.3.5 L´evy-Feldheim motion
    • 3.4 Fractional Brownian motion
      • 3.4.1 Differential Brownian motion process
      • 3.4.2 Integral Brownian motion process
      • 3.4.3 Fractional Brownian motion
      • 3.4.4 Fractional Gaussian noises
      • 3.4.5 Barnes-Allan model
      • 3.4.6 Fractional L´evy motion
    • 3.5 Fractional Poisson motion
      • 3.5.1 Renewal processes
      • 3.5.2 Selfsimilar renewal processes
      • 3.5.3 Three forms of fractal dust generator
      • 3.5.4 The nth arrival time distribution
      • 3.5.5 Limit fractional Poisson distributions
      • 3.5.6 An alternative models of fPp
      • 3.5.7 Compound Poisson process
    • 3.6 L´evy flights and L´evy walks
      • 3.6.1 L´evy Flights
      • 3.6.2 Asymptotic solution of the LF problem
      • 3.6.3 Continuous time random walk
      • 3.6.4 Some special cases
      • 3.6.5 Speed limit effect
      • 3.6.6 Moments of spatial distribution
      • 3.6.7 Exact solution for one-dimensional walk
    • 3.7 Diffusion on fractals
      • 3.7.1 Diffusion on the Sierpinski gasket
      • 3.7.2 Equation for diffusion on fractals
      • 3.7.3 Diffusion on comb-structures
      • 3.7.4 Some more on a one-dimensional fractal dust
      • 3.7.5 Flights on a single sample
      • 3.7.6 Averaging over the whole fractal ensemble
    • References
  • Part II Theory
  • 4 Fractional Differentiation
    • 4.1 Riemann-Liouville fractional derivatives
    • 4.2 Properties of R-L fractional derivatives
      • 4.2.1 Elementary properties
      • 4.2.2 The law of exponents
      • 4.2.3 Inverse operators
      • 4.2.4 Differentiation of a power function
      • 4.2.5 Term-by-term differentiation
      • 4.2.6 Differentiation of a product
      • 4.2.7 Differentiation of an integral
      • 4.2.8 Generalized Taylor series
      • 4.2.9 Expression of fractional derivatives through the integers
      • 4.2.10 Indirect differentiation: the chain rule
      • 4.2.11 Asymptotic behavior as x!a
      • 4.2.12 Asymptotic behavior of a f (n )(x) as x!¥
      • 4.2.13 The Marchaud derivative
    • 4.3 Compositions and superpositions of fractional operators
      • 4.3.1 Fractional operators
      • 4.3.2 The Gerasimov-Caputo derivative
      • 4.3.3 Hilfer’s interpolation R-L and G-C fractional
      • derivatives
      • 4.3.4 Weighted compositions of fractional operators
      • 4.3.5 Fractional derivatives of distributed orders
    • 4.4 Generalized functions approach
      • 4.4.1 Generalized functions
      • 4.4.2 Basic properties
      • 4.4.3 Regularization of power functions
      • 4.4.4 Marchaud derivative as a result of regularization
    • 4.5 Integral transformations
      • 4.5.1 The Laplace transformation
      • 4.5.2 The Mellin transform
      • 4.5.3 The Fourier transform
    • 4.6 Potentials and fractional derivatives
      • 4.6.1 The Riesz potentials on a straight line
      • 4.6.2 The Fourier transforms of the Riesz potentials
      • 4.6.3 The Riesz derivatives
      • 4.6.4 The Fourier transforms of the Riesz derivatives
      • 4.6.5 The Feller potential
    • 4.7 Fractional operators in multidimensional spaces
      • 4.7.1 The Riesz potentials and derivatives
      • 4.7.2 Directional derivatives and gradients
      • 4.7.3 Various fractionalizing grad, div, and curl operators
    • 4.8 Concluding remarks
      • 4.8.1 Leibniz’s definition
      • 4.8.2 Euler-Lacroix’s definition
      • 4.8.3 The Fourier definitions
      • 4.8.4 The Liouville definitions
      • 4.8.5 Riemann’s definition with complementary function
      • 4.8.6 From Sonin’s to Nishimoto’s fractional operators
      • 4.8.7 Local fractional derivatives
      • 4.8.8 The Jumarie nonstandard approach
    • References
  • 5 Equations and Solutions
    • 5.1 Ordinary equations
      • 5.1.1 Initialization
      • 5.1.2 Reduction to an integral equation
      • 5.1.3 Solution of inhomogeneous R-L fractional equation
      • 5.1.4 Solution of the inhomogeneous G-C fractional equation
      • 5.1.5 Indicial polynomial method
      • 5.1.6 Power series method
      • 5.1.7 Series expansion of inverse differential operators
      • 5.1.8 Method of integral transformations
      • 5.1.9 Green’s function method
      • 5.1.10 The Adomian decomposition method
      • 5.1.11 Equations with compositions of fractional operators
      • 5.1.12 Equations with superpositions of fractional operators
      • 5.1.13 Equations with varying coefficients
      • 5.1.14 Nonlinear ordinary equations
    • 5.2 Partial fractional equations
      • 5.2.1 Super-ballistic equation
      • 5.2.2 Subballistic equation
      • 5.2.3 Subdiffusion equation
      • 5.2.4 The normalization problem
      • 5.2.5 Subdiffusion on a half-axis
      • 5.2.6 The signalling problem
      • 5.2.7 The telegraph equation
      • 5.2.8 Multidimensional subdiffusion: the Schneider-Wyss solution
      • 5.2.9 One-dimensional symmetric superdiffusion
      • 5.2.10 Equations with L´evy-superposition of R-L operators
      • 5.2.11 Equations with the Feller, Riesz, and Marchaud operators
      • 5.2.12 L´evy-Feldheim motion equation
      • 5.2.13 Fractional Poisson motion
      • 5.2.14 L´evy-Poisson motion
      • 5.2.15 Fractional compound Poisson motion
      • 5.2.16 The link between solutions
      • 5.2.17 Subordinated L´evy motion
      • 5.2.18 Diffusion in a bounded domain
      • 5.2.19 Equation for diffusion on fractals
      • 5.2.20 Equation for flights on a fractal dust
      • 5.2.21 Equation for percolation
      • 5.2.22 Nonlinear equations
    • References
  • 6 Numerical Methods
    • 6.1 Gr¨unwald-Letnikov derivatives
      • 6.1.1 Fractional differences
      • 6.1.2 The G-L derivatives of integer orders
      • 6.1.3 The G-L derivatives of negative fractional orders
      • 6.1.4 The G-L derivatives on a semi-axis
    • 6.2 Finite-differences methods
      • 6.2.1 Numerical approximation of R-L and G-C derivatives
      • 6.2.2 Numerical approximation of G-L derivatives
      • 6.2.3 Estimation of accuracy
      • 6.2.4 Approximation of the Riesz-Feller derivatives
      • 6.2.5 Predictor-corrector method
      • 6.2.6 The linear scheme
      • 6.2.7 The quadratic and cubic schemes
      • 6.2.8 The collocation spline method
      • 6.2.9 The GMMP method
      • 6.2.10 The CL method
      • 6.2.11 The YA method
      • 6.2.12 Galerkin’s method
      • 6.2.13 Equation with the Riesz fractional derivatives
      • 6.2.14 Equation with Riesz-Feller derivatives
    • 6.3 Monte Carlo technique
      • 6.3.1 The inverse function method
      • 6.3.2 Density estimation
      • 6.3.3 Simulation of stable random variables
      • 6.3.4 Simulation of fractional exponential distribution
      • 6.3.5 Fractional R-L integral
      • 6.3.6 Simulation of a fractal dust in d-dimensional space
      • 6.3.7 Multidimensional Riesz potential
      • 6.3.8 Bifractional diffusion equation
    • 6.4 Variations, Homotopy and Differential Transforms
      • 6.4.1 Variational iteration method
      • 6.4.2 Homotopy analysis method
      • 6.4.3 Differential transform method
    • References
  • Index

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