Functional Fractional Calculus
When a new extraordinary and outstanding theory is stated, it has to face criticism and skeptism, because it is beyond the usual concept. The fractional calculus though not new, was not discussed or developed for a long time, particularly for lack of its
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Shantanu Das
Functional Fractional Calculus
ABC
Author Shantanu Das Scientist Reactor Control Division BARC Mumbai 400085 India E-mail: [email protected]
ISBN 978-3-642-20544-6
e-ISBN 978-3-642-20545-3
DOI 10.1007/978-3-642-20545-3 Library of Congress Control Number: 2011926534 c 2011 Springer-Verlag Berlin Heidelberg 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. Typeset & Cover Design: Scientific Publishing Services Pvt. Ltd., Chennai, India. Printed on acid-free paper 987654321 springer.com
This work is dedicated to my blind father (Late) Sri Soumendra Kumar Das, my mother Purabi, my wife Nita, my son Sankalan, my sister Shantasree, brother-in-law Hemant, and to my little niece Ishita
Preface
This book is aimed at to entice multidisciplinary pure and applied science researchers to think differently and to discover and describe nature and natural phenomena (sociological, economical, biological, physical or chemical processes) with wonderful tool of mathematics called Fractional Calculus, which is in fact a generalization of the classical Newtonian Calculus, what we all are used to. The greatest discovery by Newton was to tell the world is ‘that nature follows mathematics’; well the question is “which mathematics”? This question of finding “which mathematics” nature should follow is never ending quest for the pure and applied science researchers and this book will entice them. This work is aimed at, to make this subject popular and acceptable to engineering and science community to appreciate the universe of wonderful mathematics, which is in between classical integer order differentiation and integration, which till now is not much acknowledged, and is hidden from scientists and engineers. This book will generate ‘Physical and Engineering Essence of Fractional Calculus’. This work is inspired by thought to have overall fuel-efficient nuclear plant control system. I picked up the topic more than a decade ago while deriving reactor control laws, which aimed at fuel efficiency. Controlling the nuclear reactor close to its natural behavior by concept of exponent shape governor, ratio control and use of logarithmic logic, aims at the fuel efficiency. The power maneuvering trajectory is obtained by normalized shaped
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