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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
版面字数：
490千字
开本：
16开
装帧形式：
精装
版次：
1
最新版次
印刷时间：
2012年
ISBN：
978-7-04-032235-4
物料号：
32235-00
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。

  • 目录
    • 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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