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
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