Lecture Notes in Mathematics 2004. Berlin: Springer (ISBN 978-3-642-14573-5/pbk; 978-3-642-14574-2/ebook). viii, 247 p. EUR 44.95/net; £ 40.99; SFR 70.00 (2010).

This monograph is intended for use by graduate students, mathematicians and applied scientists who have an interest in fractional differential equations. The Caputo derivative is the main focus of the book, because of its relevance to applications. The author also describes the classical Riemann-Liouville derivative and points out the ways in which the theoretical results are different in this case. The monograph may be regarded as a fairly self-contained reference work and a comprehensive overview of the current state of the art. It contains many results and insights brought together for the first time, including some new material that has not, to my knowledge, appeared elsewhere. The material selected for the book is fairly classical, in the sense that key concepts such as existence and uniqueness theory, stability of solutions under perturbations, and both initial and boundary value problems are discussed. The author states that much of the material has been used in a course for graduate students, for whom it would be ideally suited alongside the other target audiences.