Fractional calculus and waves in linear viscoelasticity. An introduction to mathematical models.

*(English)*Zbl 1210.26004
Hackensack, NJ: World Scientific (ISBN 978-1-84816-329-4/hbk). xx, 347 p. (2010).

This very interesting book written by a well-known expert in Fractional Analysis and its applications, reveals and explains an interplay between mathematical tools used in Fractional Analysis and practical problems in Engineering. The main goal of the book is to investigate the connections among fractional calculus, linear viscoelasticity and wave motion. It is written mainly with the reflection of the research and approaches of the author and his students and includes his pioneering contributions on the applications of special functions of the Mittag-Leffler and Wright types; the book also contains a bibliographic list with about 1 000 entries, which makes the book a valuable bibliographical source for many applied scientists.

The author shows how the ideas and methods of fractional calculus help to describe dynamical properties of linear viscoelastic media, in particular problems of wave propagation and diffusion. The apparatus of some special functions is an indispensable tool for this aim. The book contains a large number of figures illustrating plots of some special functions important in FC, as well as mechanical representation of various concrete problems. The book also includes a number of Appendices, where many basic properties of various important special functions are gathered together with their plots, which makes the book convenient for the reader who can avoid a search for informations dispersed in other sources.

The main audience for this book is first of all applied scientists and engineers. Because of this, the book is purposefully written with an emphasis on problems and ways of their effective solution, rather than writing the text in the form of theorems accompanied by proofs.

The contents are as follows.

1. Preface Acknowledgements List of figures.

1. Essentials of fractional calculus; 2. Essentials of linear viscoelasticity; 3. Fractional viscoelastic models; 4. Waves in linear viscoelastic media: dispersion and dissipation; 5. Waves in linear viscoelastic media: asymptotic representation; 6. Diffusion and wave propagation via fractional calculus.

Appendix A. The Eulerian functions; Appendix B. The Bessel functions; Appendix C. The error functions; Appendix D. The exponential integral functions; Appendix E. The Mittag-Leffler functions; Appendix F. The Wright functions.

The book is written in an easy to read language and vivid manner, with an updated representation of the scientific results. The reviewer strongly recommends it for everybody interested in applications of fractional calculus.

The author shows how the ideas and methods of fractional calculus help to describe dynamical properties of linear viscoelastic media, in particular problems of wave propagation and diffusion. The apparatus of some special functions is an indispensable tool for this aim. The book contains a large number of figures illustrating plots of some special functions important in FC, as well as mechanical representation of various concrete problems. The book also includes a number of Appendices, where many basic properties of various important special functions are gathered together with their plots, which makes the book convenient for the reader who can avoid a search for informations dispersed in other sources.

The main audience for this book is first of all applied scientists and engineers. Because of this, the book is purposefully written with an emphasis on problems and ways of their effective solution, rather than writing the text in the form of theorems accompanied by proofs.

The contents are as follows.

1. Preface Acknowledgements List of figures.

1. Essentials of fractional calculus; 2. Essentials of linear viscoelasticity; 3. Fractional viscoelastic models; 4. Waves in linear viscoelastic media: dispersion and dissipation; 5. Waves in linear viscoelastic media: asymptotic representation; 6. Diffusion and wave propagation via fractional calculus.

Appendix A. The Eulerian functions; Appendix B. The Bessel functions; Appendix C. The error functions; Appendix D. The exponential integral functions; Appendix E. The Mittag-Leffler functions; Appendix F. The Wright functions.

The book is written in an easy to read language and vivid manner, with an updated representation of the scientific results. The reviewer strongly recommends it for everybody interested in applications of fractional calculus.

Reviewer: Stefan G. Samko (Faro)

##### MSC:

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

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

26A33 | Fractional derivatives and integrals |

35Q74 | PDEs in connection with mechanics of deformable solids |