##
**Theory and applications of fractional differential equations.**
*(English)*
Zbl 1092.45003

North-Holland Mathematics Studies 204. Amsterdam: Elsevier (ISBN 0-444-51832-0/hbk). xv, 523 p. (2006).

This book presents a nice and systematic treatment of the theory and applications of fractional differential equations. The following are the main features of this book. It contains eight chapters.

Chapter 1 gives an account of some basic properties of certain topics of mathematical analysis, such as function spaces, gamma function, Mittag-Leffler function, Wright function, H-function, Laplace, Mellin and Fourier integral transforms and fixed point theorems etc. Chapter 2 deals with the definitions and useful properties of several different families of fractional integral operators and fractional derivative operators.

Chapter 3 provides the fundamental existence and uniqueness theorems for ordinary fractional differential equations including Cauchy-type problems. In chapter 4, explicit and numerical solutions are developed for the fractional differential equations and boundary value problems. The techniques employed in deriving the solutions involve reduction to Volterra integral equations, composition relations and operational methods.

Explicit solutions of linear differential equations associated with Liouville, Caputo and Riesz fractional derivatives with constant coefficients are investigated in chapter 5, by the application of the classical Laplace, Fourier and Mellin integral transforms. Chapter 6 is devoted to an interesting survey of the developments and results in the field of partial differential equations. Applications of Laplace and Fourier integral transforms are demonstrated by constructing the solutions in closed form of Cauchy type problems for fractional diffusion-wave equations. In the same way, solutions of Cauchy type problems for multidimensional fractional-diffusion wave equations are also obtained in closed forms.

Chapter 7 deals with linear differential equations of sequential and non-sequential fractional orders as well as systems of linear differential equations involving Riemann-Liouville and Caputo derivatives. Chapter 8 starts with a brief historical review of the applications of fractional calculus in complex systems. It is noteworthy that the main reason for the utility of fractional derivative operators is their strong relationship with fractional Brownian motion, the continuous time random walk method, the Levy stable distributions, and the generalized central limit theorem. Further the representation of the long-memory and non-local dependence of many anomalous processes are allowed by the fractional derivative operators. Finally, a new model involving a generalized Liouville fractional derivative operator, which stimulates both sub-diffusion as well as super-fast diffusion processes, is also investigated. There is an exhaustive bibliography containing nine hundred twenty eight references. A subject-index is also given at the end.

In conclusion, it may be mentioned that it is a fast growing area with a vast potential of its applications in fractional reaction-diffusion models arising in complex physical and biophysical systems. It may be used as a text or a reference book for research workers in various disciplines of physical, chemical and biological sciences.

Chapter 1 gives an account of some basic properties of certain topics of mathematical analysis, such as function spaces, gamma function, Mittag-Leffler function, Wright function, H-function, Laplace, Mellin and Fourier integral transforms and fixed point theorems etc. Chapter 2 deals with the definitions and useful properties of several different families of fractional integral operators and fractional derivative operators.

Chapter 3 provides the fundamental existence and uniqueness theorems for ordinary fractional differential equations including Cauchy-type problems. In chapter 4, explicit and numerical solutions are developed for the fractional differential equations and boundary value problems. The techniques employed in deriving the solutions involve reduction to Volterra integral equations, composition relations and operational methods.

Explicit solutions of linear differential equations associated with Liouville, Caputo and Riesz fractional derivatives with constant coefficients are investigated in chapter 5, by the application of the classical Laplace, Fourier and Mellin integral transforms. Chapter 6 is devoted to an interesting survey of the developments and results in the field of partial differential equations. Applications of Laplace and Fourier integral transforms are demonstrated by constructing the solutions in closed form of Cauchy type problems for fractional diffusion-wave equations. In the same way, solutions of Cauchy type problems for multidimensional fractional-diffusion wave equations are also obtained in closed forms.

Chapter 7 deals with linear differential equations of sequential and non-sequential fractional orders as well as systems of linear differential equations involving Riemann-Liouville and Caputo derivatives. Chapter 8 starts with a brief historical review of the applications of fractional calculus in complex systems. It is noteworthy that the main reason for the utility of fractional derivative operators is their strong relationship with fractional Brownian motion, the continuous time random walk method, the Levy stable distributions, and the generalized central limit theorem. Further the representation of the long-memory and non-local dependence of many anomalous processes are allowed by the fractional derivative operators. Finally, a new model involving a generalized Liouville fractional derivative operator, which stimulates both sub-diffusion as well as super-fast diffusion processes, is also investigated. There is an exhaustive bibliography containing nine hundred twenty eight references. A subject-index is also given at the end.

In conclusion, it may be mentioned that it is a fast growing area with a vast potential of its applications in fractional reaction-diffusion models arising in complex physical and biophysical systems. It may be used as a text or a reference book for research workers in various disciplines of physical, chemical and biological sciences.

Reviewer: Ram Kishore Saxena (Jodhpur)

### MSC:

34A08 | Fractional ordinary differential equations |

26A33 | Fractional derivatives and integrals |

34-02 | Research exposition (monographs, survey articles) pertaining to ordinary differential equations |

35A22 | Transform methods (e.g., integral transforms) applied to PDEs |

34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |

45J05 | Integro-ordinary differential equations |

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

65R20 | Numerical methods for integral equations |

45-02 | Research exposition (monographs, survey articles) pertaining to integral equations |

65L10 | Numerical solution of boundary value problems involving ordinary differential equations |

45D05 | Volterra integral equations |

45G10 | Other nonlinear integral equations |