Fractal Elements and their Applications
This book describes a new type of passive electronic components, called fractal elements, from a theoretical and practical point of view. The authors discuss in detail the physical implementation and design of fractal devices for application in fractional
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Anis Kharisovich Gil’mutdinov Pyotr Arkhipovich Ushakov Reyad El-Khazali
Fractal Elements and their Applications
Analog Circuits and Signal Processing Series Editors: Mohammed Ismail, Dublin, USA Mohamad Sawan, Montreal, Canada
More information about this series at http://www.springer.com/series/7381
Anis Kharisovich Gil’mutdinov Pyotr Arkhipovich Ushakov • Reyad El-Khazali
Fractal Elements and their Applications Ed. by A.Kh. Gil’mutdinov
Anis Kharisovich Gil’mutdinov Kazan National Research Technical University-KAI (KNRTU-KAI) Kazan, Russia
Pyotr Arkhipovich Ushakov Kalashnikov State Technical University Izhevsk, Russia
Reyad El-Khazali Khalifa Univ. of Science, Tech. & Research Abu Dhabi, United Arab Emirates
ISSN 1872-082X ISSN 2197-1854 (electronic) Analog Circuits and Signal Processing ISBN 978-3-319-45248-7 ISBN 978-3-319-45249-4 (eBook) DOI 10.1007/978-3-319-45249-4 Library of Congress Control Number: 2017930413 © Springer International Publishing Switzerland 2017 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 This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
We dedicate this book To the blessed memory Of Rashid Shakirovich Nigmatullin, The founder of the Kazan scientific school Of fractional operators
Foreword
Today we can say with certainty that both scientists and engineers have widely recognized the need to use the fractals theory along with the theory of fractional integral-differential operators and fractal treatment to solve a variety of problems that emerge in various fields of modern science and technology. Terms “fractals” and “fractal” reflect the modern view of the physical nature of real objects and processes; this view was firmly established after publication of the pioneering works of B. Mandelbrot about fractal geometry of the Nature. The “fractional operators” reflec
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