The Analysis of Fractional Differential Equations An Application-Ori
Fractional calculus was first developed by pure mathematicians in the middle of the 19th century. Some 100 years later, engineers and physicists have found applications for these concepts in their areas. However there has traditionally been little interac
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2004
Kai Diethelm
The Analysis of Fractional Differential Equations An Application-Oriented Exposition Using Differential Operators of Caputo Type
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Kai Diethelm GNS Gesellschaft f¨ur Numerische Simulation mbH Am Gaußberg 2 38114 Braunschweig Germany [email protected]
ISBN: 978-3-642-14573-5 e-ISBN: 978-3-642-14574-2 DOI: 10.1007/978-3-642-14574-2 Springer Heidelberg Dordrecht London New York Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2010933969 Mathematics Subject Classification (2010): 34A08, 34A12, 34-02, 34-01, 26A33, 33E12 c Springer-Verlag Berlin Heidelberg 2010 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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. Cover design: SPi Publisher Services Printed on acid-free paper springer.com
Preface
There is a universe of mathematics lying in between the complete differentiations and integrations. — O. Heaviside
This book is devoted to some questions in Fractional Calculus, that is, the theory of differential and integral operators of non-integer order, and in particular to differential equations containing such operators. Even though the first steps of the theory itself date back to the first half of the nineteenth century, the subject only really came to life over the last few decades. A particular feature is that engineers and scientists have developed new models that involve fractional differential equations. These models have been applied successfully, e.g., in mechanics (theory of viscoelasticity and viscoplasticity), (bio-)chemistry (modelling of polymers and proteins), electrical engineering (transmission of ultrasound waves), medicine (modelling of human tissue under mechanical loads), etc. The mathematical theory seems to be lagging behind the needs of those applications but the wealth of applications indeed indicates the truth of the above quote from Heaviside [93, §437]. There are some books dealing with the aspects that can be summarized as the “pure mathematical” side of the problems without taking into consideration those questions that arise in the applications mentioned above, and some that the engineer’s point of view without a rigorous mathematical justification of the ideas. This book attempts to fill the
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