Advances in Fractional Calculus Theoretical Developments and Applica
In the last two decades, fractional (or non integer) differentiation has played a very important role in various fields such as mechanics, electricity, chemistry, biology, economics, control theory and signal and image processing. For example, in the last
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Advances in Fractional Calculus Theoretical Developments and Applications in Physics and Engineering
edited by
J. Sabatier Université de Bordeaux I Talence, France
O. P. Agrawal Southern Illinois University Carbondale, IL, USA and
J. A. Tenreiro Machado Institute of Engineering of Porto Portugal
A C.I.P. Catalogue record for this book is available from the Library of Congress.
ISBN-13 978-1-4020-6041-0 (HB) ISBN-13 978-1-4020-6042-7 (e-book) Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com
Printed on acid-free paper
The views and opinions expressed in all the papers of this book are the authors’ personal one. The copyright of the individual papers belong to the authors. Copies cannot be reproduced for commercial profit. All Rights Reserved © 2007 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work.
We dedicate this book to the honorable memory of our colleague and friend Professor Peter W. Krempl
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Table of Contents Preface.......................................................................................................xi
1. Analytical and Numerical Techniques................ 1 Three Classes of FDEs Amenable to Approximation Using a Galerkin Technique ...................................................................................................3 S. J. Singh, A. Chatterjee Enumeration of the Real Zeros of the Mittag-Leffler Function ED(z), 1 < D < 2....................................................................................................15 J. W. Hanneken, D. M. Vaught, B. N. Narahari Achar The Caputo Fractional Derivative: Initialization Issues Relative to Fractional Differential Equations ..........................................................27 B. N. Narahari Achar, C. F. Lorenzo, T. T. Hartley Comparison of Five Numerical Schemes for Fractional Differential Equations ..................................................................................................43 O. P. Agrawal, P. Kumar Suboptimum H2 Pseudo-rational Approximations to Fractionalorder Linear Time Invariant Systems ........................................................ 61 D. Xue, Y. Chen Linear Differential Equations of Fractional Order.....................................77 B. Bonilla, M. Rivero, J. J. Trujillo Riesz Potentials as Centred Derivatives ....................................................93 M. D. Ortigueira
2. Classical Mechanics and Particle Physics........ 113 On Fractional Variational Principles ....................................................... 115 D. Baleanu, S. I. Muslih
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Fractional Kinetics in Pseudochaotic Systems and Its Applications ......
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