物理及工程中的分数维微积分：应用（第2卷 英文版） 作者：（俄罗斯）尤查金 著 出版时间：2013年版 内容简介 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）（精）》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。本书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 Mechanics 7.1 Tautochrone problem 7.1.1 Non-relativistic case 7.1.2 Relativistic case 7.2 Inverse problems 7.2.1 Finding potential from a period-energy dependence 7.2.2 Finding potential from scattering data 7.2.3 Stellar systems 7.3 Motion through a viscous fluid 7.3.1 Entrainment of fluid by a moving wall 7.3.2 Newton's equation with fractional term 7.3.3 Solution by the Laplace transform method 7.3.4 Solution by the Green functions method 7.3.5 Fractionalized fall process 7.4 Fractional oscillations 7.4.1 Fractionalized harmonic oscillator 7.4.2 Linear chain of fractional oscillators 7.4.3 Fractionalized waves 7.4.4 Fractionalized Frenkel-Kontorova model 7.4.5 Oscillations of bodies in a viscous fluid 7.5 Dynamical control problems 7.5.1 PID controller and its fractional generalization 7.5.2 Fractional transfer functions 7.5.3 Fractional optimal control problem 7.6 Analytical fractional dynamics 7.6.1 Euler-Lagrange equation 7.6.2 Discrete system Hamiltonian 7.6.3 Potentials of non-concervative forces 7.6.4 Hamilton-Jacobi mechanics 7.6.5 Hamiltonian formalism for field theory References Continuum Mechanics 8.1 Classical hydrodynamics 8.1.1 A simple hydraulic problem 8.1.2 Liquid drop oscillations 8.1.3 Sound radiation 8.1.4 Deep water waves 8.2 Turbulent motion 8.2.1 Kolmogorov's model of turbulence 8.2.2 From Kolmogorov's hypothesis to the space-fractional equation 8.2.3 From Boltzmann's equation to the time-fractional telegraph one 8.2.4 Turbulent diffusion in a viscous fluid 8.2.5 Navier-Stokes equation 8.2.6 Reynolds' equation 8.2.7 Diffusion in lane flows 8.2.8 Subdiffusion in a random compressible flow 8.3 Fractional models of viscoelasticity 8.3.1 Two first models of fractional viscoelasticity 8.3.2 Fractionalized Maxwell model 8.3.3 Fractionalized Kelvin-Voigt model 8.3.4 Standard model and its generalization 8.3.5 Bagley-Torvik model 8.3.6 Hysteresis loop 8.3.7 Rabotnov's model 8.3.8 Compound mechanical models 8.3.9 The Rouse model of polymers 8.3.10 Hamiltonian dynamic approach 8.4 Viscoelastic fluids motion 8.4.1 Gerasimov's results 8.4.2 E1-Shahed-Salem solutions 8.4.3 Fractional Maxwell fluid： plain flow 8.4.4 Fractional Maxwell fluid： longitudinal flow in a cylinder 8.4.5 Magnetohydrodynamic flow 8.4.6 Burgers' equation 8.5 Solid bodies 8.5.1 Viscoelastic rods 8.5.2 Local fractional approach 8.5.3 Nonlocal approach Reference Porous Media 9.1 Diffusion 9.1.1 Main concepts of anomalous diffusion 9.1.2 Granular porosity 9.1.3 Fiber porosity 9.1.4 Filtration 9.1.5 MHD flow in porous media 9.1.6 Advection-diffusion model 9.1.7 Reaction-diffusion equations 9.2 Fractional acoustics 9.2.1 Lokshin-Suvorova equation 9.2.2 Schneider-Wyss equation 9.2.3 Matignon et al. equation 9.2.4 Viscoelastic loss operators 9.3 Geophysical applications 9.3.1 Water transport in unsaturated soils 9.3.2 Seepage flow 9.3.3 Foam Drainage Equation 9.3.4 Seismic waves 9.3.5 Multi-degree-of-freedom system of devices 9.3.6 Spatial-temporal distribution of aftershocks References 10 Thermodynamics 10.1 Classical heat transfer theory 10.1.1 Heat flux through boundaries 10.1.2 Flux through a spherical surface 10.1.3 Splitting inhomogeneous equations 10.1.4 Heat transfer in porous media 10.1.5 Hyperbolic heat conduction equation 10.1.6 Inverse problems 10.2 Fractional heat transfer models 10.2.1 Fractional heat conduction laws 10.2.2 Fractional equations for heat transport 10.2.3 Application to thermoelasticity 10.2.4 Some irreversible processes 10.3 Phase transitions 10.3.1 Ornstein-Zernicke equation 10.3.2 Fractional Ginzburg-Landau equation 10.3.3 Classification of phase transitions 10.4 Around equilibrium 10.4.1 Relaxation to the thermal equilibrium 10.4.2 Fractionalization of the entropy References 11 Electrodynamics 11.1 Electromagnetic field 11.1.1 Maxwell equations 11.1.2 Fractional multipoles 11.1.3 A link between two electrostatic images 11.1.4 "Intermediate" waves 11.2 Optics 11.2.1 Fractional differentiation method 11.2.2 Wave-diffusion model of image transfer 11.2.3 Superdiffusion transfer 11.2.4 Subdiffusion and combined （bifractional） diffusion transfer models 11.3 Laser optics 11.3.1 Laser beam equation 11.3.2 Propagation of laser beam through fractal medium 11.3.3 Free electron lasers 11.4 Dielectrics 11.4.1 Phenomenology of relaxation 11.4.2 Cole-Cole process： macroscopic view 11.4.3 Microscopic view 11.4.4 Memory phenomenon 11.4.5 Cole-Davidson process 11.4.6 Havriliak-Negami process 11.5 Semiconductors 11.5.1 Diffusion in semiconductors 11.5.2 Dispersive transport： transient current curves 11.5.3 Stability as a consequence of self-similarity 11.5.4 Fractional equations as a consequence of stability 11.6 Conductors 11.6.1 Skin-effect in a good conductor 11.6.2 Electrochemistry 11.6.3 Rough surface impedance 11.6.4 Electrical line 11.6.5 Josephson effect References 12 Quantum Mechanics 12.1 Atom optics 12.1.1 Atoms in an optical lattice 12.1.2 Laser cooling of atoms 12.1.3 Atomic force microscopy 12.2 Quantum particles 12.2.1 Kinetic-fractional Schodinger equation 12.2.2 Potential-fractional Schrodinger equation 12.2.3 Time-fractional Schrodinger equation …… 13 Plasma Dynamics 14 Cosmic Rays 15 Closing Chapter Appendix A Some Special Functions Appendix B Fractional Stable Densities Appendix C Fractional Operators： Symbols and Formulas Index
