Fractional calculus : an introduction for physicists

500 Pages·2014·11.479 MB·English

by  Richard Herrmann

Fractional Calculus
An Introduction For Physicists
2nd Edition
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Fractional Calculus
An Introduction For Physicists
2nd Edition
Richard Herrmann
GigaHedron, Germany
World Scientific
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI
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Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
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Library of Congress Cataloging-in-Publication Data
Herrmann, Richard, author.
Fractional calculus : an introduction for physicists / Richard Herrmann, GigaHedron,
Germany. -- 2nd edition.
pages cm
Includes bibliographical references and index.
ISBN 978-9814551076 (hardcover : alk. paper)
1. Fractional calculus. I. Title.
QC20.7.F75H4775 2014
515'.83--dc23
2013048544
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2014 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
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is not required from the publisher.
In-house Editor: Rhaimie Wahap
Printed in Singapore
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Intentio vero nostra est
manifestare in hoc libro
ea, que sunt, sicut sunt,
et ad artis certitudinem redigere,
quorum nullus scientiam habuit
hactenus neque artem.
Fridericus Romanorum Imperator Secundus (1194–1250)
De arte venandi cum avibus
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Preface to the Second Edition
WearepleasedthatthefirsteditionofFractionalCalculus-AnIntroduction
for Physicists and its reprint has found many friends among students and
scientists.
Withthenecessityforasecondeditionarising,weusedtheopportunity
to implement several improvements, expansions and explanatory remarks,
which enhance comprehensibility and are a reflection of helpful discussions
and hints from colleagues and readers all over the world.
This new, extended edition reflects the broadening range of develop-
ments and demonstrations of the fractional approach in different branches
of physics, in particular:
Imageprocessing-presentingafractionalapproachresultinginastable
algorithmfor 3D-shape recoveryfrom sequences of aperture afflicted slides
obtained in confocal microscopy or autofocus algorithms.
Folded potentials in cluster physics, where a new interpretation of the
Riesz fractional integral is given in terms of a smooth transition from
Coulomb to Yukawa-potentials.
Infrared spectroscopy, with a first practical application of the frac-
tional quantum harmonic oscillator to describe rot-vib spectra in diatomic
molecules from a generalized point of view.
Numerical solutions of the fractional Schro¨dinger equation show the
influence of nonlocal concepts in quantum mechanics.
Triaxiality as a ground state property of nuclei is investigated using an
analytical fractional model.
Localaspects for alternativefractionalderivative definitions andconse-
quences are discussed.
Fractional calculus with variable order is discussed and the impact of
genetic strategies based on evolutionary differential equations to describe
cosmologicalmodels is investigated.
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viii Fractional Calculus
As a new feature we present exercises with elaborated solutions, which
helptogetacquaintedwithpresentedstrategiesandallowforadirectinves-
tigation and application of new concepts developed within the framework
of fractional calculus. We discuss questions, which help to gain a deeper
understanding of general aspects of the theory. This book will also be a
valuable addition for courses devoted to this subject.
We hope, that this new edition will motivate students and scientists
to explore the fascinating, still mysterious world of fractional calculus and
apply these new ideas in their own researcharea, too.
WethankAnkeFriedrichforhercontinuoussupportandthepublishing
team of World Scientific for their help.
Richard Herrmann
Dreieich, Germany
Autumn, 2013
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Preface to the First Edition
Theoretical physics has evolved into a multi-faceted science. The sheer
amount of knowledge accumulated over centuries forces a lecturer to focus
on the mere presentation of derived results. The presentation of strategies
or the long path to find appropriate tools and methods often has been
neglected.
As a consequence, from a students point of view, theoretical physics
seems to be a construct of axiomatic completeness, where apparently is no
spaceleftforspeculationsandnewapproaches. Thelimitationsofpresented
conceptsandstrategiesarenotobviousuntilthesetoolsareappliedtonew
problems.
Especially the creative process of research, which evolves beyond the
state of mere reception of already known interrelations is a particular am-
bition of most students.
Oneprerequisitetoreachthatgoalisapermanentreadiness,toquestion
even well-established results and check the validity of common statements
time and time again. Already Roger Baconin his opus majus pointed out,
that the main causes of error are the belief in false authority, the force of
habit, the ignorance of others and pretended knowledge [Bacon (1267)].
New concepts and new methods depend on each other. The unified
theory of electro-magnetism by James Clerk Maxwell was motivated by
Faraday’sexperiments. Thedefiniteformulationwasthenrealizedinterms
ofanewtheoryofpartialdifferentialequations. Einstein’sgeneralrelativity
is based on experiments on the constancy of the speed of light; the elegant
presentation was made possible with Riemann’s tensor calculus.
This book too takes up an at first simple idea: due to the fact that
differential equations play such a central role in physics since the times of
Newton and since the first and second derivative denote such fundamental
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