ADVANCES IN FRACTIONAL CALCULUS docx

Table of Contents 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.. 61 The Caputo Frac

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

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

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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

We dedicate this book to the honorable memory of our colleague and friend Professor Peter W Krempl

Table of Contents

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

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

Transport in Porous Media 199

Modelling and Identification of Diffusive Systems using Fractional

A Benchellal, T Poinot, J C Trigeassou

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

viii 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

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

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

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

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

Table of Contents

x

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

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,

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, mostscientists 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

xii 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

publication We hope that readers will find this book useful and valuable in the advancement of their knowledge and their field

Part 1

Analytical and

Numerical Techniques

we demonstrate how that approximation can be used to find accurate numerical solutions of three different classes of fractional differential equations (FDEs), where

order greater than one An example of a traveling point load on an infinite beam resting on an elastic, fractionally damped, foundation is studied The second class generalized Basset’s equation are studied The third class contains FDEs where the

other means In each case, the Galerkin approximation is found to be very good We conclude that the Galerkin approximation can be used with confidence for a variety

of FDEs, including possibly nonlinear ones for which analytical solutions may be difficult or impossible to obtain.

Abstract

We have recently elsewhere a Galerkin approximation scheme for fractional order derivatives, and used it to obtain accurate numerical solutions

presented

of second-order (mechanical) systems with fractional-order damping terms Here,

contains FDEs where the highest derivative has order 1 Examples of the so-called highest derivative is the fractional-order derivative itself Two specific examples are

Fractional derivative, Galerkin, finite element, Basset’s problem, relaxation,

J Sabatier et al (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications

Satwinder Jit Singh and Anindya Chatterjee

is exactly equal to 1, or is a fraction between 0 and 1.

In this article, we will demonstrate three strategies for these three classes ofFDEs, whereby a new Galerkin technique [4] for fractional derivatives can beapproximation scheme of [4] involves two calculations:

where A and B are n × n matrices (specified by the scheme; see [4]), c is an

, and a is an n × 1 vector n internalvariables that approximate the infinite-dimensional dynamics of the actual

As will be seen below, the first category of FDEs (section 2) poses no realproblem over and above the examples already considered in [4] That is, in[4], the highest derivatives in the examples considered had order 2; while inthe example considered in section 2 below, the highest derivative will be orinfinite domain Our approximation scheme provides significant advantages forthis problem The second category of FDEs (section 3) also leads to numericalsolution of ODEs (not FDEs) The specific example considered here is relevant

to the physical problem of a sphere falling slowly under gravity through aviscous liquid, but not yet at steady state Again, the approximation schemeleads to an algorithmically simple, quick and accurate solution However, theequations are stiff and suitable for a routine that can handle stiff systems,such as Matlab’s “ode23t” Finally, the third category of FDEs (section 4)solved simply and accurately using an index one DAE solver such as Matlab’s

“ode23t”

1

which involve fractional-order derivatives of the dependent variable(s) areDifferential equations with a single-independent variable (usually “time”),

whether the highest-order derivative in the FDE is an integer greater than 1,

used to obtain simple, quick, and accurate numerical solutions The Galerkin

fractional order derivative The T superscript in Eq (3) denotes matrix pose

trans-order 4 However, the example of section 2 is a boundary-value problem on an

leads to a system of differential algebraic equations (DAEs), which can be

A Maple-8 worksheet to compute the matrices A , B, and c is available on [5] Singh and Chatterjee

We emphasize that we have deliberately chosen linear examples below

so that analytical or semi-analytical alternative solutions are available forcomparing with our results using the Galerkin approximation However, itwill be clear that the Galerkin approximation will continue to be useful for

a variety of nonlinear problems where alternative solution techniques mightrun into serious difficulties

2 Traveling Load on an Infinite Beam

The governing equation for an infinite beam on a fractionally damped elasticfoundation, and with a moving point load (see Fig 1), is

conditions of interest are

where a is now a function of both x and t, and the overdot denotes a partialderivative with respect to t Changing variables to ξ = x − vt and τ = t toshift to a steadily moving coordinate system, we get

In the above, steady state is achieved as τ → ∞, and we get

Substituting y = ξ + v w above for later convenience, we get

Now, seeking a steady-state solution, Eqs (5) and (6) become

Without the Galerkin approximation, the

written as

fractional term in Eq (4) can be

substituting in Eq (9) we get (with incomplete incorporation of steady stateconditions)

Singh and Chatterjee

ξ = 0 These jump/continuity conditions provide as many equations as thereare state variables; and these equations can be used to solve for the samenumber of unknown coefficients of eigenvectors in the solution The overallprocedure is straightforward, and can be implemented in, say, a few lines ofMatlab code Numerical results obtained will be presented below.

Equation (10) cannot, as far as we know, be solved in closed form It can

be solved numerically using Fourier transforms The Fourier transform of u(ξ)

2.4 Results

Fig 2 The Galerkin approximation is very good

The agreement between the two solutions (Galerkin and Fourier) providessupport for the correctness of both In a problem with several unequally spaced

Thus, the steady state version of Eq (4) without approximation is

Solutions, with Galerkin and without

Solution of Eq (7) and (8) is straightforward and quick An algebraic value problem is solved and a jump condition imposed The details are asfollows For ξ = 0, the system reduces to a homogeneous first-order system

traveling loads, the Galerkin technique will remain straightforward while theFourier approach will become more complicated Our point here is not that theFourier solution is intellectually inferior (we find it elegant) Rather, straight-forward application of the Galerkin technique requires less problem-specificingenuity and effort

Fig 2 Numerical results for a traveling point load on an infinite beam at steady state.

3 Off Spheres Falling Through Viscous Liquids

A sphere falling slowly under its own weight through a viscous liquid willapproach a steady speed [6] The approach is described by a FDE wherethe highest derivative has order 1 Here, we study no fluid mechanics issues.Rather, we consider two such FDEs with, for simplicity, zero initial conditions.Such problems have been referred to as examples of the generalized Basset’s

Singh and Chatterjee

problem [7] Our aim is to demonstrate the use of our Galerkin approximationfor such problems.

Consider

0 < α < 1 Here, for demonstration, we will consider α = 1/2 and 1/3 Thesolution methods discussed below will work for any reasonable α between 0and 1

(in particular, suppose we consider s values on a vertical line in the complexplane, we are prepared to choose that line as far into the right half plane asneeded) The series we obtain is

The Laplace transform of the solution to Eq (12) is given by

Series solution using Laplace transforms

with initial conditions v(0) = 0 and a(0) = 0 Recall that, for any value of

3.3 Results

Results for the above problem are shown in Fig 3 The Galerkin

using MAPLE (fewer than 150 terms may have worked; more were surely notneeded)

150 ) term Right: 150

) term.

4 FDEs With Highest Derivative Fractional

Consider

damping and under slow loading (where inertia plays a negligible role), such as

in creep tests Here, we concentrate on demonstrating the use of our Galerkintechnique for this class of problems

4.1

duce ˙x(t) by taking a 1 − α order derivative, but such differentiation requires

tion matches well with the series solutions of Eq (12) for α = 1/2 and1/3 The sum in Eq (14) was taken upto the O(t

Fig 3 Comparison between Laplace transform and 15-element Galerkin mation solutions: Left: α = 1/2 and sum in Eq (14) upto O(t

approxi-α = 1/3 and sum in Eq (14) upto O(t

Equations of this form are called relaxation fractional Eq [8] Theseequations have relevance to, e.g., mechanical systems with fractional-order

Adaptation of the Galerkin approximation

intro-Our usual Galerkin approximation strategy will not work here directly,because

Singh and Chatterjee

the forcing function f (t) to have such a derivative, and we avoid such entiation here Instead, we adopt the Galerkin approximation through con-

differ-of x(t) in equation (3) We interpret the above as follows If the forcing wassome general function h(t) instead of ˙x(t); and if h(t) was integrable, i.e.,h(t) = ˙g(t) for some function g(t); and if, in addition, g(t) was continuous at

t = 0, then by adding a constant to g(t) we could ensure that g(0) = 0 whilestill satisfying h(t) = ˙g(t) Further, the forcing of h(t) (in place of ˙x(t)) in

Eq (2) would result in an α order derivative of g(t) (in place of x(t)) in

Eq (3) In other words, if

Compute matrices A , B, and c for 1 − α order derivatives instead of αReplace Eq (15) by the following system:

then easily solved using standardObserve that x˙(t) forcing in Eq (2) results in an α order derivative

or

We used α = 1/2 and 1/3 for numerical simulations The index of theDAEs here (see [9] for details) is one For both values of α, DAEs (18) areinitial conditions are calculated as x(0) = 0 , a(0) = 0 and y(0) = 1; a guessfor corresponding initial slopes, which is an optional input to “ode23t,” is

The solution of Eq (15) can be obtained using Laplace transforms For

The Laplace transform of the solution to Eq (15) for α = 1/3 is given by

Singh and Chatterjee

4.3 Results

Numerical results are shown in Fig 4 The Galerkin approximation matches

Fig 4.

solutions Left: α = 1/2 Right: α = 1/3 For α = 1/3, the series is summed up to O(t 150

).

5 Discussion and Conclusions

We have identified three classes of FDEs that are amenable to solution usingdeveloped recently in other work [4] To showcase the effectiveness of theanalytically (if only in the form of power series) However, more general andnonlinear problems which are impossible to solve analytically are also expected

to be equally effectively solved using this approximation technique

The approximation technique used here, as discussed in [4], involves merical evaluation of certain matrices For approximation of a derivative of

nu-a given frnu-actionnu-al order between 0 nu-and 1, nu-and with nu-a given number of shnu-apefunctions in the Galerkin approximation, these matrices need be calculatedonly once They can then be used in any problem where a derivative of thesame order appears A MAPLE file which calculates these matrices is avail-able on the web We hope that this technique will serve to provide a simple,reliable, and routine method of numerically solving FDEs in a wide range ofapplications

the exact solutions well in both cases The sum in Eq (23) is taken upto theO(t

Comparison between analytical and 15-element Galerkin approximation

approximation technique, we have used linear FDEs, which could also be solved

a new Galerkin approximation for the fractional-order derivative, that was

References

1 Samko SG, Kilbas AA, Marichev OI (1993) Fractional Integrals and Derivatives: Theory and Applications Gordon and Breach, Amsterdam

2 Oldham KB (1974) The Fractional Calculus Academic Press, New York

3 Koh CG, Kelly JM (1990) Earthquake Eng Struc Dyn., 19:229–241

4 Singh SJ, Chatterjee A (2005) Nonlinear Dynamics (in press)

5 http://www.geocities.com/dynamics_iisc/SystemMatrices.zip

6 Basset AB (1910) Quart J Math 41:369–381

7 Mainardi F, Pironi P, Tampieri F (1995) On a Generalization of Basset Problem via Fractional Calculus, in: Proceedings CANCAM 95

8 Mainardi F (1996) Chaos, Solitons Fractals, 7(9):1461–1477

9 Hairer E, Wanner G (1991) Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems Springer, Berlin

Singh and Chatterjee

E (z) for any value of in the range 1 < < 2 and some specific results are tabulated

k

zz

E > 0, z C (1)

2John W Hanneken , David M Vaught , and B N Narahari Achar

Mittag-Leffler functions, zeros, fractional calculus

© 2007 Springer

J Sabatier et al (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications

in Physics and Engineering, 15–26

15

University of Memphis, Physics Department, Memphis, TN 38152; Tel: 901.678.2417, University of Memphis, Physics Department, Memphis, TN 38152; DHawkeye10@aol.com Fax: 901.678.4733, E-mail: jhannekn@memphis.edu

described by differential and/or integral equations of fractional order Conse- quently, the zeros of E (z) and their distribution are of fundamental impor- tance and play a significant role in the dynamic solutions The Mittag- Lefflerfunction E (z) is known to have a finite number of real zeros in the range

University of Memphis, Physics Department, Memphis, TN 38152; Tel: 901.678.3122, Fax: 901.678.4733, E-mail: nachar@memphis.edu

1 < < 2

OF THE MITTAG-LEFFLER FUNCTION E (z),

k ȕ

,

zz

E , 0, z C (2)

It may be noted that when = 1, E ,1(z) = E (z) Properties of the

Mittag-others have considered complex [8,9] and complex [10], the present work is restricted to real and The Mittag-Leffler functions are natural extensions of

are often expressed in terms of Mittag-Leffer functions in much the same way that solutions of many integer order differential equations may be expressed in terms of exponential functions Consequently, the zeros of E ,1(z), which play a significant role in the dynamic solutions, are of intrinsic interest

Except for the special case of = 1, in general E ,1(z) has an infinite number

of zeros [11,12] and all complex zeros of E (z) appear as pairs of complex conjugates [13] To facilitate the discussion of the zeros, the domain of values

2 based on the nature of the zeros, but E ,1(z) and its zeros exhibit similar properties within each range For 0 1, E (z) has no real zeros [14] and thus must have an infinite number of complex zeros For = 1, E1,1(z) can be

1

E (z) has a finite number of zeros on the negative real axis [5,8,9,11,14] and must in addition have an infinite number of complex zeros [11,15] For 2,

,1(z) has no positive real zeros Thus, for convenience, the variable x will be used to represent a positive real number so that E ,1

argument Real zeros occur only in the ranges 1 2, and 2 The range

1 2 is the range for which the least is known and yet is quite relevant for many physical problems [6,17] The objective of this paper is to determine the exact number of real zeros for E ,1

These results will be discussed later in connection with an asymptotic formula for the number of real zeros valid near = 2 [14] The first requirement is a discussion of how to calculate E ,1

Hanneken, Vaught, and Achar

Leffler functions have been summarized in several references [5–7] Although

the exponential function and solutions of fractional-order differential equations

written as E ( z) = exp(z), which has no zeros real or complex For 1 2,

complex zeros [8–10,16] Note that regardless of the range of , E

(–x) clearly has a negative real

(–x) for arbitrary in the range 1 2

(–x) accurately

can be conveniently divided into four ranges: 0 1, = 1, 1 2, and

E (z) has an infinite number of zeros that are real, negative, and simple and no

2 Numerical Evaluation of E ,

Numerical values of E , (z) are easily calculated using the power series given in

Eq (2) when the argument z is not too large However, for large arguments this method is impractical because of the extremely slow convergence of the series Instead, use will be made of the representation of E ,(z) as a Laplace inversion integral [6]

ds

zs

seiʌz

ȕ Į Br

s ȕ

,

Į 21 (3)

where Br denotes the Bromwich path Using standard techniques in the theory

of calculus of residues [18], E , ( z ) can be decomposed into two parts [14] For the special case of a negative real argument, the result is given by:

EĮ,ȕ x gĮ,ȕ x fĮ,ȕ x (4a)

Į / 1 ȕ

Į / 1 Į

/ 1 ȕ

ȕ1ʌcosĮ

ʌcosxexpĮ2x

Į / 1 ȕ 0

Į Į 2

Į ȕ Į Į / 1

ʌĮcosr2r

Įȕʌsinʌȕsinrrrxexpʌ1

ȕ1ʌcosĮ

ʌcosxexpĮ

2x

1 Į

1 1

1 Į Į / 1 1

,

1ʌĮcosr2r

ʌĮsinrrxexpʌ

1x

,1

were in agreement to better than 40 significant digits with the values calculated directly from Eq (1) for small values of the argument As an alternative to the

infinite series as follows[14]

1 n n

1 n 2

1

,

Į

Įn1x

1Į

21x

1Į

1x

1x

This series is particularly useful when both x and the gamma function are large and the series converges very quickly The value of the gamma function approaches infinity as its argument approaches a negative integer Thus, Eq (6)

is most useful for close to 2 and x large

3 Zeros of E ,1

Critical to the derivation of a formula for the number of real zeros is an understanding of the nature of the zeros and this is best done by examining the graphs of E ,1 ,1(0) = 1 and for large x values E ,1negative and asymptotically approaches zero governed predominately by f ,1

Eq (5c), with the exponentially decreasing oscillations of g ,1

superimposed The fact that the curves of E ,1

ultimately become negative for large x implies that E ,1

,1The curve exhibits only one zero at x 2.293 and for larger x remains negative with the superimposed oscillation of g ,1

scale The rate of exponential decay of g ,1

x1/ cos( / ), the cos( / ) being negative in the range 1 2 As increases this exponent decreases resulting in larger amplitude oscillations This is illustrated in the graph of E ,1

amplitude oscillations of g ,1

24.243 in addition to the one at x 2.110

Hanneken, Vaught, and Achar

(–x) were computed primarily from Eqs (5a–c) using Mathematica [19] with the integration performed using the built-in function NIntegrate The values computed using Eqs (5a–c)

(–x) can be written in an asymptotic

(–x) of Multiplicity 2

Numerical values of the Mittag-Leffler function E

(–x),

(–x) are positive at x = 0 and

(–x) is determined by the exponent (–x) imperceptible on this

(–x) for = 1.5 also shown in Fig 1 The larger (–x) give rise to a relative maximum at x 17.472

(–x), Eq (5b),

(–x) can only cross the x-axis an odd number of times[5] This is illustrated in the plot of E (–x) for = 1.3 shown in Fig 1

x

extending above the x-axis and yielding two more zeros at x 13.765 and

of E ,1

1.422190690801 separates the range of values where E,1

zero from the range where E ,1

,1 x)the next section depends essentially on the existence of these values of where the curve of E ,1

,1tangent to the x-axis have been numerically determined A few selected values are given in Table 1 These values will be most useful in section 5 to establish

7 < 9 , 9 zeros for 9 < 11 , , 11281 zeros for 11281 < 11283

Fig 1 Plots of E (–x) for various values of

(–x) is exactly tangent to the x-axis This is illustrated in the graph

(–x) for 1.422190690801 also shown in Fig 1 This curve has a zero

(–x) has only one

at x 2.145 and is tangential at x 16.724 where it has a zero of multiplicity

of 2 still yielding an odd total number of zeros It may be noted that for = 1.3 the curve crosses the x-axis only once yielding one zero and for = 1.5 the curve crosses the x-axis 3 times yielding 3 zeros Thus, the value of

(–x) has three zeros The next larger value of

ranges of reliability for the iteration results for < 1.999 In reading Table 1, for example, is the lowest value of for which E (–x)

is tangent to the x-axis Thus, E

where the curve is tangent to the x-axis is at 1.5718839229424 where E has five zeros The iteration formula for the number of real zeros described in

Hanneken, Vaught, and Achar

(–x) is tangent to the x-axis

4 Iteration Formula

Two conditions must be satisfied for E ,1

both the function and its derivative must be zero, or

E ,1 E x 0

dx

d1 ,

1 Į

1

Įi1x

1Į

ʌsinxcosĮ

ʌcosxexp

d

Į , Į 1

, Į

Į

1 Į

1

Į ,

Į

x

Į

ʌsinxĮ

ʌcosĮ

ʌcosxexpĮ2x

The asymptotic expansion of Eq (4c) with = is given by[14]:

Į3

xĮ2

xĮ

xxf

4 3

2 Į

ʌ

1

(12)

(–x) to be tangent to the x-axis, namely

From Eq (5a), E (–x) = g (–x) +f (–x) the condition, Eq (7a), requires

(–x) Substituting Eq (5b) for g (–x) and Eq (6) for f

(9)

and thus the second condition, Eq (7b), requires E (–x) = 0 Since from

(–x) = g (–x) +f (–x) it follows that g (–x) where (–x) is given by Eq (4b) which for = becomes

the expansion in Eq (11) will be negligibly small and consequently f (–x) 0

(–x) = 0 is approximately satisfied when g (–x) in

Eq (10) equals zero, or

(–x) = 0 and

(–x) = –f

Equation (12) is satisfied when the cosine argument is given by /2 + 2m ,

m 0,1,2,3,

Į/ʌsin

ʌm2Į/ʌ2/ʌx

Į

(13)

Note that although Eq (12) is also satisfied at 3 /2 +2m , E,

zero when cosine is negative Substituting Eq (13) into Eq (8) and solving for

m yields:

4

1Į2

1Į

/ʌcotʌ2

Į12/ʌĮ/ʌcosmʌ2

Į/ʌsinĮln

m

Į Į 1 Į

Į

Į/ʌcotʌ2

A1lnĮm2

1m4

11lnĮ

1 i i

(14)

where

,3,2,1iĮ1iʌm2Į/ʌ2/ʌ1i

Į/ʌsinĮ1A

Į i

Į i i

1/ sin( / )] term in g ,1,1

g,1

Hanneken, Vaught, and Achar

with m = 0, 1, 2, 3, Solving Eq (12) for x yields

–15attempt to satisfy both Eqs (7a and 7b), the iteration process converges to a

value Note that m represents the number of relative maxima of E

(–x) as a sum of two functions g(–x) (–x) is negative for all x and is a completely and f(–x) The function f

(–x) is larger than (–x) it gives rise to a relative maximum in E (–x) above the x-axis This

(–x) cannot be

x

value of m such that x given by Eq (13) is just beyond the largest zero for that

(–x) from

results in two zeros for when cosine is positive and g

(–x) is larger than that of process continues as long as the magnitude of

(–x)

The A ’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 process until consecutive values of

nentially Each full period oscillation of the cos[x

(–x) exhibits oscillations with an amplitude which decays expo-

in only one zero during this interval Equations (14) and (15) are the main results of this paper

5 Accuracy of the Iteration Results

,1arbitrary in the range 1 < < 2 with some restrictions based on the number of significant digits in These restrictions result because of the approximate

< 1.42 but in this range E ,1

,1significant digits in are specified As gets closer to 2, can be specified to

an increasing number of significant digits However, an increased number of significant digits in does not guarantee the correct number of zeros, as

predict that E ,1

1.9796276 Thus, if is specified to 8 significant digits, must be 1.9796277

If is specified to a certain number of significant digits, Table 2 gives the range

of real zeros

(–x) < f (–x) their sum is less than zero and no more

(–x), n, is then given by

(–x) has decayed to less than f (–x), resulting

number of real zeros in this case can be easily enumerated by a brute-force

Using Eqs (14) and (15), the number of real zeros of E (–x) can be calculated for

Eqs (14) and (15) become more accurate When the value of deviated further from

2, the results from Eqs (14) and (15) become less accurate However, the total

illustrated by the following example For = 1.9796275, Eqs (14) and (15) correctly

(–x) will have 349 zeros However, at = 1.9796276, Eqs (14) and (15) incorrectly predict 349 zeros instead of the correct 351 At = 1.9796276,

to be guaranteed that Eqs (14) and (15) will predict the correct number of real zeros

of that will guarantee that the results of Eqs (14) and (15) yield the correct number

in Eq (11) improves as approaches 2 and consequently the results of using

technique described later Equations (14) and (15) do not yield reliable results for

1 < (–x) has only one real zero Equations (14) and (15)

do give the correct number of real zeros of E (–x) for 1.42 when at most 3

the approximations used in deriving Eqs (14) and (15) are not accurate enough

to discriminate between 349 zeros at = 1.9796275 and 351 zeros at =

6 Results and Conclusions

,1for various values of all of which have been verified by the brute force counting method Table 4 extends Table 3 to values of closer to 2 where the

,1arbitrary provided the restrictions on the number of significant digits specified

in are observed (Table 2)

Hanneken, Vaught, and Achar

Table 2 Ranges of reliability for the results

digits in results from Eqs (14) and (15)

Range of for reliable

Table 3 gives the number of real zeros of E (–x) computed from Eqs (14) and (15)

results of Eqs (14) and (15) are most accurate For values of not listed in either table, Eqs (14) and (15) correctly predict the number of real zeros of E (–x) for any

Table 3 Number of real zeros of E (–x)

Table 4 Number of real zeros of E (–x) for > 1.999

Number of real zeros 2

4

–– –5

10–6

–7 –8 –9 –10 –11 –12 –13 –14 –15 –16 –17 –18 –19 –20

de l’Academie des Sciences, Paris Series II, Vol 137, pp 554–558

Nazionale dei Lincei Series V, Vol 13, pp 3–5

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Riemann-Liouville Fractional Calculus, in: Kilbas AA (ed.), Boundary Value Problems, Special Functions and Fractional Calculus Belarusian State University,

26

29:217–234

10 Ostrovskii V, Peresyokkova IN (1997) Nonasymptotic results on distribution

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12 Sedletskii AM (2000) On zeros of functions of Mittag-Leffler type, Math Notes 68(5):602–613

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19 Mathematica Software System, Version 4, Wolfram Research, Champaign, IL

Hanneken, Vaught, and Achar

INITIALIZATION ISSUES RELATIVE TO

FRACTIONAL DIFFERENTIAL EQUATIONS

B N Narahari Achar1, Carl F Lorenzo2, and Tom T Hartley3

commonly held belief that the Caputo formulation of fractional derivatives properly accounts for the initialization effects is not generally true when applied

to the solution of fractional differential equations

be shown that the commonly held belief that the Caputo derivative properly

University of Memphis, Memphis, TN 38152; Tel: (901)678-3122, Fax: (901)678-4733, NASA Glenn Research Center, Cleveland, OH 44135

past (history) by means of an initialization function for the Riemann–Liouville

Lorenzo and Hartley (LH) [1,2] have clearly established the importance of time-

considered both the Riemann–Liouville (RL) and the Grunwald formulations of fractional calculus [3–6] in developing the initialization function [7] This paper

© 2007 Springer

and the Grunwald formulations of fractional calculus The present work add- resses this issue for the Caputo fractional derivative and cautions that the

Caputo fractional derivatives, initialization issues

in Physics and Engineering, 27– 42

27fractional derivative from the perspective of the Lorenzo–Hartley scheme It will E-mail: nachar@memphis.edu

J Sabatier et al (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications

Differintegral

Consider the following qth order fractional integrals of f t , the first integral

starting at time t a, and the second, starting at time t c a, respectively:

,,

)()()(

1)

q t f d

t

a

q q

t

and

.,

)()()(

1)

q t f d

t

c

q q

t

It is assumed that the function f t is zero for all t athe time interval between t a and t c being considered to be the “history” of the fractional integral c d t q f (t) Initialization consists in adding a function to the integral starting at time t c so that the result of fractional integration starting at time

c t d f t

q

c

a

the start time t c, will be considered here Then the generalized fractional

Achar, Lorenzo, and Hartley

2 Initialization of the Riemann–Liouville Fractional

t c,

integral, for arbitrary, real, and nonnegative values of v is defined by

section, initialization limitations of the Caputo derivative when applied to solu- tions of fractional differential equations are discussed

initialization”, in which case the integral can only be initialized prior to

Of the two types of initializations described by LH [7], only the “terminal