Computational methods in the fractional calculus of variations.

*(English)*Zbl 1322.49001
London: Imperial College Press (ISBN 978-1-78326-640-1/hbk). xii, 266 p. (2015).

The monograph is devoted to generalizations of the classic calculus of variations and optimal control to the case in which derivatives and integrals are understood as fractional ones of arbitrary order. Because of the lack of analytic methods to solve such fractional problems, numerical techniques are developed. The authors, who are well-known experts in the area of fractional calculus, mainly investigate the approximation of fractional operators by means of series of integer-order derivatives and generalized finite differences. Upper bounds for the error of the proposed approximations and their efficiency are discussed. Direct and indirect methods for solving fractional variational problems are studied in detail. Optimality conditions for different types of unconstrained and constrained fractional variational problems and for fractional optimal control problems are presented. The numerical methods are then used to solve some illustrative examples. MATLAB codes for the examples are included.

The theory developed in the book can be very useful in applications. An epidemic model of dengue disease, in which fractional derivatives were used successfully, can serve as an example.

This well-written monograph presents nontrivial generalizations of the calculus of variations and optimal control that opens doors to new and interesting modern scientific problems. Summing up, the book is suitable for graduate students in mathematics, physics and engineering, as well as for researchers interested in fractional calculus.

Editorial remark: this book is a companion volume to [A. B. Malinowska and D. F. M. Torres, Introduction to the fractional calculus of variations. London: Imperial College Press (2012; Zbl 1258.49001)].

The theory developed in the book can be very useful in applications. An epidemic model of dengue disease, in which fractional derivatives were used successfully, can serve as an example.

This well-written monograph presents nontrivial generalizations of the calculus of variations and optimal control that opens doors to new and interesting modern scientific problems. Summing up, the book is suitable for graduate students in mathematics, physics and engineering, as well as for researchers interested in fractional calculus.

Editorial remark: this book is a companion volume to [A. B. Malinowska and D. F. M. Torres, Introduction to the fractional calculus of variations. London: Imperial College Press (2012; Zbl 1258.49001)].

Reviewer: Wiesław Kotarski (Sosnowiec)

##### MSC:

49-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control |

49M25 | Discrete approximations in optimal control |

49J15 | Existence theories for optimal control problems involving ordinary differential equations |

49K15 | Optimality conditions for problems involving ordinary differential equations |

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

26A33 | Fractional derivatives and integrals |