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**Fractional dynamics. Applications of fractional calculus to dynamics of particles, fields and media.**
*(English)*
Zbl 1214.81004

Nonlinear Physical Science. Berlin: Springer; Beijing: Higher Education Press (ISBN 978-3-642-14002-0/hbk; 978-7-04-029473-6/hbk). xv, 504 p. (2010).

The monograph consists of five parts. The first part is devoted to the fractional continuous models of fractal distributions of particls and fields. After definitions and properties of Riemann-Liouville and Riesz fractional integrals, introduction of Hausdorff measure and Hausdorff dimension, presentation of fractals as metric sets with non-integer Hausdorff dimension here the author considers some their physical (mass distribution of fractal systems, mass distribution on fractals and mass of fractal media, electric charge of fractal distribution, fractal distribution of particles, elementary models of fractal distributions) and mathematical (functions and integrals on fractals, integration over non-integer-dimensional space and multi-variable integration on fractals, fractional integral and measure (mass) on the real axis, probability on fractals) applications. Further follows hydrodynamics of fractal metric (Chapter 2) fractal rigid body dynamics (Chapter 3), electrodynamics of fractal distributions of charges and fields (Chapter 4), Ginzburg-Landau equation for fractal media (Chapter 5), Fokker-Planck equation for fractal distributions of probability (Chapter 6) and statistical mechanics of fractal phase space distributions (Chapter 7).

The subject of Part II is fractional dynamics of media with long-range interaction. In Chapter 8 chains and lattices with long-range interactions, and continuous limits of these discrete systems are considered. Complex Ginzburg-Landau equation describes a lot of phenomena, such as nonlinear waves, second-order phase transitions and superconductivity. It is used to describe the evolution of unstable modes amplitudes for any processes exhibiting PoincarĂ©-Andronov-Hopf bifurcation in large class of nonlinear wave phenomena and to describe synchronization and collective oscillations in complex media. Chapter 9 is devoted to fractional Ginzburg-Landau equation. In the Psi-series approach the existence of Laurent series for each different variables is considered. The existence of Laurent series is closely connected with the singularity analysis of differential equations. In Chapter 10 the psi-series approach is considered to fractional equations with the aim to describe nonlocal properties of the complex media.

In Part III (Fractional spatial dynamics) the author suggests the fractal vector calculus (Chapter 11), fractal exterior calculus and fractional differential forms (Chapter 12), fractional calculus of variations (Chapter 14) for the description of fractional dynamical systems (DS), fractional generalization of gradient and Hamiltonian Systems (Chapter 13) together with the investigation of fractal stability of their solutions. In particular fractional statistical mechanics and kinetics (Liouville, Bogolyubov and Vlasov equations, Fokker-Planck equations in Chapter 15), fractional electrodynamics (Chapter 12) taking into account their non-local properties on spatial variables are considered.

In Part IV the author describes the fractional temporal dynamics, where derivatives with respect to time variable have non-integer orders. The nonholonomic systems with generalized constraints to describe a long-ferm memory are considered in Chapter 17. The electrodynamics of dielectric media is described as a fractional temporal electrodynamics (Chapter 16). The discrete maps with memory are obtained from the fractional differential equations of kicked DS (Chapter 18).

Applications of fractional derivatives in quantum dynamics are considered in last Part V. These derivatives are defined as fractional powers of selfadjoint derivatives. Fractional generalization of quantum Markovian dynamics is suggested together with the quantization of different fractional derivatives and fractal functions. The text is self-contained. Every chapter is equipped by the list of numerous recent publications.

The subject of Part II is fractional dynamics of media with long-range interaction. In Chapter 8 chains and lattices with long-range interactions, and continuous limits of these discrete systems are considered. Complex Ginzburg-Landau equation describes a lot of phenomena, such as nonlinear waves, second-order phase transitions and superconductivity. It is used to describe the evolution of unstable modes amplitudes for any processes exhibiting PoincarĂ©-Andronov-Hopf bifurcation in large class of nonlinear wave phenomena and to describe synchronization and collective oscillations in complex media. Chapter 9 is devoted to fractional Ginzburg-Landau equation. In the Psi-series approach the existence of Laurent series for each different variables is considered. The existence of Laurent series is closely connected with the singularity analysis of differential equations. In Chapter 10 the psi-series approach is considered to fractional equations with the aim to describe nonlocal properties of the complex media.

In Part III (Fractional spatial dynamics) the author suggests the fractal vector calculus (Chapter 11), fractal exterior calculus and fractional differential forms (Chapter 12), fractional calculus of variations (Chapter 14) for the description of fractional dynamical systems (DS), fractional generalization of gradient and Hamiltonian Systems (Chapter 13) together with the investigation of fractal stability of their solutions. In particular fractional statistical mechanics and kinetics (Liouville, Bogolyubov and Vlasov equations, Fokker-Planck equations in Chapter 15), fractional electrodynamics (Chapter 12) taking into account their non-local properties on spatial variables are considered.

In Part IV the author describes the fractional temporal dynamics, where derivatives with respect to time variable have non-integer orders. The nonholonomic systems with generalized constraints to describe a long-ferm memory are considered in Chapter 17. The electrodynamics of dielectric media is described as a fractional temporal electrodynamics (Chapter 16). The discrete maps with memory are obtained from the fractional differential equations of kicked DS (Chapter 18).

Applications of fractional derivatives in quantum dynamics are considered in last Part V. These derivatives are defined as fractional powers of selfadjoint derivatives. Fractional generalization of quantum Markovian dynamics is suggested together with the quantization of different fractional derivatives and fractal functions. The text is self-contained. Every chapter is equipped by the list of numerous recent publications.

Reviewer: Boris V. Loginov (Ul’yanovsk)

### MSC:

81-02 | Research exposition (monographs, survey articles) pertaining to quantum theory |

00A79 | Physics |

26A33 | Fractional derivatives and integrals |

81V25 | Other elementary particle theory in quantum theory |

49S05 | Variational principles of physics |

81Q35 | Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices |

28A80 | Fractals |

34A08 | Fractional ordinary differential equations |

60G22 | Fractional processes, including fractional Brownian motion |

35R11 | Fractional partial differential equations |