Fractional calculus. Models and numerical methods.

*(English)*Zbl 1248.26011
Series on Complexity, Nonlinearity and Chaos 3. Hackensack, NJ: World Scientific (ISBN 978-981-4355-20-9/hbk; 978-981-4355-21-6/ebook). xxiv, 400 p. (2012).

The main focus of this book is the application of fractional calculus to ordinary and partial differential equations, the development of fractional models, and the exposition of numerical methods for these models.

The book is divided into seven chapters, an appendix, and an extensive bibliography. The first chapter is introductory in nature. It presents several definitions of fractional derivatives and integrals such as Riemann-Liouville, Caputo, Hadamard, Marchaud, Grünwald-Letnikov, and lists some of their properties and interrelations. The theorems in this chapter are stated without proofs; the reader is referred to the literature for further details.

In Chapter 2, the reader is exposed to a survey of numerical solution methods for fractional ordinary and partial differential equations. For instance, direct and indirect methods, and linear multistep methods are presented for fractional ODEs. The last section in this chapter deals with time-fractional diffusion-wave equations.

Chapter 3, entitled Efficient Numerical Methods, presents the reader with several methodologies that are applicable to a wide variety of problems, and which go deeper into issues such as non-locality, parallel computation, stability, and robustness. The two sections deal with ODEs and PDEs, respectively.

Generalizations of the classical theory of Stirling numbers of the first and second kind in the context of fractional calculus are presented in Chapter 4. The important role that Stirling numbers play in the computation of finite difference schemes and numerical approximation methods is exposed. This chapter displays the current state-of-the-art of Stirling numbers in fractional calculus.

In Chapter 5, the authors consider variational principles based on fractional calculus. The first half of this chapter deals with fractional Euler-Lagrange equations, whereas the second half concentrates on fractional Hamiltonian dynamics. Each of the two sections introduces and surveys current methodologies.

A short introduction to continuous time random walks and fractional diffusion models is given in Chapter 6. The highlights of this chapter are limit theorems for fractional diffusion.

The final Chapter 7 applies continuous time random walks in the context of fractional calculus to finance and economics. In particular, simulation of continuous time random walks, option pricing, and price fluctuations in financial markets are considered.

In the Appendix, the authors list the source codes, written in FORTRAN77, for several numerical methods: Adams-Bashforth-Moulton, Lubich’s fractional backwards differentiaion formulae, time-fractional diffusion equations, as well as the computation of Mittag-Leffler functions and a Monte Carlo simulation for continuous time random walks.

The book is clearly intended for scientists who would like to learn about the applicability of fractional calculus to a wide range of problems. The authors have succeeded in providing and elucidating several numerical methods based on fractional models and have thus expanded the tool box of applied scientists.

The book is divided into seven chapters, an appendix, and an extensive bibliography. The first chapter is introductory in nature. It presents several definitions of fractional derivatives and integrals such as Riemann-Liouville, Caputo, Hadamard, Marchaud, Grünwald-Letnikov, and lists some of their properties and interrelations. The theorems in this chapter are stated without proofs; the reader is referred to the literature for further details.

In Chapter 2, the reader is exposed to a survey of numerical solution methods for fractional ordinary and partial differential equations. For instance, direct and indirect methods, and linear multistep methods are presented for fractional ODEs. The last section in this chapter deals with time-fractional diffusion-wave equations.

Chapter 3, entitled Efficient Numerical Methods, presents the reader with several methodologies that are applicable to a wide variety of problems, and which go deeper into issues such as non-locality, parallel computation, stability, and robustness. The two sections deal with ODEs and PDEs, respectively.

Generalizations of the classical theory of Stirling numbers of the first and second kind in the context of fractional calculus are presented in Chapter 4. The important role that Stirling numbers play in the computation of finite difference schemes and numerical approximation methods is exposed. This chapter displays the current state-of-the-art of Stirling numbers in fractional calculus.

In Chapter 5, the authors consider variational principles based on fractional calculus. The first half of this chapter deals with fractional Euler-Lagrange equations, whereas the second half concentrates on fractional Hamiltonian dynamics. Each of the two sections introduces and surveys current methodologies.

A short introduction to continuous time random walks and fractional diffusion models is given in Chapter 6. The highlights of this chapter are limit theorems for fractional diffusion.

The final Chapter 7 applies continuous time random walks in the context of fractional calculus to finance and economics. In particular, simulation of continuous time random walks, option pricing, and price fluctuations in financial markets are considered.

In the Appendix, the authors list the source codes, written in FORTRAN77, for several numerical methods: Adams-Bashforth-Moulton, Lubich’s fractional backwards differentiaion formulae, time-fractional diffusion equations, as well as the computation of Mittag-Leffler functions and a Monte Carlo simulation for continuous time random walks.

The book is clearly intended for scientists who would like to learn about the applicability of fractional calculus to a wide range of problems. The authors have succeeded in providing and elucidating several numerical methods based on fractional models and have thus expanded the tool box of applied scientists.

Reviewer: Peter Massopust (München)

##### MSC:

26-02 | Research exposition (monographs, survey articles) pertaining to real functions |

26A33 | Fractional derivatives and integrals |

34A08 | Fractional ordinary differential equations and fractional differential inclusions |

35R11 | Fractional partial differential equations |

60G22 | Fractional processes, including fractional Brownian motion |