sabatier agrawal machado advances in fractional calculus 9 pot

ADVANCES IN FRACTIONAL CALCULUS Advances in Fractional Calculus J. Sabatier Talence, France O. P. Agrawal Southern Illinois University Carbondale, IL, USA J. A. Tenreiro Machado Institute of Engineering of Porto Portugal Theoretical Developments and Applications in Physics and Engineering edited by and Université de Bordeaux I A C.I.P. Catalogue record for this book is available from the Library of Congress. Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands. www.springer.com Printed on acid-free paper All Rights Reserved 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. © 2007 Springer ISBN-13 978-1-4020-6041-0 (HB) ISBN-13 978-1-4020-6042-7 (e-book) No part of this work may be reproduced, stored in a retrieval system, or transmitted 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. iii We dedicate this book to the honorable memory of our colleague and friend Professor Peter W. Krempl Table of Contents 1. Analytical and Numerical Techniques 1 Three Classes of FDEs Amenable to Approximation Using a Galerkin Technique 3 Enumeration of the Real Zeros of the Mittag-Leffler Function E  (z), J. W. Hanneken, D. M. Vaught, B. N. Narahari Achar 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 2 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 1 < < 2 15 Suboptimum H order Linear Time Invariant Systems 61 The Caputo Fractional Derivative: Initialization Issues Relative to Fractional Differential Equations 27 Pseudo-rational Approximations to Fractional- vii Preface xi D. Baleanu, S. I. Muslih S. J. Singh, A. Chatterjee G. M. Zaslavsky P. W. Krempl Integral Type 155 R. R. Nigmatullin, J. J. Trujillo 3. Diffusive Systems 169 Boundary 171 N. Krepysheva, L. Di Pietro, M. C. Néel K. Logvinova, M. C. Néel Transport in Porous Media 199 Modelling and Identification of Diffusive Systems using Fractional A. Benchellal, T. Poinot, J. C. Trigeassou 4. Modeling 227 Identification of Fractional Models from Frequency Data 229 D. Valério, J. Sá da Costa Driving Force 243 B. N. Narahari Achar, J. W. Hanneken M. Haschka, V. Krebs Fractional Kinetics in Pseudochaotic Systems and Its Applications 127 Semi-integrals and Semi-derivatives in Particle Physics 139 Mesoscopic Fractional Kinetic Equations versus a Riemann–Liouville Solute Spreading in Heterogeneous Aggregated Porous Media 185 F. San Jose Martinez, Y. A. Pachepsky, W. J. Rawls A Direct Approximation of Fractional Cole–Cole Systems by Ordinary First-order Processes 257 2viii TableofContents Enhanced Tracer Diffusion in Porous Media with an Impermeable Fractional Advective-Dispersive Equation as a Model of Solute Models 213 Dynamic Response of the Fractional Relaxor–Oscillator to a Harmonic Pattern 271 L. Sommacal, P. Melchior, J. M. Cabelguen, A. Oustaloup, A. Ijspeert Application in Vibration Isolation 287 P. Serrier, X. Moreau, A. Oustaloup 5. Electrical Systems 303 C. Reis, J. A. Tenreiro Machado, J. B. Cunha Electrical Skin Phenomena: A Fractional Calculus Analysis 323 Gate Arrays 333 J. L. Adams, T. T. Hartley, C. F. Lorenzo 6. Viscoelastic and Disordered Media 361 Fractional Derivative Consideration on Nonlinear Viscoelastic Statical and Dynamical Behavior under Large Pre-displacement 363 H. Nasuno, N. Shimizu, M. Fukunaga Quasi-Fractals: New Possibilities in Description of Disordered Media 377 R. R. Nigmatullin, A. P. Alekhin Mechanical Systems 403 G. Catania, S. Sorrentino Fractional Multimodels of the Gastrocnemius Muscle for Tetanus Implementation of Fractional-order Operators on Field Programmable C. X. Jiang, J. E. Carletta, T. T. Hartley Analytical Modelling and Experimental Identification of Viscoelastic 2 ix TableofContents Limited-Bandwidth Fractional Differentiator: Synthesis and A Fractional Calculus Perspective in the Evolutionary Design of Combinational Circuits 305 J. K. Tar J. A. Tenreiro Machado, I. S. Jesus, A. Galhano, J. B. Cunha, Complex Order-Distributions Using Conjugated order Differintegrals 347 Fractional Damping: Stochastic Origin and Finite Approximations 389 S. J. Singh, A. Chatterjee 7. Control 417 LMI Characterization of Fractional Systems Stability 419 M. Moze, J. Sabatier, A. Oustaloup Calculus 435 M. Kuroda V. Feliu, B. M. Vinagre, C. A. Monje D. Valério, J. Sá da Costa Tracking Design 477 P. Melchior, A. Poty, A. Oustaloup Flatness Control of a Fractional Thermal System 493 P. Melchior, M. Cugnet, J. Sabatier, A. Poty, A. Oustaloup P. Lanusse, A. Oustaloup Generation CRONE Controller 527 P. Lanusse, A. Oustaloup, J. Sabatier J. Liang, W. Zhang, Y. Chen, I. Podlubny Fractional-order Control of a Flexible Manipulator 449 Tuning Rules for Fractional PIDs 463 2 TableofContentsx Active Wave Control for Flexible Structures Using Fractional Frequency Band-Limited Fractional Differentiator Prefilter in Path Robustness Comparison of Smith Predictor-based Control and Fractional-Order Control 511 Wave Equations with Delayed Boundary Measurement Using the Smith Predictor 543 Robust Design of an Anti-windup Compensated 3rd- Robustness of Fractional-order Boundary Control of Time Fractional Preface Fractional Calculus is a field of applied mathematics that deals with derivatives and integrals of arbitrary orders (including complex orders), and their applications in science, engineering, mathematics, economics, and other fields. It is also known by several other names such as Generalized name “Fractional Calculus” is holdover from the period when it meant calculus of ration order. The seeds of fractional derivatives were planted over 300 years ago. Since then many great mathematicians (pure and applied) of their times, such as N. H. Abel, M. Caputo, L. Euler, J. Fourier, A. K.  not being taught in schools and colleges; and others remain skeptical of this for fractional derivatives were inconsistent, meaning they worked in some cases but not in others. The mathematics involved appeared very different applications of this field, and it was considered by many as an abstract area containing only mathematical manipulations of little or no use. Nearly 30 years ago, the paradigm began to shift from pure mathematical Fractional Calculus has been applied to almost every field of science, has made a profound impact include viscoelasticity and rheology, electrical engineering, electrochemistry, biology, biophysics and bioengineering, signal and image processing, mechanics, mechatronics, physics, and control theory. Although some of the mathematical issues remain unsolved, most of the difficulties have been overcome, and most of the documented key mathematical issues in the field have been resolved to a point where many Marichev (1993), Kiryakova (1994), Carpinteri and Mainardi (1997), Podlubny (1999), and Hilfer (2000) have been helpful in introducing the field to engineering, science, economics and finance, pure and applied field. There are several reasons for that: several of the definitions proposed engineering, and mathematics. Some of the areas where Fractional Calculus Oustaloup (1991, 1994, 1995), Miller and Ross (1993), Samko, Kilbas, and from that of integer order calculus. There were almost no practical formulations to applications in various fields. During the last decade mathematics communities. The progress in this field continues. Three Integral and Differential Calculus and Calculus of Arbitrary Order. The Grunwald, J. Hadamard, G. H. Hardy, O. Heaviside, H. J. Holmgren, P. S. Laplace, G. W. Leibniz, A. V. Letnikov, J. Liouville, B. Riemann M. Riesz, and H. Weyl, have contributed to this field. However, most scientists and engineers remain unaware of Fractional Calculus; it is of the mathematical tools for both the integer- and fractional-order calculus are the same. The books and monographs of Oldham and Spanier (1974), xi recent books in this field are by West, Grigolini, and Bologna (2003), One of the major advantages of fractional calculus is that it can be believe that many of the great future developments will come from the applications of fractional calculus to different fields. For this reason, we symposium on Fractional Derivatives and Their Applications (FDTAs), ASME-DETC 2003, Chicago, Illinois, USA, September 2003; IFAC first workshop on Fractional Differentiations and its Applications (FDAs), Bordeaux, France, July 2004; Mini symposium on FDTAs, ENOC-2005, Eindhoven, the Netherlands, August 2005; the second symposium on FDTAs, ASME-DETC 2005, Long Beach, California, USA, September 2005; and IFAC second workshop on FDAs, Porto, Portugal, July 2006) and published several special issues which include Signal Processing, Vol. 83, No. 11, 2003 and Vol. 86, No. 10, 2006; Nonlinear dynamics, Vol. 29, No. further advance the field of fractional derivatives and their applications. In spite of the progress made in this field, many researchers continue to ask: “What are the applications of this field?” The answer can be found right here in this book. This book contains 37 papers on the applications of within the boundaries of integral order calculus, that fractional calculus is indeed a viable mathematical tool that will accomplish far more than what integer calculus promises, and that fractional calculus is the calculus for the future. FDTAs, ASME-DETC 2005, Long Beach, California, USA, September 2005. We sincerely thank the ASME for allowing the authors to submit modified versions of their papers for this book. We also thank the authors for submitting their papers for this book and to Springer-Verlag for its Kilbas, Srivastava, and Trujillo (2005), and Magin (2006). considered as a super set of integer-order calculus. Thus, fractional calculus has the potential to accomplish what integer-order calculus cannot. We are promoting this field. We recently organized five symposia (the first 1–4, 2002 and Vol. 38, No. 1–4, 2004; and Fractional Differentiations and its Applications, Books on Demand, Germany, 2005. This book is an attempt to Fractional Calculus. These papers have been divided into seven categories based on their themes and applications, namely, analytical and numerical believe that researchers, new and old, would realize that we cannot remain Eindhoven, The Netherlands, August 2005, and the second symposium on 2xii Preface techniques, classical mechanics and particle physics, diffusive systems, viscoelastic and disordered media, electrical systems, modeling, and control. Applications, theories, and algorithms presented in these papers are contemporary, and they advance the state of knowledge in the field. We the papers presented at the Mini symposium on FDTAs, ENOC-2005, Most of the papers in this book are expanded and improved versions of [...]... 112 69 11271 11273 11275 11277 112 79 11281 11283 n 1 .99 899 4787610 1 .99 899 494 8054 1 .99 899 5108443 1 .99 899 5268780 1 .99 899 54 290 62 1 .99 899 55 892 90 1 .99 899 57 494 65 1 .99 899 590 9586 1 .99 899 60 696 54 1 .99 899 62 296 67 1 .99 899 63 896 27 1 .99 899 65 495 34 1 .99 899 67 093 87 1 .99 899 68 691 86 1 .99 899 702 893 2 1 .99 899 7188625 1 .99 899 7348263 1 .99 899 75078 49 1 .99 899 7667381 1 .99 899 7826860 1 .99 899 798 6285 1 .99 899 8145657 1 .99 899 830 497 6 1 .99 899 8464241... 1 .99 899 7188625 1 .99 899 7348263 1 .99 899 75078 49 1 .99 899 7667381 1 .99 899 7826860 1 .99 899 798 6285 1 .99 899 8145657 1 .99 899 830 497 6 1 .99 899 8464241 1 .99 899 8623453 1 .99 899 8782612 1 .99 899 894 1718 1 .99 899 9100770 1 .99 899 92 597 70 1 .99 899 9418716 1 .99 899 95776 09 1 .99 899 9736450 1 .99 899 9 895 237 1 .99 900005 397 1 REAL ZEROS OF THE MITTAG-LEFFLER FUNCTION 21 4 Iteration Formula Two conditions must be satisfied for E ,1(–x) to be tangent... 1.87718 792 1171 1.88 196 8 294 552 1.886348272721 1. 890 378331112 1. 894 100 597 857 1. 897 55053 793 1 1 .90 0758240821 1 .90 37 494 17 395 1 .90 6546180470 1 .90 91676623 39 1 .91 163050 799 9 1 .91 394 9272538 1 .91 613674 390 3 1 .91 82042070 29 1 .92 0161661487 1 .92 201800 199 4 1 .92 37811 690 33 1 .92 5458275243 n 11217 112 19 11221 11223 11225 11227 112 29 11231 11233 11235 11237 112 39 11241 11243 11245 11247 112 49 11251 11253 11255 11257 112 59 11261 11263... 1.500 1.600 1.700 1.800 1 .90 0 # of zeros 0 1 1 1 1 3 5 9 17 45 1 .90 0 1 .91 0 1 .92 0 1 .93 0 1 .94 0 1 .95 0 1 .96 0 1 .97 0 1 .98 0 1 .99 0 # of zeros 45 53 61 73 91 115 153 2 19 357 815 1 .99 0 1 .99 1 1 .99 2 1 .99 3 1 .99 4 1 .99 5 1 .99 6 1 .99 7 1 .99 8 1 .99 9 # of zeros 815 92 3 1,0 59 1,237 1,4 79 1,825 2,357 3,273 5,181 1,1281 ... 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 (truncated) at which E ,1(–x) is tangent to the x-axis n 1.422 190 690 801 1.57188 392 294 2 1.6 490 68237342 1. 698 516223760 1.733 693 032768 1.760338811725 1.781 392 651685 1. 798 543344750 1.81284 194 9070 1.82 498 2270661 1.835443517675 1.844568817828 1.852611186687 1.8 597 61810886 1.866168176867 1.87 194 6 096 560 1.87718 792 1171... http://www.geocities.com/dynamics_iisc/SystemMatrices.zip Basset AB ( 191 0) Quart J Math 41:3 69 381 Mainardi F, Pironi P, Tampieri F ( 199 5) On a Generalization of Basset Problem via Fractional Calculus, in: Proceedings CANCAM 95 Mainardi F ( 199 6) Chaos, Solitons Fractals, 7 (9) :1461–1477 Hairer E, Wanner G ( 199 1) Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems Springer, Berlin ENUMERATION OF THE REAL ZEROS OF THE... 6 7 8 9 Samko SG, Kilbas AA, Marichev OI ( 199 3) Fractional Integrals and Derivatives: Theory and Applications Gordon and Breach, Amsterdam Oldham KB ( 197 4) The Fractional Calculus Academic Press, New York Koh CG, Kelly JM ( 199 0) Earthquake Eng Struc Dyn., 19: 2 29 241 Singh SJ, Chatterjee A (2005) Nonlinear Dynamics (in press) http://www.geocities.com/dynamics_iisc/SystemMatrices.zip Basset AB ( 191 0)... Significant digits in 3 4 5 6 7 8 9 10 11 Range of for reliable results from Eqs (14) and (15) 1.42 . ( 199 3), Kiryakova ( 199 4), Carpinteri and Mainardi ( 199 7), Podlubny ( 199 9), and Hilfer (2000) have been helpful in introducing the field to engineering, science, economics and finance, pure and. the definitions proposed engineering, and mathematics. Some of the areas where Fractional Calculus Oustaloup ( 199 1, 199 4, 199 5), Miller and Ross ( 199 3), Samko, Kilbas, and from that of integer. ADVANCES IN FRACTIONAL CALCULUS Advances in Fractional Calculus J. Sabatier Talence, France O. P. Agrawal Southern Illinois University Carbondale, IL, USA J. A. Tenreiro Machado Institute