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

Kilbas, Anatoly A.; Srivastava, Hari M.; Trujillo, Juan J.

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.

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

Keywords:

textbook; fractional differential equations; Fourier transform; Laplace transform; Mellin transform; Mittag-Leffler function; H-function; Riemann-Liouville fractional integrals and derivatives; Caputo fractional derivative; Dirac equations; Green function; ordinary diffusion equation; Cauchy-type problems; boundary value problems; Volterra integral equations; linear differential equations; fractional diffusion-wave equations; fractional Brownian motion; continuous time random walk method; central limit theorem; Liouville fractional derivative; fractional reaction-diffusion models