Advanced Methods in the Fractional Calculus of Variations
This brief presents a general unifying perspective on the fractional calculus. It brings together results of several recent approaches in generalizing the least action principle and the Euler–Lagrange equations to include fractional derivatives.The depend
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Agnieszka B. Malinowska Tatiana Odzijewicz Delfim F.M. Torres
Advanced Methods in the Fractional Calculus of Variations
SpringerBriefs in Applied Sciences and Technology
More information about this series at http://www.springer.com/series/8884
Agnieszka B. Malinowska Tatiana Odzijewicz Delfim F.M. Torres •
Advanced Methods in the Fractional Calculus of Variations
123
Agnieszka B. Malinowska Faculty of Computer Science Bialystok University of Technology Białystok Poland
Delfim F.M. Torres Department of Mathematics University of Aveiro Aveiro Portugal
Tatiana Odzijewicz Department of Mathematics and Mathematical Economics Warsaw School of Economics Warsaw Poland
ISSN 2191-530X ISSN 2191-5318 (electronic) SpringerBriefs in Applied Sciences and Technology ISBN 978-3-319-14755-0 ISBN 978-3-319-14756-7 (eBook) DOI 10.1007/978-3-319-14756-7 Library of Congress Control Number: 2014960038 Springer Cham Heidelberg New York Dordrecht London © The Author(s) 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com)
Preface
Fractional differentiation means “differentiation of arbitrary order.” Its origin goes back to more than 300 years, when in 1695 L’Hopital asked Leibniz the meaning of dn y 1 dxn for n ¼ 2. After that, many famous mathematicians, like J. Fourier, N.H. Abel, J. Liouville, B. Riemann, among others, contributed to the development of Fractional Calculus (Hilfer 2000; Podlubny 1999; Samko et al. 1993). It is possible to introduce fractional differentiation in several different ways, e.g., by following Riemann– Liouville, Grünwald–Letnikov, Caputo or Miller–Ross. During three centuries, the theory of fractional derivatives has developed as a pure theoretical field of mathematics. In the last few decades, however, it has been shown that fractional differentiation can be useful in various fields: physics (classic and quantum mechan
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