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Introduction to the fractional calculus of variations. (English) Zbl 1258.49001
London: Imperial College Press (ISBN 978-1-84816-966-1/hbk). xvi, 275 p. (2012).
Calculus of Variations is an old field of mathematics generated by real world applications. Its development is also in direct connection with various problems occuring in biology, chemistry, control theory, dynamics, economics, engineering, physics, etc. The founders of Calculus of Variations were Leonhard Euler (1707-1783) and Joseph Louis Lagrange (1736-1813). The name of the field was given by Euler in his “Elementa Calculi Variationum.” Subseqently considerable attention has been paid to this field, including the current development based on modern analysis techniques.
This book deals with a special subject, the Fractional Calculus of Variations, that has been developed in the last two decades. It is a combination of the calculus of variations and fractional calculus. This combination is not artificial, it comes naturally from more accurate descriptions of physical phenomena. Fractional Calculus is also an old branch of mathematics that deals with derivatives and integrals of a non-integer order. Its origin goes back three centuries ago. There are several names that are associated with Fractional Calculus, such as: L’Hopital, Euler, Laplace, Fourier, Abel, Liouville, Riemann, Weyl, Caputo (in chronological order).
As mentioned by the authors of this book, “theory of the fractional calculus of variations started only in 1996 with the work of Riewe, in order to better describe non-conservative systems in mechanics (Riewe, 1996, 1997). Riewe formulated the problem of the calculus of variations with fractional derivatives, and obtained the respective Euler-Lagrange equation, combining both conservative and non-conservative cases. The inclusion of non-conservatism in the theory of calculus of variations is extremely important from the point of view of applications.”
To have an idea about the contents of this book, which is an introductory one, let us just mention the chapters of the book: The Classical Calculus of Variations; Fractional Calculus of Variations via Riemann-Liouville Operators; Fractional Calculus of Variations via Caputo Operators; Other Approaches to the Fractional Calculus of Variations; Towards a Combined Fractional Mechanics and Quantization. An extensive Bibligraphy and an Index are also included.
The book is well organized and displays the most important results in the field, so I predict that the book will have a positive impact on a large audience.

49-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control
49J15 Existence theories for optimal control problems involving ordinary differential equations
34A08 Fractional ordinary differential equations and fractional differential inclusions
26A33 Fractional derivatives and integrals