Sabatier Agrawal Machado Advances in Fractional Calculus Episode 1 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 [...]... 11 217 11 219 11 2 21 112 23 11 225 11 227 11 229 11 2 31 112 33 11 235 11 237 11 239 11 2 41 112 43 11 245 11 247 11 249 11 2 51 112 53 11 255 11 257 11 259 11 2 61 112 63 11 265 11 267 11 269 11 2 71 112 73 11 275 11 277 11 279 11 2 81 112 83 n 1. 998994787 610 1. 998994948054 1. 99899 510 8443 1. 998995268780 1. 998995429062 1. 998995589290 1. 998995749465 1. 998995909586 1. 998996069654 1. 998996229667 1. 998996389627 1. 998996549534 1. 998996709387 1. 99899686 918 6... 1. 798543344750 1. 812 8 419 49070 1. 8249822706 61 1.835443 517 675 1. 844568 817 828 1. 852 611 186687 1. 8597 618 10886 1. 86 616 817 6867 1. 8 719 46096560 1. 87 718 79 211 71 1.8 819 68294552 1. 8863482727 21 1.8903783 311 12 1. 89 410 0597857 1. 8975505379 31 1.9007582408 21 1.903749 417 395 1. 90654 618 0470 1. 90 916 7662339 1. 911 630507999 1. 913 949272538 1. 916 136743903 1. 918 204207029 1. 92 016 16 614 87 1. 922 018 0 019 94 1. 9237 811 69033 1. 925458275243 n 11 217 11 219 ... specified in are observed (Table 2) Table 3 Number of real zeros of E ,1( –x) 1. 000 1. 100 1. 200 1. 300 1. 400 1. 500 1. 600 1. 700 1. 800 1. 900 # of zeros 0 1 1 1 1 3 5 9 17 45 1. 900 1. 910 1. 920 1. 930 1. 940 1. 950 1. 960 1. 970 1. 980 1. 990 # of zeros 45 53 61 73 91 115 15 3 219 357 815 1. 990 1. 9 91 1.992 1. 993 1. 994 1. 995 1. 996 1. 997 1. 998 1. 999 # of zeros 815 923 1, 059 1, 237 1, 479 1, 825 2,357 3,273 5 ,18 1 1, 12 81 ... E ,1( –x) has 5 zeros for 5 < 9 , 9 zeros for 9 < 11 , , 11 2 81 zeros for 11 2 81 < 11 283 7 Hanneken, Vaught, and Achar 20 Table 1 Values of n 3 5 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. 42 219 06908 01 1.5 718 83922942 1. 649068237342 1. 698 516 223760 1. 733693032768 1. 760338 811 725 1. 7 813 926 516 85 1. 798543344750... 1. 998996549534 1. 998996709387 1. 99899686 918 6 1. 998997028932 1. 99899 718 8625 1. 998997348263 1. 998997507849 1. 9989976673 81 1.998997826860 1. 998997986285 1. 99899 814 5657 1. 998998304976 1. 9989984642 41 1.998998623453 1. 998998782 612 1. 9989989 417 18 1. 99899 910 0770 1. 998999259770 1. 998999 418 716 1. 998999577609 1. 998999736450 1. 998999895237 1. 9990000539 71 REAL ZEROS OF THE MITTAG-LEFFLER FUNCTION 21 4 Iteration Formula Two conditions... yields: ln m sin 2 1 m cos 2 cot ln 1 1 4m / / /2 1 1 2 / 1 2m ln 1 1 4 Ai i 1 2 cot 1i sin / 2m i (14 ) / where Ai i 1 /2 / i i 1, 2,3, i 1 The Ai ’s come from keeping terms beyond i = 1 in the infinite series in Eq (8) In Eq (14 ), m cannot be solved explicitly, but can be determined iteratively by guessing a value of m and using this value of m in Eq (14 ) to calculate a new guess for m and repeating the... decomposed into two parts [14 ] For the special case of a negative real argument, the result is given by: E 2 g , x , 0 sin 2r cos 1 E ,1 f ,1 dr (4c) 1/ < 1, g , (–x) = 0 For the special case of where + 1 > and for (4a–c) reduce to 2 (4b) r sin 2 x x x 1 / sin 1/ exp x 1 / r r r (4a) x , 1 x x f cos x , g g exp x 1 / cos 1 f x , x ,1 g ,1 x 1 exp x x cos cos f ,1 1 x (5a) 1 x sin exp x 1 / r r 1 sin 1 0 r2... zeros, fractional calculus 1 Introduction The single parameter Mittag-Leffler function E (z) is defined over the entire complex plane by E z k 0 zk k 1 > 0, z C (1) 15 J Sabatier et al (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, 15 –26 © 2007 Springer 16 Hanneken, Vaught, and Achar and is named after Mittag-Leffler who introduced it in 19 03... 24.243 in addition to the one at x 2 .11 0 REAL ZEROS OF THE MITTAG-LEFFLER FUNCTION 19 0.05 0.00 -0.05 z -16 .724 E ,1( z) -0 .10 = 1. 3 -0 .15 -0.20 1. 42 219 -0.25 = 1. 5 -0.30 -25 -20 -15 -10 -5 0 z Fig 1 Plots of E ,1( –x) for various values of Clearly, there is a value of between = 1. 3 and = 1. 5 for which the curve of E ,1( –x) is exactly tangent to the x-axis This is illustrated in the graph 1. 42 219 06908 01. .. 1/ 3 is given by X(s) = [1 − (−s 1/ 3 )] 1 1 = s (1 + s1/3 ) s4/3 ( 21) On expanding the numerator above (assuming |s| > 1) and simplifying, we get ∞ ( 1) n (22) X(s) = sn/3 n=4 The above series is absolutely convergent for |s| > 1 Inverting gives ∞ x(t) = ( 1) n tn/3 1 Γ (n/3) n=4 (23) THREE CLASSES OF FDEs AMENABLE 13 11 4.3 Results Numerical results are shown in Fig 4 The Galerkin approximation matches . Physics 11 3 On Fractional Variational Principles 11 5 1 < < 2 15 Suboptimum H order Linear Time Invariant Systems 61 The Caputo Fractional Derivative: Initialization Issues Relative to Fractional. 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. defini- – © 2007 Springer. in Physics and Engineering, 3 14 . TO APPROXIMATION USING AGALERKIN TECHNIQUE Mechanical Engineering Department, Indian Institute of Science, Bangalore 560 012 , India for simplicity