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Fractional Differential EquationsGuest Editors: Fawang Liu, Mark M. Meerschaert, Shaher Momani, Nikolai N. Leonenko, Wen Chen, and Om P. Agrawal

International Journal of Differential Equations

Fractional Differential Equations

International Journal of DifferentialEquations

Fractional Differential Equations

Guest Editors: Fawang Liu, Mark M. Meerschaert, Shaher Momani,

Nikolai N. Leonenko, Wen Chen, and Om P. Agrawal

Copyright q 2010 Hindawi Publishing Corporation. All rights reserved.

This is an issue published in volume 2010 of “International Journal of Differential Equations.” All articles are openaccess articles distributed under the Creative Commons Attribution License, which permits unrestricted use, distribu-tion, and reproduction in any medium, provided the original work is properly cited.

Editorial BoardOm Agrawal, USABashir Ahmad, Saudi ArabiaCherif Amrouche, FranceSabri Arik, TurkeyDumitru Baleanu, TurkeyNicola Bellomo, ItalyVieri Benci, ItalyElena Braverman, CanadaAlberto Cabada, SpainJinde Cao, ChinaDer Chen Chang, USAYang Chen, UKGui Qiang Chen, USAFengde Chen, ChinaCharles E. Chidume, ItalyY. J. E. Cho, Republic of KoreaI. D. Chueshov, UkraineShangbin Cui, ChinaToka Diagana, USAJinqiao Duan, USAM. El-Gebeily, Saudi ArabiaAhmed M. El-Sayed, EgyptKhalil Ezzinbi, MoroccoZhaosheng Feng, USADaniel Franco, SpainDavood Domiri Ganji, IranWeigao Ge, ChinaFritz Gesztesy, USAYoshikazu Giga, JapanJaume Gine, SpainJerome A. Goldstein, USASaid R. Grace, EgyptJohn R. Graef, USAMaurizio Grasselli, ItalyTasawar K. Hayat, PakistanJihuan He, ChinaEmmanuel Hebey, FranceHelge Holden, NorwayMayer Humi, USA

Elena I. Kaikina, MexicoQingkai Kong, USADe-Xing Kong, ChinaM. O. Korpusov, RussiaA. M. Krasnosel’skii, RussiaMiroslav Krstic, USAP. A. Krutitskii, RussiaM. R. S. Kulenovıc, USAKarl Kunisch, AustriaAlexander Kurganov, USAJose A. Langa, SpainPhilippe G. Lefloch, FranceNikolai N. Leonenko, UKYuji Liu, ChinaFawang Liu, AustraliaEduardo Liz, SpainWen Xiu Ma, USARuyun Ma, ChinaT. R. Marchant, AustraliaMarco Marletta, UKR. V. N. Melnik, CanadaS. A. Messaoudi, Saudi ArabiaStanisław Migorski, PolandAndro Mikelic, FranceShaher Momani, JordanGaston M. N’Guerekata, USAJuan Jose Nieto, SpainMuhammad A. Noor, PakistanSotiris K. Ntouyas, GreeceDonal O’Regan, IrelandJong Yeoul Park, KoreaPablo Pedregal, SpainKanishka Perera, USARodrigo Lopez Pouso, SpainRamon Quintanilla, SpainPatrick J. Rabier, USAYoussef N. Raffoul, USAThemistocles M. Rassias, GreeceYuri V. Rogovchenko, Sweden

Samir H. Saker, EgyptMartin Schechter, USAWilliam E. Schiesser, USALeonid Shaikhet, UkraineZhiqiang Shao, ChinaQin Sheng, USAJunping Shi, USAPanayiotis D. Siafarikas, GreecePeter L. Simon, HungaryStevo Stevic, SerbiaI. G. Stratis, GreeceJian-Ping Sun, ChinaGuido H. Sweers, GermanyNasser Eddine Tatar, Saudi ArabiaRoger M. Temam, USAGunther A. Uhlmann, USAJan Van Neerven, USAAghalaya S. Vatsala, USAPei Guang Wang, ChinaMingxin Wang, ChinaLihe Wang, USAMingxin Wang, ArgentinaZhi-Qiang Wang, USAThomas P. Witelski, USAGershon Wolansky, IsraelP.J.Y. Wong, SingaporeJen Chih Yao, TaiwanJingxue Yin, ChinaJianshe S. Yu, ChinaVyacheslav A. Yurko, RussiaQi S. Zhang, USASining Zheng, ChinaSongmu Zheng, ChinaYong Zhou, ChinaFeng Zhou, ChinaWenming Zou, ChinaXingfu Zou, Canada

Contents

Fractional Differential Equations, Fawang Liu, Mark M. Meerschaert, Shaher Momani,Nikolai N. Leonenko, Wen Chen, and Om P. AgrawalVolume 2010, Article ID 215856, 2 pages

The M-Wright Function in Time-Fractional Diffusion Processes: A Tutorial Survey,Francesco Mainardi, Antonio Mura, and Gianni PagniniVolume 2010, Article ID 104505, 29 pages

Stability and Convergence of an Effective Numerical Method for the Time-Space FractionalFokker-Planck Equation with a Nonlinear Source Term, Qianqian Yang, Fawang Liu,and Ian TurnerVolume 2010, Article ID 464321, 22 pages

He’s Variational Iteration Method for Solving Fractional Riccati Differential Equation,H. Jafari and H. TajadodiVolume 2010, Article ID 764738, 8 pages

Solitary Wave Solutions for a Time-Fraction Generalized Hirota-Satsuma Coupled KdVEquation by a New Analytical Technique, Majid Shateri and D. D. GanjiVolume 2010, Article ID 954674, 11 pages

Time-Optimal Control of Systems with Fractional Dynamics, Christophe Tricaud andYangQuan ChenVolume 2010, Article ID 461048, 16 pages

On the Speed of Spread for Fractional Reaction-Diffusion Equations, Hans EnglerVolume 2010, Article ID 315421, 16 pages

Riesz Potentials for Korteweg-de Vries Solitons and Sturm-Liouville Problems,Vladimir VarlamovVolume 2010, Article ID 193893, 18 pages

Linear Fractionally Damped Oscillator, Mark NaberVolume 2010, Article ID 197020, 12 pages

Positive Solution to Nonzero Boundary Values Problem for a Coupled System of NonlinearFractional Differential Equations, Jinhua Wang, Hongjun Xiang, and Zhigang LiuVolume 2010, Article ID 186928, 12 pages

The Periodic Solutions of the Compound Singular Fractional Differential System with Delay,XuTing Wei and XuanZhu LuVolume 2010, Article ID 509286, 9 pages

The Use of Fractional B-Splines Wavelets in Multiterms Fractional Ordinary DifferentialEquations, X. Huang and X. LuVolume 2010, Article ID 968186, 13 pages

Solvability of Nonlinear Langevin Equation Involving Two Fractional Orders with DirichletBoundary Conditions, Bashir Ahmad and Juan J. NietoVolume 2010, Article ID 649486, 10 pages

On the Selection and Meaning of Variable Order Operators for Dynamic Modeling,Lynnette E. S. Ramirez and Carlos F. M. CoimbraVolume 2010, Article ID 846107, 16 pages

Hindawi Publishing CorporationInternational Journal of Differential EquationsVolume 2010, Article ID 215856, 2 pagesdoi:10.1155/2010/215856

EditorialFractional Differential Equations

Fawang Liu,1 Mark M. Meerschaert,2 Shaher Momani,3Nikolai N. Leonenko,4 Wen Chen,5 and Om P. Agrawal6

1 School of Mathematical Sciences, Queensland University of Technology, P.O. Box 2434,Brisbane, Qeensland 4001, Australia

2 Department of Statistics and Probability, Michigan State University, A416 Wells Hall,East Lansing, MI 48823, USA

3 Department of Mathematics, Mu’tah University, P.O. Box 7, Mu’tah 61710, Jordan4 School of Mathematics, Cardiff University, Cardiff CF2 4YH, UK5 Department of Engineering Mechanics, Hohai University, Xikang Road Number 1,Nanjing, Jiangsu 210098, China

6 Department of Mechanical Engineering and Energy Processes, Southern Illinois University,Carbondale, IL 62901, USA

Correspondence should be addressed to Fawang Liu, [email protected]

Received 13 April 2010; Accepted 13 April 2010

Copyright q 2010 Fawang Liu et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

It is my pleasure to present this special issue of the International Journal of DifferentialEquations dedicated to Fractional Differential Equations (FDEs).

Fractional differential equations are generalizations of ordinary differential equationsto an arbitrary (noninteger) order. Fractional differential equations have attracted consider-able interest because of their ability to model complex phenomena. These equations capturenonlocal relations in space and time with power-law memory kernels. Due to the extensiveapplications of FDEs in engineering and science, research in this area has grown significantlyall around the world.

This special issue of Fractional Differential Equations consists of one review article(paper 1) and 12 original articles covering various aspects of FDEs and their applications bythe prominent researchers in the field.

The review article surveys the properties of a transcendental function of the Wrighttype, nowadays known as the M-Wright function, entering as a probability densityin a relevant class of self-similar stochastic processes. The second paper presents acomputationally effective numerical method for the time-space fractional Fokker-Planckequations with a nonlinear source term, together with a stability and convergence analysisof the numerical method.

2 International Journal of Differential Equations

The third paper uses He’s variational iteration method for solving the fractional Riccatidifferential equation. The fourth paper proposes a fractional iteration method and has beenapplied to study the Hirota-Satsuma coupled KdV of time-fractional-order equation.

The fifth paper develops a formulation for fractional time-optimal control problems.The sixth paper derives conditions for the speed of spread of solutions of fractionalscalar reaction-diffusion equations to be finite or infinite. The seventh paper studies theRiesz fractional derivatives for the Korteweg-de Vries solitons and proves that these Rieszpotentials and their Hilbert transforms are linearly independent solutions of a Sturm-Liouville problem.

The eighth paper gives the analytical solution of the linear fractionally dampedoscillator equation by means of Laplace transform. The ninth paper proves the existenceand uniqueness of a positive solution to the nonzero boundary value problem for a coupledsystem of fractional differential equations. The tenth paper gives sufficient conditions onthe existence of a periodic solution for a class of compound singular fractional differentialsystems with delay, involving Nishimoto fractional derivative. The eleventh paper discussesthe existence and uniqueness of the solutions of the nonhomogeneous linear differentialequations of arbitrary positive real order by using the fractional B-Splines wavelets andthe Mittag-Leffler function. The twelfth paper proves the existence of solutions of thefractional Langevin equation with two different fractional orders in a Banach space usingthe contraction mapping principle and Krasnoselskii’s fixed point theorem. The final paperaddresses the selection and meaning of variable-order operators for dynamic modelling.

Thus, this special issue provides a wide spectrum of current research in the area ofFDEs, and I hope that experts in this and related fields would find it useful.

Finally, I would like to acknowledge the guest editors Mark M. Meerschaert, ShaherMomani, Nikolai N. Leonenko, We Chen, and Om P. Agrawal for their assistance inproducing this special edition within the specified timeframe.

Fawang LiuMark M. Meerschaert

Shaher MomaniNikolai N. Leonenko

Wen ChenOm P. Agrawal


Review ArticleThe M-Wright Function in Time-FractionalDiffusion Processes: A Tutorial Survey

Francesco Mainardi,1 Antonio Mura,2 and Gianni Pagnini3

1 Department of Physics, University of Bologna and INFN, Via Irnerio 46, 40126 Bologna, Italy2 CRESME Ricerche S.p.A, Viale Gorizia 25C, 00199 Roma, Italy3 CRS4, Centro Ricerche Studi Superiori e Sviluppo in Sardegna, Polaris Building 1,09010 Pula, Italy

Correspondence should be addressed to Francesco Mainardi, [email protected]

Received 13 September 2009; Accepted 8 November 2009

Academic Editor: Fawang Liu

Copyright q 2010 Francesco Mainardi et al. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

In the present review we survey the properties of a transcendental function of the Wright type,nowadays known as M-Wright function, entering as a probability density in a relevant class ofself-similar stochastic processes that we generally refer to as time-fractional diffusion processes.Indeed, the master equations governing these processes generalize the standard diffusion equationby means of time-integral operators interpreted as derivatives of fractional order. When thesegeneralized diffusion processes are properly characterized with stationary increments, the M-Wright function is shown to play the same key role as the Gaussian density in the standard andfractional Brownian motions. Furthermore, these processes provide stochastic models suitable fordescribing phenomena of anomalous diffusion of both slow and fast types.

1. Introduction

By time-fractional diffusion processes, we mean certain diffusion-like phenomena governedby master equations containing fractional derivatives in time whose fundamental solutioncan be interpreted as a probability density function (pdf) in space evolving in time. Itis well known that, for the most elementary diffusion process, the Brownian motion, themaster equation, is the standard linear diffusion equation whose fundamental solution isthe Gaussian density with a spatial variance growing linearly in time. In such case we speakabout normal diffusion, reserving the term anomalous diffusion when the variance growsdifferently. A number of stochastic models for explaining anomalous diffusion have beenintroduced in literature; among them we like to quote the fractional Brownian motion; see, forexample, [1, 2], the Continuous Time Random Walk; see, for example, [3–6], the Levy flights;see, for example, [7], the Schneider grey Brownian motion; see [8, 9], and, more generally,


random walk models based on evolution equations of single and distributed fractional orderin time and/or space; see, for example, [10–18].

In this survey paper we focus our attention on modifications of the standard diffusionequation, where the time can be stretched by a power law (t → tα, 0 < α < 2) and the first-order time derivative can be replaced by a derivative of noninteger order β (0 < β ≤ 1). Inthese cases of generalized diffusion processes the corresponding fundamental solution stillkeeps the meaning of a spatial pdf evolving in time and is expressed in terms of a specialfunction of the Wright type that reduces to the Gaussian when β = 1. This transcendentalfunction, nowadays known as M-Wright function, will be shown to play a fundamentalrole for a general class of self-similar stochastic processes with stationary increments, whichprovide stochastic models for anomalous diffusion, as recently shown by Mura et al. [19–22].

In Section 2 we provide the reader with the essential notions and notations concerningthe integral transforms and fractional calculus, which are necessary in the rest of the paper.In Section 3 we introduce in the complex plane C the series and integral representations ofthe general Wright function denoted by Wλ,μ(z) and of the two related auxiliary functionsFν(z), Mν(z), which depend on a single parameter. In Section 4 we consider our auxiliaryfunctions in real domain pointing out their main properties involving their integrals and theirasymptotic representations. Mostly, we restrict our attention to the second auxiliary function,which we call M-Wright function, when its variable is in R

+ or in all of R but extended insymmetric way. We derive a fundamental formula for the absolute moments of this functionin R

+, which allows us to obtain its Laplace and Fourier transforms. In Section 5 we considersome types of generalized diffusion equations containing time partial derivatives of fractionalorder and we express their fundamental solutions in terms of the M-Wright functionsevolving in time with a given self-similarity law. In Section 6 we stress how the M-Wrightfunction emerges as a natural generalization of the Gaussian probability density for a classof self-similar stochastic processes with stationary increments, depending on two parameters(α, β). These processes are defined in a unique way by requiring the determination of anymultipoint probability distribution and include the well-known standard and fractionalBrownian motion. We refer to this class as the generalized grey Brownian motion (ggBm),because it generalizes the grey Brownian motion (gBm) introduced by Schneider [8, 9].Finally, a short concluding discussion is drawn. In Appendix A we derive the fundamentalsolution of the time-fractional diffusion equation. In Appendix B we outline the relevance ofthe M-Wright function in time-fractional drift processes entering as subordinators in time-fractional diffusion.

2. Notions and Notations

2.1. Integral Transforms Pairs

In our analysis we will make extensive use of integral transforms of Laplace, Fourier, andMellin types; so we first introduce our notation for the corresponding transform pairs. Wedo not point out the conditions of validity and the main rules, since they are given in anytextbook on advanced mathematics.

Let

˜f(s) = L{

f(r); r −→ s}

=∫∞

0e−srf(r)dr (2.1)

International Journal of Differential Equations 3

be the Laplace transform of a sufficiently well-behaved function f(r) with r ∈ R+, s ∈ C, and

let

f(r) = L−1{

˜f(s); s −→ r}

=1

2πi

∫

Bre+sr ˜f(s)ds (2.2)

be the inverse Laplace transform, where Br denotes the so-called Bromwich path, a straight

line parallel to the imaginary axis in the complex s-plane. Denoting by L←→ the justapositionof the original function f(r) with its Laplace transform ˜f(s), the Laplace transform pair reads

f(r) L←→ ˜f(s). (2.3)

Let

f(κ) = F{

f(x);x −→ κ}

=∫+∞

−∞e+iκxf(x)dx (2.4)

be the Fourier transform of a sufficiently well-behaved function f(x) with x ∈ R, κ ∈ R, andlet

f(x) = F−1{

f(κ);κ −→ x}

=1

2π

∫+∞

−∞e−iκx f(κ)dκ (2.5)

be the inverse Fourier transform. Denoting byF←→ the justaposition of the original function

f(x) with its Fourier transform f(κ), the Fourier transform pair reads

f(x)F←→ f(κ). (2.6)

Let

f∗(s) =M{

f(r); r −→ s}

=∫∞

0r s−1f(r)dr (2.7)

be the Mellin transform of a sufficiently well-behaved function f(r) with r ∈ R+, s ∈ C, and let

f(r) =M−1{f∗(s); s −→ r}

=1

2πi

∫

Brr−s f∗(s)ds (2.8)

be the inverse Mellin transform. Denoting by M←→ the justaposition of the original functionf(r) with its Mellin transform f∗(s), the Mellin transform pair reads

f(r) M←→ f∗(s). (2.9)


2.2. Essentials of Fractional Calculus with Support in R+

Fractional calculus is the branch of mathematical analysis that deals with pseudodifferentialoperators that extend the standard notions of integrals and derivatives to any positivenoninteger order. The term fractional is kept only for historical reasons. Let us restrictour attention to sufficiently well-behaved functions f(t) with support in R

+. Two mainapproaches exist in the literature of fractional calculus to define the operator of derivativeof noninteger order for these functions, referred to Riemann-Liouville and to Caputo. Bothapproaches are related to the so-called Riemann-Liouville fractional integral defined for anyorder μ > 0 as

Jμt f(t) :=

1Γ(

μ)

∫ t

0(t − τ)μ−1f(τ)dτ. (2.10)

We note the convention J0t = I (Identity) and the semigroup property

Jμt J

νt = Jνt J

μt = Jμ+νt , μ ≥ 0, ν ≥ 0. (2.11)

The fractional derivative of order μ > 0 in the Riemann-Liouville sense is defined as theoperator Dμ

t which is the left inverse of the Riemann-Liouville integral of order μ (in analogywith the ordinary derivative), that is,

Dμt J

μt = I, μ > 0. (2.12)

If m denotes the positive integer such that m − 1 < μ ≤ m, we recognize, from (2.11) and(2.12), Dμ

t f(t) := Dmt J

m−μt f(t); hence,

Dμt f(t) =

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

dm

dtm

[

1Γ(

m − μ)

∫ t

0

f(τ)dτ

(t − τ)μ+1−m

]

, m − 1 < μ < m,

dm

dtmf(t), μ = m.

(2.13)

For completeness we define D0t = I.

On the other hand, the fractional derivative of order μ > 0 in the Caputo sense is definedas the operator ∗D

μt such that ∗D

μt f(t) := J

m−μt Dm

t f(t); hence,

∗Dμt f(t) =

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

1Γ(

m − μ)

∫ t

0

f (m)(τ)dτ

(t − τ)μ+1−m , m − 1 < μ < m,

dm

dtmf(t), μ = m.

(2.14)


We note the different behavior of the two derivatives in the limit μ → (m − 1)+. In fact,

μ → (m − 1)+

⎧

⎨

⎩

Dμt f(t) −→ Dm

t J1t f(t) = D

(m−1)t f(t),

∗Dμt f(t) −→ J1

t Dmt f(t) = D

(m−1)t f(t) −D(m−1)

t f(0+),(2.15)

where the limit for t → 0+ is taken after the operation of derivation.Furthermore, recalling the Riemann-Liouville fractional integral and derivative of the

power law for t > 0,

Jμt t

γ =Γ(

γ + 1)

Γ(

γ + 1 + μ) tγ+μ,

Dμt tγ =

Γ(

γ + 1)

Γ(

γ + 1 − μ) tγ−μ,

μ > 0, γ > −1, (2.16)

we find the relationship between the two types of fractional derivative

Dμ

[

f(t) −m−1∑

k=0

tk

k!f (k)(0+)

]

= ∗Dμt f(t). (2.17)

We note that the Caputo definition for the fractional derivative incorporates the initial valuesof the function and of its integer derivatives of lower order. The subtraction of the Taylorpolynomial of degree m − 1 at t = 0+ from f(t) is a sort of regularization of the fractionalderivative. In particular, according to this definition, the relevant property that the derivativeof a constant is zero is preserved for the fractional derivative.

Let us finally point out the rules for the Laplace transform with respect to the fractionalintegral and the two fractional derivatives. These rules are expected to properly generalizethe well-known rules for standard integrals and derivatives.

For the Riemann-Liouville fractional integral, we have

L{

Jμt f(t); t −→ s

}

=˜f(s)sμ

, μ ≥ 0. (2.18)

For the Caputo fractional derivative, we consequently get

L{

∗Dμt f(t); t −→ s

}

= sμ ˜f(s) −m−1∑

k=0

sμ−1−kf (k)(0+), m − 1 < μ ≤ m, (2.19)

where f (k)(0+) := limt→ 0+f(k)(t). The corresponding rule for the Riemann-Liouville fractional

derivative is more cumbersome and it reads

L{

Dμt f(t); t −→ s

}

= sμ ˜f(s) −m−1∑

k=0

[

Dkt J

(m−μ)t

]

f(0+)sm−1−k, m − 1 < μ ≤ m, (2.20)


where the limit for t → 0+ is understood to be taken after the operations of fractionalintegration and derivation. As soon as all the limiting values f (k)(0+) are finite and m − 1 <μ < m, formula (2.20) for the Riemann-Liouville derivative is simplified into

L{

Dμt f(t); t −→ s

}

= sμ ˜f(s), m − 1 < μ < m. (2.21)

In the special case f (k)(0+) = 0 for k = 0, 1, m − 1, we recover the identity between thetwo fractional derivatives. The Laplace transform rule (2.19) was practically the key resultof Caputo [23, 24] in defining his generalized derivative in the late sixties. The two fractionalderivatives have been well discussed in the 1997 survey paper by Gorenflo and Mainardi[25]; see also [26], and in the 1999 book by Podlubny [27]. In these references the authorshave pointed out their preference for the Caputo derivative in physical applications whereinitial conditions are usually expressed in terms of finite derivatives of integer order.

For further reading on the theory and applications of fractional calculus, werecommend the recent treatise by Kilbas et al. [28].

3. The Functions of the Wright Type

3.1. The General Wright Function

The Wright function, which we denote by Wλ,μ,(z), is so named in honor of E. MaitlandWright, the eminent British mathematician, who introduced and investigated this functionin a series of notes starting from 1933 in the framework of the asymptotic theory of partitions;see [29–31]. The function is defined by the series representation, convergent in the wholez-complex plane

Wλ,μ(z) :=∞∑

n=0

zn

n! Γ(

λn + μ) , λ > −1, μ ∈ C. (3.1)

Originally, Wright assumed that λ ≥ 0, and, only in 1940 [32], he considered −1 < λ < 0.We note that in Chapter 18 of Vol. 3 of the handbook of the Bateman Project [33], devoted toMiscellaneous Functions, presumably for a misprint, the parameter λ of the Wright function isrestricted to be nonnegative. When necessary, we propose to distinguish the Wright functionsin two kinds according to λ ≥ 0 (first kind) and −1 < λ < 0 (second kind).

For more details on Wright functions the reader can consult, for example, [34–41] andreferences therein.

The integral representation of the Wright function reads

Wλ,μ(z) =1

2πi

∫

Haeσ+zσ

−λ dσ

σμ, λ > −1, μ ∈ C, (3.2)

where Ha denotes the Hankel path. We remind that the Hankel path is a loop that starts from−∞ along the lower side of the negative real axis, encircles the circular area around the originwith radius ε → 0 in the positive sense, and ends at −∞ along the upper side of the negative


real axis. The equivalence of the series and integral representations is easily proved usingHankel formula for the Gamma function

1Γ(ζ)

=∫

Haeuu−ζdu, ζ ∈ C (3.3)

and performing a term-by-term integration. In fact,

Wλ,μ(z) =1

2πi

∫

Haeσ+zσ

−λ dσ

σμ=

12πi

∫

Haeσ[

∞∑

n=0

zn

n!σ−λn

]

dσ

σμ

=∞∑

n=0

zn

n!

[

12πi

∫

Haeσσ−λn−μdσ

]

=∞∑

n=0

zn

n! Γ[

λn + μ] .

(3.4)

It is possible to prove that the Wright function is entire of order 1/(1 + λ); hence, it is ofexponential type only if λ ≥ 0 (which corresponds to Wright function of the first kind). Thecase λ = 0 is trivial since W0,μ(z) = e z/Γ(μ), provided that μ/= 0,−1,−2, . . . .

3.2. The Auxiliary Functions of the Wright Type

Mainardi, in his first analysis of the time-fractional diffusion equation [42, 43], aware of theBateman handbook [33], but not yet of the 1940 paper by Wright [32], introduced the two(Wright-type) entire auxiliary functions,

Fν(z) :=W−ν,0(−z), 0 < ν < 1,

Mν(z) :=W−ν,1−ν(−z), 0 < ν < 1,(3.5)

interrelated through

Fν(z) = νzMν(z). (3.6)

As a matter of fact, functions Fν(z) and Mν(z) are particular cases of the Wrightfunction of the second kind Wλ,μ(z) by setting λ = −ν and μ = 0 or μ = 1, respectively.

Hereafter, we provide the series and integral representations of the two auxiliaryfunctions derived from the general formulas (3.1) and (3.2), respectively.

The series representations for the auxiliary functions read

Fν(z) :=∞∑

n=1

(−z)n

n! Γ(−νn) =1π

∞∑

n=1

(−z)n−1

n!Γ(νn + 1) sin(πνn), (3.7)

Mν(z) :=∞∑

n=0

(−z)n

n! Γ[−νn + (1 − ν)] =1π

∞∑

n=1

(−z)n−1

(n − 1)!Γ(νn) sin(πνn). (3.8)

The second series representations in (3.7)-(3.8) have been obtained by using the reflectionformula for the Gamma function Γ(ζ) Γ(1 − ζ) = π/ sin πζ.


As an exercise, the reader can directly prove that the radius of convergence of thepower series in (3.7)-(3.8) is infinite for 0 < ν < 1 without being aware of Wright’s results, asit was shown independently by Mainardi [42]; see also [27].

Furthermore, we have Fν(0) = 0 and Mν(0) = 1/Γ(1 − ν). We note that relation (3.6)between the two auxiliary functions can be easily deduced from (3.7)-(3.8), by using the basicproperty of the Gamma function Γ(ζ + 1) = ζ Γ(ζ).

The integral representations for the auxiliary functions read

Fν(z) :=1

2πi

∫

Haeσ−zσ

ν

dσ, (3.9)

Mν(z) :=1

2πi

∫

Haeσ−zσ

ν dσ

σ1−ν . (3.10)

We note that relation (3.6) can be obtained also from (3.9)-(3.10) with an integration by parts.In fact,

Mν(z) =∫

Haeσ−zσ

ν dσ

σ1−ν =∫

Haeσ(

− 1νz

d

dσe−zσ

ν

)

dσ

=1νz

∫

Haeσ−zσ

ν

dσ =Fν(z)νz

.

(3.11)

The equivalence of the series and integral representations is easily proved by using theHankel formula for the Gamma function and performing a term-by-term integration.

3.3. Special Cases

Explicit expressions of Fν(z) and Mν(z) in terms of known functions are expected for someparticular values of ν. Mainardi and Tomirotti [43] have shown that for ν = 1/q, whereq ≥ 2 is a positive integer, the auxiliary functions can be expressed as a sum of simpler (q− 1)entire functions. In the particular cases q = 2 and q = 3, we find

M1/2(z) =1√π

∞∑

m=0(−1)m

(

12

)

m

z2m

(2m)!=

1√π

exp

(

−z2

4

)

, (3.12)

M1/3(z) =1

Γ(2/3)

∞∑

m=0

(

13

)

m

z3m

(3m)!− 1Γ(1/3)

∞∑

m=0

(

23

)

m

z3m+1

(3m + 1)!

= 32/3 Ai(

z

31/3

)

,

(3.13)

where Ai denotes the Airy function.


Furthermore, it can be proved that M1/q(z) satisfies the differential equation of orderq − 1

dq−1

dzq−1M1/q(z) +

(−1)q

qzM1/q(z) = 0, (3.14)

subjected to the q − 1 initial conditions at z = 0, derived from (3.14), such that

M(h)1/q(0) =

(−1)h

Γ[

1 − (h + 1)/q] =

(−1)h

πΓ[

h + 1q

]

sin[

π(h + 1)q

]

, (3.15)

with h = 0, 1, . . . , q − 2. We note that, for q ≥ 4, (3.14) is akin to the hyper-Airy differentialequation of order q − 1; see, for example, [44]. Consequently, the auxiliary function Mν(z)could be considered a sort of generalized hyper-Airy function. However, in view of furtherapplications in stochastic processes, we prefer to consider it as a natural (fractional)generalization of the Gaussian function, similarly as the Mittag-Leffler function is knownto be the natural (fractional) generalization of the exponential function.

To stress the relevance of the auxiliary function Mν(z), it was also suggested thespecial name M-Wright function, a terminology that has been followed in literature to someextent.

Some authors including Podlubny [27], Gorenflo et al. [34, 35], Hanyga [45], Balescu[46], Chechkin et al. [12], Germano et al. [47], and Kiryakova [48, 49] refer to the M-Wrightfunction as the Mainardi function. It was Professor Stankovic, during the presentation of thepaper by Mainardi and Tomirotti [43] at the Conference of Transform Methods and SpecialFunctions, Sofia 1994, who informed Mainardi, being aware only of the Bateman Handbook[33], that the extension for −1 < λ < 0 had been already made just by Wright himself in1940 [32], following his previous papers published in the thirties. Mainardi, in the paper[50] devoted to the 80th birthday of Prof. Stankovic, used the occasion to renew his personalgratitude to Prof. Stankovic for this earlier information that led him to study the originalpapers by Wright and work also in collaboration on the functions of the Wright type forfurther applications; see, for example, [34, 35, 51].

Moreover, the analysis of the limiting cases ν = 0 and ν = 1 requires special attention.For ν = 0 we easily recognize from the series representations (3.7)-(3.8)

F0(z) ≡ 0, M0(z) = e−z. (3.16)

The limiting case ν = 1 is singular for both auxiliary functions as expected from the definitionof the general Wright function when λ = −ν = −1. Later we will deal with this singular casefor the M-Wright function when the variable is real and positive.

4. Properties and Plots of the Auxiliary Wright Functionsin Real Domain

Let us state some relevant properties of the auxiliary Wright functions, with special attentionto the Mν function in view of its role in time-fractional diffusion processes.


4.1. Exponential Laplace Transforms

We start with the Laplace transform pairs involving exponentials in the Laplace domain.These were derived by Mainardi in his earlier analysis of the time-fractional diffusionequation; see, for example, [42, 52],

1rFν

(

1rν

)

=ν

rν+1Mν

(

1rν

)

L←→ e−sν

, 0 < ν < 1, (4.1)

1νFν

(

1rν

)

=1rνMν

(

1rν

)

L←→ e−sν

s1−ν , 0 < ν < 1. (4.2)

We note that the inversion of the Laplace transform of the exponential exp(−sν) is relevantsince it yields for any ν ∈ (0, 1) the unilateral extremal stable densities in probability theory,denoted by L−νν (r) in [53]. As a consequence, the nonnegativity of both auxiliary Wrightfunctions when their argument is positive is proved by the Bernstein theorem. We refer toFeller’s treatise [54, 55] for Laplace transforms, stable densities and Bernstein theorem. TheLaplace transform pair in (4.1) has a long history starting from a formal result by Humbert[56] in 1945, of which Pollard [57] provided a rigorous proof one year later. Then, in 1959Mikusinski [58] derived a similar result on the basis of his theory of operational calculus. In1975, albeit unaware of the previous results, Buchen and Mainardi [59] derived the result in aformal way. We note that all the above authors were not informed about the Wright functions.To our actual knowledge the former author who derived the Laplace transforms pairs (4.1)-(4.2) in terms of Wright functions of the second kind was Stankovic in 1970; see [39].

Hereafter we would like to provide two independent proofs of (4.1) carrying out theinversion of exp(−sν), either by the complex Bromwich integral formula following [42], or bythe formal series method following [59]. Similarly we can act for the Laplace transform pair(4.2). For the complex integral approach we deform the Bromwich path Br into the Hankelpath Ha, that is equivalent to the original path, and we set σ = sr. Recalling the integralrepresentation (3.9) for the Fν function and (3.6), we get

L−1[exp(−sν); s −→ r]

=1

2πi

∫

Bresr−s

ν

ds =1

2πi r

∫

Haeσ−(σ/r)

ν

dσ

=1rFν

(

1rν

)

=ν

rν+1Mν

(

1rν

)

.

(4.3)

Expanding in power series the Laplace transform and inverting term by term, we formallyget

L−1[exp(−sν)]

=∞∑

n=0

(−1)n

n!L−1[sνn] =

∞∑

n=1

(−1)n

n!r−νn−1

Γ(−νn)

=1rFν

(

1rν

)

=ν

rν+1Mν

(

1rν

)

,

(4.4)

where now we have used the series representation (3.7) for the function Fν along with therelationship formula (3.6).


4.2. Asymptotic Representation for Large Argument

Let us point out the asymptotic behavior of the function Mν(r) when r → ∞. Choosing avariable r/ν rather than r, the computation of the desired asymptotic representation by thesaddle-point approximation is straightforward. Mainardi and Tomirotti [43] have obtained

Mν

(

r

ν

)

∼ a(ν)r(ν−1/2)/(1−ν) exp[

−b(ν)r1/(1−ν)],

a(ν) =1

√

2π(1 − ν)> 0, b(ν) =

1 − νν

> 0.(4.5)

The above evaluation is consistent with the first term in the asymptotic series expansionprovided by Wright with a cumbersome and formal procedure for his general function Wλ,μ

when −1 < λ < 0; see [32]. In 1999 Wong and Zhao have derived asymptotic expansions ofthe Wright functions of the first and second kind in the whole complex plane following a newmethod for smoothing Stokes’ discontinuities; see [40, 41], respectively.

We note that, for ν = 1/2 as (4.5) provides the exact result consistent with (3.12),

M1/2(2r) =1√π

e−r2 ⇐⇒M1/2(r) =

1√π

e−r2/4. (4.6)

We also note that in the limit ν → 1− the function Mν(r) tends to the Dirac generalizedfunction δ(r − 1), as can be recognized also from the Laplace transform pair (4.1).

4.3. Absolute Moments

From the above considerations we recognize that, for the M-Wright functions, the followingrule for absolute moments in R

+ holds

∫∞

0rδMν(r)dr =

Γ(δ + 1)Γ(νδ + 1)

, δ > −1, 0 ≤ ν < 1. (4.7)

In order to derive this fundamental result, we proceed as follows on the basis of the integralrepresentation (3.10):

∫∞

0rδMν(r)dr =

∫∞

0rδ[

12πi

∫

Haeσ−rσ

ν dσ

σ1−ν

]

dr

=1

2πi

∫

Haeσ[∫∞

0e−rσ

ν

rδdr

]

dσ

σ1−ν

=Γ(δ + 1)

2πi

∫

Ha

eσ

σνδ+1dσ =

Γ(δ + 1)Γ(νδ + 1)

.

(4.8)


Above we have legitimized the exchange between integrals and used the identity

∫∞

0e−rσ

ν

rδdr =Γ(δ + 1)

(σν)δ+1, (4.9)

along with the Hankel formula of the Gamma function. Analogously, we can compute all themoments of Fν(r) in R

+.

4.4. The Laplace Transform of the M-Wright Function

Let the Mittag-Leffler function be defined in the complex plane for any ν ≥ 0 by the followingseries and integral representation; see, for example, [33, 60]:

Eν(z) =∞∑

n=0

zn

Γ(νn + 1)=

12πi

∫

Ha

ζν−1 e ζ

ζν − z dζ, ν > 0, z ∈ C. (4.10)

Such function is entire of order 1/α for α > 0 and reduces to the function exp(z) for ν > 0 andto 1/(1 − z) for ν = 0. We recall that the Mittag-Leffler function for ν > 0 plays fundamentalroles in applications of fractional calculus like fractional relaxation and fractional oscillation;see, for example, [25, 26, 61, 62], so that it could be referred as the Queen function of fractionalcalculus. Recently, numerical routines for functions of Mittag-Leffler type have been provided,e.g., by Freed et al. [63], Gorenflo et al. [64] (with MATHEMATICA), Podlubny [65] (withMATLAB), Seybold and Hilfer [66].

We now point out that the M-Wright function is related to the Mittag-Leffler functionthrough the following Laplace transform pair:

Mν(r)L←→ Eν(−s), 0 < ν < 1. (4.11)

For the reader’s convenience, we provide a simple proof of (4.11) by using two differentapproaches. We assume that the exchanges between integrals and series are legitimate inview of the analyticity properties of the involved functions. In the first approach we use theintegral representations of the two functions obtaining

∫∞

0e−srMν(r)dr =

12πi

∫∞

0e−s r[∫

Haeσ−rσ

ν dσ

σ 1−ν

]

dr

=1

2πi

∫

Haeσσν−1

[∫∞

0e−r(s+σ

ν)dr

]

dσ

=1

2πi

∫

Ha

eσσν−1

σν + sdσ = Eν(−s).

(4.12)


In the second approach we develop in series the exponential kernel of the Laplace transformand we use the expression (4.7) for the absolute moments of the M-Wright function arrivingto the following series representation of the Mittag-Leffler function:

∫∞

0e−srMν(r)dr =

∞∑

n=0

(−s)n

n!

∫∞

0rnMν(r)dr

=∞∑

n=0

(−s)n

n!Γ(n + 1)Γ(νn + 1)

=∞∑

n=0

(−s)n

Γ(νn + 1)= Eν(−s).

(4.13)

We note that the transformation term by term of the series expansion of the M-Wrightfunction is not legitimate because the function is not of exponential order; see [67]. However,this procedure yields the formal asymptotic expansion of the Mittag-Leffler function Eν(−s)as s → ∞ in a sector around the positive real axis; see, for example, [33, 60], that is,

∞∑

n=0

∫∞0 e−sr(−r)ndr

n!Γ(−νn + (1 − ν)) =∞∑

n=0

(−1)n

Γ(−νn + 1 − ν)1sn+1

=∞∑

m=1

(−1)m−1

Γ(−νm + 1)1sm∼ Eν(−s), s −→ ∞.

(4.14)

4.5. The Fourier Transform of the Symmetric M-Wright Function

The M-Wright function, extended on the negative real axis as an even function, is related tothe Mittag-Leffler function through the following Fourier transform pair:

Mν(|x|)F←→ 2E2ν

(

−κ2)

, 0 < ν < 1. (4.15)

Below, we prove the equivalent formula

∫∞

0cos(κr)Mν(r)dr = E2ν

(

−κ2)

. (4.16)

For the proof it is sufficient to develop in series the cosine function and use formula (4.7) forthe absolute moments of the M-Wright function:

∫∞

0cos(κr)Mν(r)dr =

∞∑

n=0(−1)n

κ2n

(2n)!

∫∞

0r2nMν(r)dr

=∞∑

n=0(−1)n

κ2n

Γ(2νn + 1)= E2ν

(

−κ2)

.

(4.17)


0

0.2

0.4

0.6

0.8

1

−5 −4 −3 −2 −1 0 1 2 3 4 5

Mν(x)

x

ν = 0ν = 1/8ν = 1/4ν = 3/8ν = 1/2

(a)

0

10−2

10−1

100

−5 −4 −3 −2 −1 0 1 2 3 4 5

Mν(x)

x

ν = 0ν = 1/8ν = 1/4ν = 3/8ν = 1/2

(b)

Figure 1: Plots of the symmetric Mν-Wright function with ν = 0, 1/8, 1/4, 3/8, 1/2 for−5 ≤ x ≤ 5: (a) linearscale, (b) logarithmic scale.

4.6. The Mellin Transform of the M-Wright Function

It is straightforward to derive the Mellin transform of the M-Wright function using result(4.7) for the absolute moments of the M-Wright function. In fact, setting δ = s − 1 in (4.7), byanalytic continuation it follows:

Mν(r)M←→ Γ(s)

Γ(ν(s − 1) + 1), 0 < ν < 1. (4.18)

4.7. Plots of the Symmetric M-Wright Function

It is instructive to show the plots of the (symmetric) M-Wright function on the real axis forsome rational values of the parameter ν. In order to have more insight of the effect of theparameter itself on the behavior close to and far from the origin, we adopt both linear andlogarithmic scale for the ordinates.

In Figures 1 and 2 we compare the plots of the Mν(x)-Wright functions in−5 ≤ x ≤ 5for some rational values of ν in the ranges ν ∈ [0, 1/2] and ν ∈ [1/2, 1], respectively. InFigure 1 we see the transition from exp(−|x|) for ν = 0 to 1/

√π exp(−x2) for ν = 1/2, whereas

in Figure 2 we see the transition from 1/√π exp(−x2) to the delta functions δ(x± 1) for ν = 1.

Because of the two symmetrical humps for 1/2 < ν ≤ 1, the Mν function appears bimodalwith the characteristic shape of the capital letter M.

In plotting Mν(x) at fixed ν for sufficiently large x, the asymptotic representation(4.5)-(4.6) is useful since, as x increases, the numerical convergence of the series in (3.8)decreases up to being completely inefficient: henceforth, the matching between the seriesand the asymptotic representation is relevant and followed by Mainardi and associates; see,for example, [38, 51, 53, 68]. However, as ν → 1−, the plotting remains a very difficulttask because of the high peak arising around x = ±1. For this we refer the reader to the1997 paper by Mainardi and Tomirotti [69], where a variant of the saddle point method hasbeen successfully used to properly depict the transition to the delta functions δ(x ± 1) as νapproaches 1. For the numerical point of view we like to highlight the recent paper by Luchko[70], where algorithms are provided for computation of the Wright function on the real axiswith prescribed accuracy.


0

0.2

0.4

0.6

0.8

1

−5 −4 −3 −2 −1 0 1 2 3 4 5

Mν(x)

x

ν = 1/2ν = 5/8ν = 3/4ν = 1

(a)

0

10−2

10−1

100

−5 −4 −3 −2 −1 0 1 2 3 4 5

Mν(x)

x

ν = 1/2ν = 5/8ν = 3/4ν = 1

(b)

Figure 2: Plots of the symmetric M-Wright function with ν = 1/2, 5/8, 3/4, 1 for−5 ≤ x ≤ 5: (a) linear scale,(b) logarithmic scale.

4.8. The M-Wright Function in Two Variables

In view of the time-fractional diffusion processes that will be considered in the next sections,it is worthwhile to introduce the function in the two variables

Mν(x, t) := t−νMν

(

xt−ν)

, 0 < ν < 1, x, t ∈ R+, (4.19)

which defines a spatial probability density in x evolving in time t with self-similarityexponent H = ν. Of course for x ∈ R we have to consider the symmetric version obtainedfrom (4.19) multiplying by 1/2 and replacing x by |x|.

Hereafter we provide a list of the main properties of this function, which can bederived from Laplace and Fourier transforms of the corresponding M-Wright function inone variable.

From (4.2) we derive the Laplace transform of Mν(x, t) with respect to t ∈ R+ as

L{Mν(x, t); t −→ s} = sν−1e−xsν

. (4.20)

From (4.10) we derive the Laplace transform of Mν(x, t) with respect to x ∈ R+ as

L{Mν(x, t);x −→ s} = Eν(−stν). (4.21)

From (4.15) we derive the Fourier transform of Mν(|x|, t) with respect to x ∈ R as

F{Mν(|x|, t);x → κ} = 2E2ν

(

−κ2tν)

. (4.22)

Moreover, using the Mellin transform, Mainardi et al. [71] derived the following integralformula:

Mν(x, t) =∫∞

0Mλ(x, τ)Mμ(τ, t)dτ, ν = λμ. (4.23)


Special cases of the M-Wright function are simply derived for ν = 1/2 and ν = 1/3 from thecorresponding ones in the complex domain; see (3.12)-(3.13). We devote particular attentionto the case ν = 1/2 for which we get from (4.6) the Gaussian density in R:

M1/2(|x|, t) =1

2√πt1/2

e−x2/(4t). (4.24)

For the limiting case ν = 1 we obtain

M1(|x|, t) =12[δ(x − t) + δ(x + t)]. (4.25)

5. Fractional Diffusion Equations

Let us now consider a variety of diffusion-like equations starting from the standard diffusionequation whose fundamental solutions are expressed in terms of the M-Wright functiondepending on space and time variables. The two variables, however, turn out to be relatedthrough a self-similarity condition.

5.1. The Standard Diffusion Equation

The standard diffusion equation for the field u(x, t) with initial condition u(x, 0) = u0(x) is

∂u

∂t= K1

∂2u

∂x2, −∞ < x <∞, t ≥ 0, (5.1)

where K1 is a suitable diffusion coefficient of dimensions [K1] = [L]2[T]−1 = cm2/sec. Thisinitial-boundary value problem can be easily shown to be equivalent to the Volterra integralequation

u(x, t) = u0(x) +K1

∫ t

0

∂2u(x, τ)∂x2

dτ. (5.2)

It is well known that the fundamental solution (usually referred as the Green function), whichis the solution corresponding to u0(x) = δ(x), is the Gaussian probability density evolvingin time with variance (mean square displacement) proportional to time. In our notation wehave

G1(x, t) =1

2√

πK1t1/2e−x

2/(4K1t), (5.3)

σ21(t) :=

∫+∞

−∞x2G1(x, t)dx = 2K1t. (5.4)

This variance law characterizes the process of normal diffusion as it emerges from Einstein’sapproach to Brownian motion (Bm); see, for example, [72].


In view of future developments, we rewrite the Green function in terms of the M-Wright function by recalling (3.12), that is,

G1(x, t) =12

1√

K1t1/2M1/2

(

|x|√

K1t1/2

)

. (5.5)

From the self-similarity of the Green function in (5.3) or (5.5), we are led to write

G1(x, t) =1

√

K1tHG1

(

|x|√

K1tH, 1

)

, (5.6)

where H = 1/2 is the similarity (or Hurst) exponent and ξ = |x|/(√

K1t1/2) acts as the

similarity variable. We refer to the one-variable function G1(ξ) as the reduced Green function.

5.2. The Stretched-Time Standard Diffusion Equation

Let us now stretch the time variable in (5.1) by replacing t with tα where 0 < α < 2. We have

∂u

∂(tα)= Kα

∂2u

∂x2, −∞ < x < +∞, t ≥ 0, (5.7)

where Kα is a sort of stretched diffusion coefficient of dimensions [Kα] = [L]2[T]−α =cm2/secα. It is easy to recognize that such equation is akin to the standard diffusion equationbut with a diffusion coefficient depending on time; K1(t) = αtα−1Kα. In fact, using the rule

∂

∂tα=

1αtα−1

∂

∂t, (5.8)

we have

∂u

∂t= αtα−1Kα

∂2u

∂x2, −∞ < x < +∞, t ≥ 0. (5.9)

The integral form corresponding to (5.7)—(5.9) reads

u(x, t) = u0(x) + αKα

∫ t

0

∂2u(x, τ)∂x2

τα−1dτ. (5.10)

The corresponding fundamental solution is the stretched-time Gaussian

Gα(x, t) =1

2√

πKαtα/2e−x

2/(4Kαtα) =

12

1√

Kαtα/2M1/2

(

|x|√

Kαtα/2

)

. (5.11)


The corresponding variance

σ2α(t) :=

∫+∞

−∞x2Gα(x, t)dx = 2Kαt

α (5.12)

is characteristic of a general process of anomalous diffusion, precisely of slow diffusion for 0 <α < 1, and of fast diffusion for 1 < α < 2.

5.3. The Time-Fractional Diffusion Equation

In literature there exist two forms of the time-fractional diffusion equation of a single order,one with Riemann-Liouvile derivative and one with Caputo derivative. These forms areequivalent if we refer to the standard initial condition u(x, 0) = u0(x), as shown in [73].

Taking a real number β ∈ (0, 1), the time-fractional diffusion equation of order β in theRiemann-Liouville sense reads

∂u

∂t= KβD

1−βt

∂2u

∂x2, (5.13)

whereas in the Caputo sense reads

∗Dβt u = Kβ

∂2u

∂x2, (5.14)

where Kβ is a sort of fractional diffusion coefficient of dimensions [Kβ] = [L]2[T]−β =cm2/secβ. Like for diffusion equations of integer order (5.1) and (5.7)-(5.9), we consider theequivalent integral equation corresponding to our fractional diffusion equations (5.13)-(5.14)as

u(x, t) = u0(x) +Kβ1

Γ(

β)

∫ t

0(t − τ)β−1 ∂

2u(x, τ)∂x2

dτ. (5.15)

The Green function Gβ(x, t) for the equivalent (5.13)–(5.15) can be expressed, also in thiscase, in terms of the M-Wright function, as shown in Appendix A by adopting two differentapproaches, as follows:

Gβ(x, t) =12

1√

Kβtβ/2Mβ/2

⎛

⎜

⎝

|x|√

Kβtβ/2

⎞

⎟

⎠. (5.16)

The corresponding variance can be promptly obtained from the general formula (5.5) forthe absolute moment of the M-Wright function. In fact, using (5.5) and (5.16) and after anobvious change of variable, we obtain

σ2β(t) :=

∫+∞

−∞x2Gβ(x, t)dx =

2Γ(

β + 1)Kβt

β. (5.17)


As a consequence, for 0 < β < 1 the variance is consistent with a process of slow diffusion withsimilarity exponent H = β/2. For further reading on time-fractional diffusion equations andtheir solutions the reader is referred, among others, to [38, 51, 53] and [74, 75].

5.4. The Stretched Time-Fractional Diffusion Equation

In the fractional diffusion equation (5.13), let us stretch the time variable by replacing t withtα/β where 0 < α < 2 and 0 < β ≤ 1. We have

∂u

∂tα/β= KαβD

1−βtα/β

∂2u

∂x2, (5.18)

namely,

∂u

∂t=α

βtα/β−1KαβD

1−βtα/β

∂2u

∂x2, (5.19)

where Kαβ is a sort of stretched diffusion coefficient of dimensions [Kαβ] = [L]2[T]−α =cm2/secα that reduces to Kα if β = 1 and to Kβ if α = β. Integration of (5.19) gives thecorresponding integral equation [21]

u(x, t) = u0(x) +Kαβ1

Γ(

β)

α

β

∫ t

0τα/β−1

(

tα/β − τα/β)β−1 ∂2u(x, τ)

∂x2dτ, (5.20)

whose Green function Gαβ(x, t) is

Gαβ(x, t) =12

1√

Kαβtα/2Mβ/2

⎛

⎜

⎝

|x|√

Kαβtα/2

⎞

⎟

⎠, (5.21)

with variance

σ2α,β(t) :=

∫+∞

−∞x2Gα,β(x, t)dx =

2Γ(

β + 1)Kαβt

α. (5.22)

As a consequence, the resulting process turns out to be self-similar with Hurst exponent H =α/2 and a variance law consistent with both slow diffusion if 0 < α < 1 and fast diffusion if1 < α < 2. We note that the parameter β does explicitly enter in the variance law (5.22) onlyas in the determination of the multiplicative constant.


It is straightforward to note that the evolution equations of this process reduce to thosefor time-fractional diffusion if α = β < 1, for stretched diffusion if α/= 1 and β = 1, and finallyfor standard diffusion if α = β = 1.

6. Fractional Diffusion Processes with Stationary Increments

We have seen that any Green function associated to the diffusion-like equations consideredin the previous section can be interpreted as the time-evolving one-point pdf of certainself-similar stochastic processes. However, in general, it is not possible to define a unique(self-similar) stochastic process because the determination of any multipoint probabilitydistribution is required; see, for example, [22].

In other words, starting from a master equation which describes the dynamicevolution of a probability density function f(x, t), it is always possible to define anequivalence class of stochastic processes with the same marginal density function f(x, t).All these processes provide suitable stochastic representations for the starting equation. It isclear that additional requirements may be stated in order to uniquely select the probabilisticmodel.

For instance, considering (5.19), the additional requirement of stationary increments,as shown by Mura et al.; see [19–22], can lead to a class {Bα,β(t), t ≥ 0}; called “generalized”grey Brownian motion (ggBm), which, by construction, is made up of self-similar processes withstationary increments and Hurst exponent H = α/2. Thus {Bα,β(t), t ≥ 0} is a special class ofH-sssi processes, which provide non-Markovian stochastic models for anomalous diffusion,of both slow type (0 < α < 1) and fast type (1 < α < 2). According to a common terminology,H-sssi stands for H-self-similar-stationary-increments, see for details [2].

The ggBm includes some well known processes, so that it defines an interestinggeneral theoretical framework. The fractional Brownian motion (fBm) appears for β = 1and is associated with (5.7); the grey Brownian motion (gBm), defined by Schneider [8, 9],corresponds to the choice α = β, with 0 < β < 1, and is associated to (5.13), (5.14), or (5.15);finally, the standard Brownian motion (Bm) is recovered by setting α = β = 1 being associatedto (5.1). We should note that only in the particular case of Bm the corresponding process isMarkovian.

In Figure 3 we present a diagram that allows to identify the elements of the ggBmclass. The top region 1 < α < 2 corresponds to the domain of fast diffusion with long-rangedependence. We remind that a self-similar process with stationary increments is said to possesslong-range dependence if the autocorrelation function of the increments tends to zero like apower function and such that it does not result integrable, see for details [2]. In this domainthe increments of the process Bα,β(t) are positively correlated, so that the trajectories tend tobe more regular (persistent). It should be noted that long-range dependence is associated toa non-Markovian process which exhibits long-memory properties. The horizontal line α = 1corresponds to processes with uncorrelated increments, which model various phenomena ofnormal diffusion. For α = β = 1 we recover the Gaussian process of the standard Brownianmotion. The Gaussian process of the fractional Brownian motion is identified by the verticalline β = 1. The bottom region 0 < α < 1 corresponds to the domain of slow diffusion. Theincrements of the corresponding process Bα,β(t) turn out to be negatively correlated and thisimplies that the trajectories are strongly irregular (antipersistent motion); the increments forma stationary process which does not exhibit long-range dependence. Finally, the diagonal line(α = β) represents the Schneider grey Brownian motion (gBm).


0

0.5

1

1.5

2

0 0.5 1

α

β

0.5

0

1

H

Bα,1(t)

Bβ,β(t)

Brownianmotion

Purely random

Long range dependencePersistent

Anti -persistent

fBmgBm

Bα,β(t)

B2−β,β(t)

Figure 3: Parametric class of generalized grey Brownian motion.

Here we want to define the ggBm by making use of the Kolmogorov extensiontheorem and the properties of the M-Wright function. According to Mura and Pagnini [21],the generalized grey Brownian motion Bα,β(t) is a stochastic process defined in a certainprobability space such that its finite-dimensional distributions are given by

fα,β(

x1, x2, . . . , xn; γα,β)

=(2π)−(n−1)/2

√

2Γ(

1 + β)n det γα,β

∫∞

0

1τn/2

M1/2

(

ξ

τ1/2

)

Mβ(τ)dτ, (6.1)

with

ξ =

⎛

⎝2Γ(

1 + β)−1

n∑

i,j=1

xiγα,β−1(ti, tj

)

xj

⎞

⎠

1/2

(6.2)

and covariance matrix

γα,β(

ti, tj)

=1

Γ(

1 + β)

(

tαi + tαj −∣

∣ti − tj∣

∣

α)

, i, j = 1, . . . , n. (6.3)

The covariance matrix (6.3) characterizes the typical dependence structure of a self-similarprocess with stationary increments and Hurst exponent H = α/2; see, for example, [2].

Using (4.23), for n = 1, (6.1) reduces to

fα,β(x, t) =1√4tα

∫∞

0M1/2

(

|x|t−α/2, τ)

Mβ(τ, 1)dτ =12t−α/2Mβ/2

(

|x|t−α/2)

. (6.4)

This means that the marginal density function of the ggBm is indeed the fundamentalsolution (5.21) of (5.18)-(5.19) with Kαβ = 1. Moreover, because M1(τ) = δ(τ − 1), for β = 1,


putting γα,1 ≡ γα, we have that (6.1) provides the Gaussian distribution of the fractionalBrownian motion

fα,1(

x1, x2, . . . , xn; γα,1)

=(2π)−(n−1)/2

√

2 det γαM1/2

⎛

⎜

⎝

⎛

⎝2n∑

i,j=1

xiγ−1α (ti, tj)xj

⎞

⎠

1/2⎞

⎟

⎠, (6.5)

which finally reduces to the standard Gaussian distribution of Brownian motion as α = 1.By the definition used above, it is clear that, fixed β, Bα,β(t) is characterized only by its

covariance structure, as shown by Mura et al. [20, 21]. In other words, the ggBm, which isnot Gaussian in general, is an example of a process defined only through its first and secondmoments, which indeed is a remarkable property of Gaussian processes. Consequently, theggBm appears to be a direct generalization of Gaussian processes, in the same way as theM-Wright function is a generalization of the Gaussian function.

7. Concluding Discussion

In this review paper we have surveyed a quite general approach to derive models foranomalous diffusion based on a family of time-fractional diffusion equations depending ontwo parameters: α ∈ (0, 2), β ∈ (0, 1].

The unifying topic of this analysis is the so-called M-Wright function by which thefundamental solutions of these equations are expressed. Such function is shown to exhibitfundamental analytical properties that were properly used in recent papers for characterizingand simulating a general class of self-similar stochastic processes with stationary incrementsincluding fractional Brownian motion and grey Brownian motion.

In this respect, the M-Wright function emerges to be a natural generalization of theGaussian density to model diffusion processes, covering both slow and fast anomalousdiffusion and including non-Markovian property. In particular, it turns out to be the mainfunction for the special H-sssi class of stochastic processes (which are self-similar withstationary increments) governed by a master equation of fractional type.

Appendices

A. The Fundamental Solution of the Time-FractionalDiffusion Equation

The fundamental solution Gβ(x, t) for the time-fractional diffusion equation can be obtainedby applying in sequence the Fourier and Laplace transforms to any form chosen among(5.13)–(5.15) with the initial condition Gβ(x, 0+) = u0(x) = δ(x). Let us devote our attentionto the integral form (5.15) using nondimensional variables by setting Kβ = 1 and adopting

the notation Jβt for the fractional integral (2.10). Then, our Cauchy problem reads

Gβ(x, t) = δ(x) + Jβt

∂2Gβ∂x2 (x, t). (A.1)


In the Fourier-Laplace domain, after applying formula (2.18) for the Laplace transform of thefractional integral and observing δ(κ) ≡ 1; see, for example, [76], we get

˜Gβ(κ, s) =1s− κ

2

sβ

˜Gβ(κ, s), (A.2)

from which

˜Gβ(κ, s) =sβ−1

sβ + κ2, 0 < β ≤ 1, R(s) > 0, κ ∈ R. (A.3)

To determine the Green function Gβ(x, t) in the space-time domain we can follow twoalternative strategies related to the order in carrying out the inversions in (A.3).

(S1) Invert the Fourier transform getting ˜Gβ(x, s) and then invert the remaining Laplacetransform.

(S2) Invert the Laplace transform getting Gβ(κ, t) and then invert the remaining Fouriertransform.

Strategy (S1)

Recalling the Fourier transform pair

a

b + κ2F←→ a

2b1/2e−|x|b

1/2, a, b > 0, (A.4)

and setting a = sβ−1, b = sβ, we get

˜Gβ(x, s) =12sβ/2−1e−|x|s

β/2. (A.5)

Strategy (S2)

Recalling the Laplace transform pair

sβ−1

sβ + cL←→ Eβ

(

−ctβ)

, c > 0, (A.6)

and setting c = κ2, we have

Gβ(κ, t) = Eβ(

−κ2tβ)

. (A.7)

Both strategies lead to the result

Gβ(x, t) =12

Mβ/2(|x|, t) =12t−β/2Mβ/2

( |x|tβ/2

)

, (A.8)


consistent with (5.16). Here we have used the M-Wright function, introduced in Section 4,and its properties related to the Laplace transform pair (4.20) for inverting (A.5) and theFourier transform pair (4.22) for inverting (A.7).

B. The Fundamental Solution of the Time-Fractional Drift Equation

Let us finally note that the M-Wright function does appear also in the fundamental solutionof the time-fractional drift equation. Writing this equation in nondimensional form andadopting the Caputo derivative, we have

∗Dβt u(x, t) = −

∂

∂xu(x, t), −∞ < x < +∞, t ≥ 0, (B.1)

where 0 < β < 1 and u(x, 0+) = u0(x). When u0(x) = δ(x),we obtain the fundamental solution(Green function) that we denote byG∗

β(x, t). Following the approach of Appendix A, we show

that

G∗β(x, t) =

⎧

⎪

⎨

⎪

⎩

t−βMβ

(

x

tβ

)

, x > 0,

0, x < 0,(B.2)

which for β = 1 reduces to the right running pulse δ(x − t) for x > 0.In the Fourier-Laplace domain, after applying formula (2.19) for the Laplace transform

of the Caputo fractional derivative and observing δ(κ) ≡ 1; see, for example, [76], we get

sβ ˜G∗β(κ, s) − s

β−1 = +iκ˜G∗β(κ, s), (B.3)

from which

˜G∗β(κ, s) =sβ−1

sβ − iκ, 0 < β ≤ 1, R(s) > 0, κ ∈ R. (B.4)

Like in Appendix A, to determine the Green function G∗β(x, t) in the space-time domain we

can follow two alternative strategies related to the order in carrying out the inversions in(B.4).

(S1) Invert the Fourier transform getting ˜Gβ(x, s) and then invert the remaining Laplacetransform.

(S2) Invert the Laplace transform getting G∗β(κ, t) and then invert the remaining Fouriertransform.


Strategy (S1)

Recalling the Fourier transform pair

a

b − iκF←→ a

be−xb, a, b > 0, x > 0, (B.5)

and setting a = sβ−1, b = sβ, we get

˜G∗β(x, s) = s

β−1e−xsβ

. (B.6)

Strategy (S2)

Recalling the Laplace transform pair

sβ−1

sβ + cL←→ Eβ

(

−ctβ)

, c > 0, (B.7)

and setting c = −iκ, we have

G∗β(κ, t) = Eβ

(

iκtβ)

. (B.8)

Both strategies lead to the result (B.2). In view of (4.1) we also recall that the M-Wrightfunction is related to the unilateral extremal stable density of index βL

β

β. Then, using our

notation stated in [53] for stable densities, we write our Green function as

G∗β(x, t) =t

βx−1−1/βL

−ββ

(

tx−1/β)

, (B.9)

To conclude this Appendix, let us briefly discuss the above results in view of their relevancein fractional diffusion processes following the recent paper by Gorenflo and Mainardi [77].Equation (B.1) describes the evolving sojourn probability density of the positively orientedtime-fractional drift process of a particle, starting in the origin at the instant zero. It hasbeen derived in [77] as a properly scaled limit for the evolution of the counting numberof the Mittag-Leffler renewal process (the fractional Poisson process). It can be given inseveral forms, and often it is cited as the subordinator (producing the operational time fromthe physical time) for space-time-fractional diffusion as in the form (B.9). For more details see[3], where simulations of space-time-fractional diffusion processes have been considered ascomposed by time-fractional and space-fractional diffusion processes.

This analysis can be compared to that described with a different language in papersby Meerschaert et al. [4, 78]. Recently, a more exhaustive analysis has been given by Gorenflo[79].


Acknowledgments

This work has been carried out in the framework of the research project Fractional CalculusModelling (http://www.fracalmo.org/). The authors are grateful to V. Kiryakova, R. Gorenfloand the anonymous referees for useful comments.

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Research ArticleStability and Convergence of an EffectiveNumerical Method for the Time-Space FractionalFokker-Planck Equation with a NonlinearSource Term

Qianqian Yang, Fawang Liu, and Ian Turner

School of Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane,QLD 4001, Australia

Correspondence should be addressed to Fawang Liu, [email protected]

Received 25 May 2009; Revised 20 August 2009; Accepted 28 September 2009

Academic Editor: Om Agrawal

Copyright q 2010 Qianqian Yang et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

Fractional Fokker-Planck equations (FFPEs) have gained much interest recently for describingtransport dynamics in complex systems that are governed by anomalous diffusion and nonex-ponential relaxation patterns. However, effective numerical methods and analytic techniques forthe FFPE are still in their embryonic state. In this paper, we consider a class of time-space fractionalFokker-Planck equations with a nonlinear source term (TSFFPENST), which involve the Caputotime fractional derivative (CTFD) of order α ∈ (0, 1) and the symmetric Riesz space fractionalderivative (RSFD) of order μ ∈ (1, 2]. Approximating the CTFD and RSFD using the L1-algorithmand shifted Grunwald method, respectively, a computationally effective numerical method is pre-sented to solve the TSFFPE-NST. The stability and convergence of the proposed numerical methodare investigated. Finally, numerical experiments are carried out to support the theoretical claims.

1. Introduction

The Fokker-Planck equation (FPE) has commonly been used to describe the Brownian motionof particles. Normal diffusion in an external force field is often modeled in terms of thefollowing Fokker-Planck equation (FPE) [1]:

∂u(x, t)∂t

=

[∂

∂x

V ′(x)mη1

+K1∂2

∂x2

]u(x, t), (1.1)

where m is the mass of the diffusing test particle, η1 denotes the fraction constantcharacterising the interaction between the test particle and its embedding, and the force is


related to the external potential through F(x) = dV (x)/dx. The FPE (1.1) is well studied fora variety of potential types, and the respective results have found wide application. In manystudies of diffusion processes where the diffusion takes place in a highly nonhomogeneousmedium, the traditional FPE may not be adequate [2, 3]. The nonhomogeneities of themedium may alter the laws of Markov diffusion in a fundamental way. In particular,the corresponding probability density of the concentration field may have a heavier tailthan the Gaussian density, and its correlation function may decay to zero at a muchslower rate than the usual exponential rate of Markov diffusion, resulting in long-rangedependence. This phenomenon is known as anomalous diffusion [4]. Fractional derivativesplay a key role in modeling particle transport in anomalous diffusion including the spacefractional Fokker-Planck (advection-dispersion) equation describing Levy flights, the timefractional Fokker-Planck equation depicting traps, and the time-space fractional equationcharacterizing the competition between Levy flights and traps [5, 6]. Different assumptionson this probability density function lead to a variety of time-space fractional Fokker-Planckequations (TSFFPEs).

TSFFPE has been successfully used for modeling relevant physical processes. Whenthe fractional differential equation is used to describe the asymptotic behavior of continuoustime random walks, its solution corresponds to the Levy walks, generalizing the Brownianmotion to the Levy motion. The following space fractional Fokker-Planck equation has beenconsidered [2, 3, 7]:

∂u(x, t)∂t

= −v∂u(x, t)∂x

+Kμ

[c+ aD

μxu(x, t) + c− xD

μ

bu(x, t)

], (1.2)

where v is the drift of the process, that is, the mean advective velocity; Kμ is the coefficient ofdispersion; aD

μx and xD

μ

b are the left and right Riemann-Liouville space fractional derivativesof order μ given by

aDμxu(x, t) =

1Γ(2 − μ

) ∂2

∂x2

∫x

a

u(ξ, t)dξ

(x − ξ)μ−1,

xDμ

bu(x, t) =1

Γ(2 − μ

) ∂2

∂x2

∫b

x

u(ξ, t)dξ

(ξ − x)μ−1;

(1.3)

c+ and c− indicate the relative weight of transition probability; Benson et al. [2, 3] took c+ =1/2+ β/2 and c− = 1/2− β/2, (−1 ≤ β ≤ 1), which indicate the relative weight forward versusbackward transition probability. If c+ = c− = −cμ = −1/2 cos(πμ/2), (1.2) can be rewritten inthe following form:

∂u(x, t)∂t

= −v∂u(x, t)∂x

+Kμ∂μu(x, t)∂|x|μ

, (1.4)

where ∂μ/∂|x|μ is the symmetric space fractional derivative of order μ (1 < μ ≤ 2). This is alsoreferred to as the Riesz derivative [8], which contains a left Riemann-Liouville derivative


(aDμx) and a right Riemann-Liouville derivative (xD

μ

b ), namely,

∂μ

∂|x|μu(x, t) = −cμ

(aD

μx + xD

μ

b

)u(x, t). (1.5)

As a model for subdiffusion in the presence of an external field, a time fractionalextension of the FPE has been introduced as the time fractional Fokker-Planck equation(TFFPE) [5, 9]:

∂u(x, t)∂t

= 0D1−αt

[∂

∂x

V ′(x)mηα

+Kα∂2

∂x2

]u(x, t), (1.6)

where the Riemann-Liouville operator 0D1−αt , (0 < α < 1) is defined through its operation:

0D1−αt u(x, t) =

1Γ(α)

∂

∂t

∫ t

0

u(x, η

)(t − η

)1−α dη. (1.7)

Yuste and Acedo [10] proposed an explicit finite difference method and a new vonNeumann-type stability analysis for the anomalous subdiffusion equation (1.6) with V ′(x) =0. However, they did not give a convergence analysis and pointed out the difficulty of thistask when implicit methods are considered. Langlands and Henry [11] also investigatedthis problem and proposed an implicit numerical L1-approximation scheme and discussedthe accuracy and stability of this scheme. However, the global accuracy of the implicitnumerical scheme has not been derived and it seems that the unconditional stability forall α in the range 0 < α ≤ 1 has not been established. Recently, Chen et al. [12] presenteda Fourier method for the anomalous subdiffusion equation, and they gave the stabilityanalysis and the global accuracy analysis of the difference approximation scheme. Zhuanget al. [13] also proposed an implicit numerical method and an analytical technique for theanomalous subdiffusion equation. Chen et al. [14] proposed implicit and explicit numericalapproximation schemes for the Stokes’ first problem for a heated generalized second gradefluid with fractional derivatives. The stability and convergence of the numerical schemeare discussed using a Fourier method. A Richardson extrapolation technique for improvingthe order of convergence of the implicit scheme is presented. However, effective numericalmethods and error analysis for the time-space fractional Fokker-Planck equation with anonlinear source term are still in their infancy and are open problems.

Equation (1.6) can be written as the following equivalent form [15]:

0Dαt u(x, t) −

u(x, 0)t−α

Γ(1 − α) =

[∂

∂x

V ′(x)mηα

+Kα∂2

∂x2

]u(x, t). (1.8)

By noting that [15]

∂αu(x, t)∂tα

= 0Dαt u(x, t) −

u(x, 0)t−α

Γ(1 − α) , (1.9)


we arrive at

∂αu(x, t)∂tα

=

[∂

∂x

V ′(x)mηα

+Kα∂2

∂x2

]u(x, t), (1.10)

where ∂αu(x, t)/∂tα is the Caputo time fractional derivative (CTFD) of order α (0 < α < 1)with starting point at t = 0 defined by [16]

∂αu(x, t)∂tα

=1

Γ(1 − α)

∫ t

0

∂u(x, η

)∂η

dη(t − η

)α . (1.11)

The time-space fractional Fokker-Plank equation (TSFFPE), which describes the competitionbetween subdiffusion and Levy flights, is given by [5]

∂u(x, t)∂t

= 0D1−αt

[∂

∂x

V ′(x)mηα

+Kμα∂μ

∂|x|μ]u(x, t), (1.12)

or

∂αu(x, t)∂tα

=[∂

∂x

V ′(x)mηα

+Kμα∂μ

∂|x|μ]u(x, t), (1.13)

where Kμα denotes the anomalous diffusion coefficient.

Schot et al. [17] investigated a fractional diffusion equation that employs time andspace fractional derivatives by taking an absorbent (or source) term and an external forceinto account, which can be described by the following time-space fractional Fokker-Plankequation with an absorbent term and a linear external force:

∂αu(x, t)∂tα

= − ∂

∂x[F(x)u(x, t)] +Kμ

α∂μ

∂|x|μu(x, t) −

∫ t

0r(t − η

)u(x, η

)dη, (1.14)

where F(x) is the external force and r(t) is a time-dependent absorbent term, which may berelated to a reaction diffusion process.

The fractional Fokker-Planck equations (FFPEs) have been recently treated by manyauthors and are presented as a useful approach for the description of transport dynamics incomplex systems that are governed by anomalous diffusion and nonexponential relaxationpatterns. The analytical solution of FFPE is only possible in simple and special cases [2, 3, 18]and the analytical solution provides a general representation in terms of Green’s functions.We note that the representation of Green’s functions is mostly expressed as convergentexpansions in negative and positive power series. These special functions are not suitablefor numerical evaluation when x is sufficiently small or sufficiently large. Therefore, a newnumerical strategy is important for solving these equations. Although numerical methodsfor the time fractional Fokker-Planck type equation, the space fractional Fokker-Plank typeequation, and the time-space fractional Fokker-Planck type equation have been considered[7, 15, 19], numerical methods and stability and convergence analysis for the FFPE are quite


limited and difficult. In fact, published papers on the numerical methods for the FFPE aresparse. We are unaware of any other published work on numerical methods for the time-space fractional Fokker-Planck type equation with a nonlinear source term. This motivatesus to consider an effective numerical method for the time-space fractional Fokker-Planckequation with a nonlinear source term and to investigate its stability and convergence.

In this paper, we consider the following time-space fractional Fokker-Planck equationwith a nonlinear source term (TSFFPE-NST):

∂αu(x, t)∂tα

=[∂

∂x

V ′(x)mηα

+Kμα∂μ

∂|x|μ]u(x, t) + s(u, x, t) (1.15)

subject to the boundary and initial conditions:

u(a, t) = u(b, t) = 0, 0 ≤ t ≤ T,

u(x, 0) = u0(x), a ≤ x ≤ b,(1.16)

where p(x) = V ′(x)/mηα is known as the drift coefficient. The nonlinear source (or absorbent)term s(u, x, t) is assumed to satisfy the Lipschitz condition:

‖s(u, x, t) − s(v, x, t)‖ ≤ L‖u − v‖. (1.17)

Let X be a Banach space with associated norm ‖u‖. We say that s : X → X is globallyLipschitz continuous if for some L > 0, we have ‖s(u) − s(v)‖ ≤ L‖u − v‖ for all u, v ∈ X, andis locally Lipschitz continuous, if the latter holds for ‖u‖, ‖v‖ ≤ M with L = L(M) for anyM > 0 [20].

Let Ω = [a, b] × [0, T]. In this paper, we suppose that the continuous problem (1.15)-(1.16) has a smooth solution u(x, t) ∈ C1+μ,2

x,t (Ω).The rest of this paper is organized as follows. In Section 2, the Caputo time fractional

derivative (CTFD) and the Riesz space fractional derivative (RSFD) are approximated bythe L1-algorithm and the shifted Grunwald method, respectively. An effective numericalmethod (ENM) for solving the TSFFPE-NST (1.15)-(1.16) is proposed. The stability andconvergence of the ENM are discussed in Sections 3 and 4, respectively. In Section 5,numerical experiments are carried out to support the theoretical analysis. Finally, someconclusions are drawn in Section 6.

2. An Effective Numerical Method for the TSFFPE-NST

In this section, we present an effective numerical method to simulate the solution behaviorof the TSFFPE-NST (1.15)-(1.16). Let xl = lh (l = 0, 1, . . . ,M) and tn = nτ (n = 0, 1, . . . ,N),where h = (b − a)/M and τ = T/N are the spatial and temporal steps, respectively.

Firstly, adopting the L1-algorithm [21], we discretize the Caputo time fractionalderivative as

∂αu(x, tn+1)∂tα

=τ−α

Γ(2 − α)

n∑j=0

bj[u(x, tn+1−j

)− u

(x, tn−j

)]+O

(τ1+α

), (2.1)

where bj = (j + 1)1−α − j1−α, j = 0, 1, 2, . . . ,N − 1.


For the symmetric Riesz space fractional derivative, we use the following shiftedGrunwald approximation [22]:

∂μu(xl, t)∂|x|μ

= − h−μ

2 cos(πμ/2

)[l+1∑i=0

wiu(xl−i+1, t) +M−l+1∑i=0

wiu(xl+i−1, t)

]+O(hq), (2.2)

where the coefficients are defined by

w0 = 1, wi = (−1)iμ(μ − 1

)· · ·

(μ − i + 1

)i!

, i = 1, 2, . . . ,M. (2.3)

This formula is not unique because there are many different valid choices for wi that lead toapproximations of different orders q [23]. The definition (2.2) provides order q = 1.

The first-order spatial derivative can be approximated by the backward differencescheme if p(x) < 0, (otherwise, the forward difference scheme can be used if p(x) > 0):

∂

∂xp(xl)u(xl, t) =

p(xl)u(xl, t) − p(xl−1)u(xl−1, t)h

+O(h). (2.4)

The nonlinear source term can be discretised either explicitly or implicitly. In thispaper, we use an explicit method and evaluate the nonlinear source term at the previoustime step:

s(u(x, tn+1), x, tn+1) = s(u(x, tn), x, tn) +O(τ). (2.5)

In this way, we avoid solving a nonlinear system at each time step and obtain anunconditionally stable and convergent numerical scheme, as shown in Section 3. However,the shortcoming of the explicit method is that it generates additional temporal error, as shownin (2.5).

Thus, using (2.1)–(2.5), we have

τ−α

Γ(2 − α)

n∑j=0

bj[u(xl, tn+1−j

)− u

(xl, tn−j

)]

=p(xl)u(xl, tn+1) − p(xl−1)u(xl−1, tn+1)

h

− Kμαh−μ

2 cos(πμ/2

)[l+1∑i=0

wiu(xl−i+1, tn+1) +M−l+1∑i=0

wiu(xl+i−1, tn+1)

]

+ s(u(xl, tn), xl, tn) +O(τ1+α + h + τ

).

(2.6)


After some manipulation, (2.6) can be written in the following form:

u(xl, tn+1) = bnu(xl, t0) +n−1∑j=0

(bj − bj+1

)u(xl, tn−j

)

+μ0

h

(p(xl)u(xl, tn+1) − p(xl−1)u(xl−1, tn+1)

)

− μ0r0

[l+1∑i=0

wiu(xl−i+1, tn+1) +M−l+1∑i=0

wiu(xl+i−1, tn+1)

]

+ μ0s(u(xl, tn), xl, tn) + Rn+1l ,

(2.7)

where μ0 = ταΓ(2 − α) > 0, r0 = Kμαh−μ/2 cos(πμ/2) < 0, and

∣∣∣Rn+1l

∣∣∣ ≤ C1τα(τ1+α + h + τ

). (2.8)

Let unl

be the numerical approximation of u(xl, tn), and let snl

be the numericalapproximation of s(u(xl, tn), xl, tn). We obtain the following effective numerical method(ENM) of the TSFFPE-NST (1.15)-(1.16):

un+1l = bnu0

l +n−1∑j=0

(bj − bj+1

)un−jl +

μ0

h

(plu

n+1l − pl−1u

n+1l−1

)

− μ0r0

[l+1∑i=0

wiun+1l−i+1 +

M−l+1∑i=0

wiun+1l+i−1

]+ μ0s

nl

(2.9)

for l = 1, 2, . . . ,M − 1, n = 0, 1, 2, . . . ,N − 1. The boundary and initial conditions can bediscretised using

un0 = unM = 0, n = 0, 1, 2, . . . ,N,

u0l = u0(lh), l = 0, 1, 2, . . . ,M.

(2.10)

Remark 2.1. If we use the implicit method to approximate the nonlinear source term, thenumerical method of the TSFFPE-NST can be written as

un+1l = bnu0

l +n−1∑j=0

(bj − bj+1

)un−jl +

μ0

h

(plu

n+1l − pl−1u

n+1l−1

)

− μ0r0

[l+1∑i=0

wiun+1l−i+1 +

M−l+1∑i=0

wiun+1l+i−1

]+ μ0s

n+1l ,

(2.11)

that is, replace snl in (2.9) with sn+1l . This numerical method is stable and convergent when

the source term s(u(x, t), x, t) satisfies the Lipschitz condition (1.17) (see, e.g., [24]).


Lemma 2.2 (see [19]). The coefficients bj satisfy

(1) bj > 0 for j = 0, 1, 2, . . . , n;

(2) 1 = b0 > b1 > · · · > bn, bn → 0 as n → ∞;

(3) when 0 < α < 1,

limj→∞

b−1j

jα= lim

j→∞

j−1(1 + j−1

)1−α − 1=

11 − α.

(2.12)

Thus, there is a positive constant C2 such that

b−1j ≤ C2j

α, j = 0, 1, 2, . . . . (2.13)

Lemma 2.3 (see [25]). The coefficients wi satisfy

(1) w0 = 0, w1 = −μ < 0, and wi > 0 for i = 2, 3, . . . ,M;

(2)∑∞

i=0 wi = 0, and∑n

i=0 wi < 0 for ∀n ∈ N.

3. Stability of the Effective Numerical Method

In this section, we analyze the stability of the ENM (2.9)-(2.10). Firstly, we rewrite (2.9) in thefollowing form:

un+1l −

μ0

h

(plu

n+1l − pl−1u

n+1l−1

)+ μ0r0

[l+1∑i=0

wiun+1l−i+1 +

M−l+1∑i=0

wiun+1l+i−1

]

= bnu0l +

n−1∑j=0

(bj − bj+1

)un−jl + μ0s

nl .

(3.1)

Let unl

be the approximate solution of the ENM (3.1), and let snl

be the approximationof snl . Setting ρnl = unl − u

nl , we obtain the following roundoff error equation:

ρn+1l −

μ0

h

(plρ

n+1l − pl−1ρ

n+1l−1

)+ μ0r0

[l+1∑i=0

wiρn+1l−i+1 +

M−l+1∑i=0

wiρn+1l+i−1

]

= bnρ0l +

n−1∑j=0

(bj − bj+1

)ρn−jl + μ0

(snl − s

nl

) (3.2)

for l = 1, 2, . . . ,M − 1; n = 0, 1, . . . ,N − 1.We suppose that p(x) ≤ 0 and that p(x) decreases monotonically on [a, b]. This is

based on the fact that physical considerations and stability dictate that p′(x) < 0 [26, 27].Assuming ‖ρn‖∞ = max1≤l≤M−1|ρnl |, and using mathematical induction, we obtain the

following theorem.


Theorem 3.1. Suppose that ρnl (l = 1, 2, . . . ,M − 1, n = 1, 2, . . . ,N) is the solution of the roundofferror equation (3.2), and the nonlinear source term s(u(x, t), x, t) satisfies the Lipschitz condition(1.17), then there is a positive constant C0, such that

∥∥ρn∥∥∞ ≤ C0

∥∥∥ρ0∥∥∥∞, n = 1, 2, . . . ,N. (3.3)

Proof. When n = 1, assume that |ρ1l0| = max{|ρ1

1|, |ρ12|, . . . , |ρ

1M−1|}. Because p(x) ≤ 0 and

decreases monotonically on [a, b], we have

0 ≤ −μ0

h

(pl0 − pl0−1

)∣∣∣ρ1l0

∣∣∣ ≤ −μ0

hpl0

∣∣∣ρ1l0

∣∣∣ + μ0

hpl0−1

∣∣∣ρ1l0−1

∣∣∣. (3.4)

Using the properties of ωi in Lemma 2.3, we have

0 ≤ μ0r0

[l0+1∑i=0

wi

∣∣∣ρ1l0

∣∣∣ +M−l0+1∑i=0

wi

∣∣∣ρ1l0

∣∣∣]

≤ μ0r0

[l0+1∑i=0

wi

∣∣∣ρ1l0−i+1

∣∣∣ +M−l0+1∑i=0

wi

∣∣∣ρ1l0+i−1

∣∣∣].

(3.5)

Combining (3.4) with (3.5), using the Lipschitz condition (1.17) and smooth solutioncondition, we obtain

∣∣∣ρ1l0

∣∣∣ ≤ ∣∣∣ρ1l0

∣∣∣ − μ0

hpl0

∣∣∣ρ1l0

∣∣∣ + μ0

hpl0−1

∣∣∣ρ1l0−1

∣∣∣+ μ0r0

[l0+1∑i=0

wi

∣∣∣ρ1l0−i+1

∣∣∣ +M−l0+1∑i=0

wi

∣∣∣ρ1l0+i−1

∣∣∣]

≤∣∣∣∣∣ρ1

l0−μ0

hpl0ρ

1l0+μ0

hpl0−1ρ

1l0−1 + μ0r0

[l0+1∑i=0

wiρ1l0−i+1 +

M−l0+1∑i=0

wiρ1l0+i−1

]∣∣∣∣∣=∣∣∣b0ρ

0l0+ μ0

(s0l0− s0

l0

)∣∣∣≤ b0

∣∣∣ρ0l0

∣∣∣ + μ0L∣∣∣ρ0

l0

∣∣∣ = (1 + μ0L

)∣∣∣ρ0l0

∣∣∣.

(3.6)

Let C = 1 + μ0L. Thus, we obtain

∥∥∥ρ1∥∥∥∞≤ C

∥∥∥ρ0∥∥∥∞. (3.7)

Now, suppose that

∥∥∥ρk∥∥∥∞≤ C

∥∥∥ρ0∥∥∥∞, k = 2, . . . , n. (3.8)


By assuming |ρn+1l0| = max{|ρn+1

1 |, |ρn+12 |, . . . , |ρn+1

M−1|}, we have that

∣∣∣ρn+1l0

∣∣∣ ≤∣∣∣∣∣ρn+1

l0−μ0

h

(pl0ρ

n+1l0− pl0−1ρ

n+1l0−1

)+ μ0r0

[l0+1∑i=0

wiρn+1l0−i+1 +

M−l0+1∑i=0

wiρn+1l0+i−1

]∣∣∣∣∣=

∣∣∣∣∣∣bnρ0l0+n−1∑j=0

(bj − bj+1

)ρn−jl0

+ μ0

(snl0 − s

nl0

)∣∣∣∣∣∣≤ bn

∣∣∣ρ0l0

∣∣∣ + n−1∑j=0

(bj − bj+1

)∣∣∣ρn−jl0

∣∣∣ + μ0L∣∣∣ρnl0

∣∣∣.

(3.9)

Using (3.7) and (3.8), we have

∣∣∣ρn+1l0

∣∣∣ ≤ bn∣∣∣ρ0l0

∣∣∣ + Cn−1∑j=0

(bj − bj+1

)∣∣∣ρ0l0

∣∣∣ + Cμ0L∣∣∣ρ0

l0

∣∣∣= bn

∣∣∣ρ0l0

∣∣∣ + C(b0 − bn)∣∣∣ρ0

l0

∣∣∣ + Cμ0L∣∣∣ρ0

l0

∣∣∣=(bnμ0L + C2

)∣∣∣ρ0l0

∣∣∣.(3.10)

Let C0 = bnμ0L + C2. Hence, we have

∥∥∥ρn+1∥∥∥∞≤ C0

∥∥∥ρ0∥∥∥∞. (3.11)

The proof of Theorem 3.1 is completed.

Applying Theorem 3.1, the following theorem of stability is obtained.

Theorem 3.2. Assuming that the nonlinear source term s(u(x, t), x, t) satisfies the Lipschitzcondition (1.17) and that the drift coefficient p(x) ≤ 0 decreases monotonically on [a, b], the ENMdefined by (2.9)-(2.10) is stable.

Remark 3.3. If p(x) > 0 and decreases monotonically on [a, b], we can use the forwarddifference method to approximate the first-order spatial derivative and apply a similaranalysis of stability.

Remark 3.4. In fact, for the case p(x) does not decrease monotonically, we can still obtain astable numerical scheme by a minor change in our current ENM. We can expand the firstterm on the RHS of (1.15) as (∂/∂x)[p(x)u(x, t)] = (dp/dx)u(x, t) +p(x)(∂u(x, t)/∂x), whichenables us to group (dp/dx)u(x, t) together with the nonlinear source term s(u, x, t) to obtaina new nonlinear source term s∗(u, x, t) = s(u, x, t) + (dp/dx)u(x, t). This way we can weakenthe assumption on p(x) and the analysis given in this section still can be used.

Remark 3.5. If we use an implicit method to approximate the nonlinear source term, as shownin Remark 2.1, we can prove that the numerical method defined in (2.11) is stable when


1 − μ0L > 0, which is independent of the spatial step. In fact, when the time step is small,the condition 1 − μ0L > 0 is generally satisfied.

4. Convergence of the Effective Numerical Method

In this section, we analyze the convergence of the ENM (2.9)-(2.10). Let u(xl, tn) be the exactsolution of the TSFFPE-NST (1.15)-(1.16) at mesh point (xl, tn), and let unl be the numericalsolution of the TSFFPE-NST (1.15)-(1.16) computed using the ENM (2.9)-(2.10). Define ηn

l=

u(xl, tn) − unl and Yn = (ηn1 , ηn1 , . . . , η

nM−1)

T . Subtracting (2.9) from (2.7) leads to

ηn+1l −

μ0

h

(plη

n+1l − pl−1η

n+1l−1

)+ μ0r0

[l+1∑i=0

wiηn+1l−i+1 +

M−l+1∑i=0

wiηn+1l+i−1

]

= bnη0l +

n−1∑j=0

(bj − bj+1

)ηn−jl + μ0

(s(u(xl, tn), xl, tn) − snl

)+ Rn+1

l ,

(4.1)

where l = 1, 2, . . . ,M − 1; n = 0, 1, . . . ,N − 1.Assuming that ‖Yn‖∞ = max1≤l≤M−1|ηnl | and using mathematical induction, we obtain

the following theorem.

Theorem 4.1. Assuming the nonlinear source term s(u(x, t), x, t) satisfies the Lipschitz condition(1.17), and the drift coefficient p(x) ≤ 0 decreases monotonically on [a, b], the ENM defined by(2.9)-(2.10) is convergent, and there exists a positive constant C∗, such that

‖Yn‖∞ ≤ C∗(τ1+α + h + τ

), n = 1, 2, . . . ,N. (4.2)

Proof. Assume |Rk0l0| = max1≤l≤M−1,1≤n≤N |Rn

l |. Following a similar argument to that presentedabove for the stability analysis of the ENM (2.9)-(2.10), when n = 1, assuming that |η1

l0| =

max{|η11|, |η

12|, . . . , |η

1M−1|}, we have

∣∣∣η1l0

∣∣∣ ≤ ∣∣∣b0η0l0+ μ0

(s(u(xl0 , t0), xl0 , t0) − s0

l0

)+ R1

l0

∣∣∣. (4.3)

Utilising Y0 = 0, the Lipschitz condition (1.17), and smooth solution condition, we obtain

∣∣∣η1l0

∣∣∣ ≤ b0

∣∣∣η0l0

∣∣∣ + μ0L∣∣∣η0

l0

∣∣∣ + ∣∣∣Rk0l0

∣∣∣ = ∣∣∣Rk0l0

∣∣∣. (4.4)

Thus,

∥∥∥Y1∥∥∥∞≤∣∣∣Rk0

l0

∣∣∣. (4.5)

Now, suppose that

∥∥∥Yk∥∥∥∞≤ b−1

k−1

∣∣∣Rk0l0

∣∣∣, k = 1, 2, . . . , n. (4.6)


Using Lemma 2.2, bk > bk+1, we have

∥∥∥Yk∥∥∥∞≤ b−1

n

∣∣∣Rk0l0

∣∣∣. (4.7)

Similarly, assuming |ηn+1l0| = max{|ηn+1

1 |, |ηn+12 |, . . . , |ηn+1

M−1|}, we have

∣∣∣ηn+1l0

∣∣∣ ≤∣∣∣∣∣∣bnη0

l0+n−1∑j=0

(bj − bj+1

)ηn−jl0

+ μ0

(s(u(xl0 , tn), xl0 , tn) − snl0

)+ Rn+1

l0

∣∣∣∣∣∣. (4.8)

Utilising Y0 = 0, the Lipschitz condition (1.17), and smooth solution condition, we obtain

∣∣∣ηn+1l0

∣∣∣ ≤ b−1n (b0 − bn)

∣∣∣Rk0l0

∣∣∣ + μ0Lb−1n

∣∣∣Rk0l0

∣∣∣ + ∣∣∣Rk0l0

∣∣∣= b−1

n

(b0 − bn + μ0L + bn

)∣∣∣Rk0l0

∣∣∣= b−1

n

(b0 + μ0L

)∣∣∣Rk0l0

∣∣∣ = Cb−1n

∣∣∣Rk0l0

∣∣∣.(4.9)

Hence,

∥∥∥Yn+1∥∥∥∞≤ Cb−1

n

∣∣∣Rk0l0

∣∣∣. (4.10)

Finally, utilising (2.8) and Lemma 2.2, b−1n ≤ C2n

α, we obtain the result on the convergence ofthe ENM (2.9)-(2.10), namely,

‖Yn‖∞ ≤ CC1C2nατα

(τ1+α + h + τ

)

≤ C∗(τ1+α + h + τ

) (4.11)

for n = 1, 2, . . . ,N.

Remark 4.2. If we use an implicit method to approximate the nonlinear source term, as shownin Remark 2.1, we can prove that the numerical method defined in (2.11) is convergent when1 − μ0L > 0, which is independent of the spatial step. In fact, when the time step is small, thecondition 1 − μ0L > 0 is generally satisfied.

5. Numerical Results

In this section, we present four numerical examples of the TSFFPE to demonstrate theaccuracy of our theoretical analysis. We also use our solution method to illustrate the changesin solution behavior that arise when the exponent is varied from integer order to fractionalorder and to identify the differences between solutions with and without the external forceterm.


Table 1: Maximum error behavior versus grid size reduction for Example 5.1 at T = 1.0.

h = τ Maximum error1/10 4.8148E-21/20 1.0111E-21/40 2.0587E-31/80 7.3019E-4

T = 5

T = 1

10.80.60.40.20

x

ENMExact

0

0.5

1

1.5

2

2.5

3u(x,t)

Figure 1: Comparison of the numerical solution with the exact solution for Example 5.1 at T = 1.0, 3.0, 5.0.T increases in the direction of the arrow.

Example 5.1. Consider the following TSFFPE:

∂αu(x, t)∂tα

=[−υ ∂

∂x+Kμ

α∂μ

∂|x|μ]u(x, t) + f(x, t),

u(a, t) = u(b, t) = 0, 0 ≤ t ≤ T,

u(x, 0) = Kμα(x − a)2(b − x)2, a ≤ x ≤ b,

(5.1)

where

f(x, t) = (1 + α)υΓ(1 + α)t(x − a)2(b − x)2

+Kμα

(Kμα + υt1+α

)2 cos

(πμ/2

) [g(x − a) + g(b − x)

]+ 2υ

(Kμα + υt1+α

)(x − a)(b − x)(a + b − 2x),

g(x) =4!

Γ(5 − μ

)x4−μ − 2(b − a) 3!Γ(4 − μ

)x3−μ

+ (b − a)2 2Γ(3 − μ

)x2−μ.

(5.2)


The exact solution of the TSFFPE (5.1) is found to be

u(x, t) =(Kμα + υt1+α

)(x − a)2(b − x)2, (5.3)

which can be verified by direct fractional differentiation of the given solution, andsubstituting into the fractional differential equation.

In this example, we take a = 0, b = 1, Kμα = 25, υ = 1, α = 0.8, and μ = 1.9. From

Figure 1, it can be seen that the numerical solution using the ENM is in good agreement withthe exact solution at different times T , with h = 1/40 and τ = 1/40. The maximum errorsof the ENM at time T = 1.0 are presented in Table 1. It can be seen that the ENM is stableand convergent for solving the TSFFPE (5.1). The errors, as our theory indicated, satisfy therelationship error ≤ (τ1+α + h + τ).

Example 5.2. Consider the following TSFFPE-NST:

∂αu(x, t)∂tα

= Kμα∂μ

∂|x|μu(x, t) −

γ

Γ(β)∫ t

0(t − ξ)β−1u(x, ξ)dξ,

u(−5, t) = u(5, t) = 0, 0 ≤ t ≤ T,

u(x, 0) = δ(x).

(5.4)

This example is a TSFFPE-NST without the external force term. In fact, it reduces tothe fractional diffusion equation with an absorbent term. The formulae to approximate theabsorbent term are presented in the appendix. Here, we take β = 0.5, γ = 1, and K

μα = 1.

Figures 2–4 show the changes in the solution profiles of the TSFFPE-NST (5.4) when α andμ are changed from integer to fraction at different times T . We see that the solution profileof the fractional order model is characterized by a sharp peak and a heavy tail. The peakheight in Figure 2 (α = 1.0 and μ = 2.0) decreases more rapidly than that in Figure 3 (α = 0.8and μ = 1.8). Furthermore, when we choose α = 0.5 and μ = 1.5, a more interesting resultcan be observed; that is, the peak height in Figure 2 decreases more slowly than that shownin Figure 4 at the early time T = 0.1, but this trend reverses for the later times T = 0.5 andT = 1.0. Hence, the TSFFPE-NST (5.4) may be useful to investigate several physical processesin the absence of an external force field by choosing appropriate α and μ.


∂αu(x, t)∂tα

=[∂

∂xp(x) +Kμ

α∂μ

∂|x|μ]u(x, t) −

γ

Γ(β)∫ t

0(t − ξ)β−1u(x, ξ)dξ,

u(−5, t) = u(5, t) = 0, 0 ≤ t ≤ T,

u(x, 0) = δ(x).

(5.5)

This example of the TSFFPE-NST incorporates the external force term with p(x) =−1 and an absorbent term. The formula to approximate the absorbent term is presented inthe appendix. Here, we take β = 0.5, γ = 1, and K

μα = 1. Figures 5–7 show the changes


543210−1−2−3−4−5

x

T = 0.1T = 0.5T = 1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

u(x,t)

Figure 2: Numerical solutions for Example 5.2 with α = 1.0 and μ = 2.0 at different times T = 0.1, 0.5, 1.0.

543210−1−2−3−4−5

x

T = 0.1T = 0.5T = 1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

u(x,t)


in the solution profiles of the TSFFPE-NST (5.5) when α and μ are changed from integerorder to fractional order at different times T . Again, we see that the solution profile of thefractional order model is characterized by a sharp peak and a heavy tail. Furthermore, dueto the presence of the external force term with p(x) = −1, the solution profiles are shifted tothe right. It is worthwhile to note that the peak of the integer order model in Figure 5 (α = 1.0


543210−1−2−3−4−5

x

T = 0.1T = 0.5T = 1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

u(x,t)


and μ = 2.0) moves to the right as time increases, but the peak of the fractional order modelin Figure 6 (α = 0.8 and μ = 1.8) and Figure 7 (α = 0.5 and μ = 1.5) does not move.

We also see that the peak heights in Figures 5 and 6 remain almost the same forincreasing time. The peak height in Figure 5 decreases more slowly than that shown inFigure 7 at the early time T = 0.1, but this trend reverses for the later times T = 0.5 andT = 1.0. Hence, the TSFFPE-NST (5.5) may be useful to investigate several physical processeswithin an external force field by choosing appropriate α and μ.


∂αu(x, t)∂tα

=[∂

∂xp(x) +Kμ

α∂μ

∂|x|μ]u(x, t) + ru(x, t)

(1 − u(x, t)

K

),

u(0, t) = u(5, t) = 0, 0 ≤ t ≤ T,

u(x, 0) = x2(5 − x)2, 0 ≤ x ≤ 5.

(5.6)

In applications to population biology, u(x, t) is the population density at location x ∈ R andtime t > 0. The nonlinear source term s(u(x, t), x, t) = ru(x, t)(1−u(x, t)/K) is Fisher’s growthterm that models population growth, where r is the intrinsic growth rate of a species and Kis the environmental carrying capacity, representing the maximum sustainable populationdensity [20, 28, 29].

In this example, we take r = 0.2, K = 1. Figure 8 shows the solution behavior whenα = 0.8, μ = 1.6 at different times T = 0.1, 0.5, 1.0, while Figure 9 shows the solution behaviorwith different values of α between 0 and 1 and fixed value of μ = 1.8 at time T = 1.0. Figure 9also shows that the system exhibits anomalous diffusion behavior and that the solution


543210−1−2−3−4−5

x

T = 0.1T = 0.5T = 1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

u(x,t)


543210−1−2−3−4−5

x

T = 0.1T = 0.5T = 1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

u(x,t)


continuously depends on the time and space fractional derivatives. Although the sourceterm for Fisher’s equation s(u(x, t), x, t) = ru(x, t)(1 − u(x, t)/K) is not globally Lipschitzcontinuous, the solution of the discrete numerical method still yields bounds on the solutionof the continuous problem and the solution of the numerical method (ENM) converges tothe unique solution of the continuous problem (5.6) as the time and space steps tend to zero[30].


543210−1−2−3−4−5

x

T = 0.1T = 0.5T = 1

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

u(x,t)


543210

x

T = 0.1T = 0.5T = 1

0

2

4

6

8

10

12

14

16

18

20

u(x,t)


6. Conclusions

In this paper, we have proposed an effective numerical method to solve the TSFFPE-NSTand proved that the ENM is stable and convergent provided that the nonlinear source termsatisfies the Lipschitz condition, the solution of the continuous problem satisfies the smooth


543210

x

α = 0.3

α = 1

0

2

4

6

8

10

12

u(x,t)

Figure 9: Numerical solutions for Example 5.4 with fixed μ = 1.8 at time T = 1.0, and different values ofα = 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0.

solution condition, and p′(x) can be either > 0 or < 0. Numerical experiments have beencarried out to support the theoretical claims. These numerical methods can also be used toinvestigate other types of fractional partial differential equations.

Appendix

Formulae for Examples 5.2 and 5.3

Let us start from (3.1), that is,

un+1l −

μ0

h

(plu

n+1l − pl−1u

n+1l−1

)+ μ0r0

[l+1∑i=0

wiun+1l−i+1 +

M−l+1∑i=0

wiun+1l+i−1

]

= bnu0l +

n−1∑j=0

(bj − bj+1

)un−jl + μ0s

nl .

(A.1)

Now setting snl= −(γ/Γ(β))

∫ tn0 (tn − ξ)

β−1u(xl, ξ)dξ, then we have

snl ≈ −γ

Γ(β) n−1∑j=0

∫ tj+1

tj

(tn − ξ)β−1u(xl, ξ)dξ. (A.2)


Applying the Mean Value Theorem (M.V.T) for integration yields

snl ≈ −γ

Γ(β) n−1∑j=0

u(xl, ξj

)∫ tj+1

tj

(tn − ξ)β−1dξ, where tj < ξj < tj+1

≈ −γ

Γ(β) n−1∑j=0

uj

l+ uj+1

l

2

⎡⎣(

tn − tj)β

β−(tn − tj+1

)ββ

⎤⎦

= −γτβ

Γ(β)· 2 · β

n−1∑j=0

(uj

l+ uj+1

l

)[(n − j

)β − (n − j − 1)β]

= −γτβ

2Γ(β + 1

) n−1∑j=0

(un−j−1l

+ un−jl

)[(j + 1

)β − jβ]

= −μ1

n−1∑j=0

qj(un−j−1l

+ un−jl

),

(A.3)

where μ1 = γτβ/2Γ(β + 1), qj = (j + 1)β − jβ, j = 0, 1, . . . .Also, we have

l+1∑i=0

wiun+1l−i+1 +

M−l+1∑i=0

wiun+1l+i−1 =

M−1∑i=0

ηliun+1i , (A.4)

where

ηli =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

wl−i+1, 1 ≤ i ≤ l − 2,

w0 +w2, i = l − 1,

2w1, i = l,

w0 +w2, i = l + 1,

wi−l+1, l + 2 ≤ i ≤M − 1.

(A.5)

Now, substituting (A.3) and (A.4) into (A.1), we obtain the numerical scheme forExample 5.2 as

un+1l + μ0r0

M−1∑i=0

ηliun+1i = bnu0

l +n−1∑j=0

[(bj − bj+1

)un−jl − μ1qj

(un−j−1l + un−jl

)], (A.6)


and the numerical scheme for Example 5.3 as

(1 −

μ0plh

)un+1l +

μ0pl−1

hun+1l−1 + μ0r0

M−1∑i=0

ηliun+1i

= bnu0l +

n−1∑j=0

[(bj − bj+1

)un−jl − μ1qj

(un−j−1l + un−jl

)].

(A.7)

Acknowledgments

This research has been supported by a Ph.D. Fee Waiver Scholarship and a School ofMathematical Sciences Scholarship, QUT. The authors also wish to thank the referees for theirconstructive comments and suggestions.

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[23] R. Lin and F. Liu, “Fractional high order methods for the nonlinear fractional ordinary differentialequation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 4, pp. 856–869, 2007.

[24] F. Liu, C. Yang, and K. Burrage, “Numerical method and analytic technique of the modifiedanomalous subdiffusion equation with a nonlinear source term,” Journal of Computational and AppliedMathematics, vol. 231, no. 1, pp. 160–176, 2009.

[25] M. M. Meerschaert and C. Tadjeran, “Finite difference approximations for fractional advection-dispersion flow equations,” Journal of Computational and Applied Mathematics, vol. 172, no. 1, pp. 65–77,2004.

[26] W. Deng, “Finite element method for the space and time fractional Fokker-Planck equation,” SIAMJournal on Numerical Analysis, vol. 47, no. 1, pp. 204–226, 2008-2009.

[27] J. I. Ramos, “Damping characteristics of finite difference methods for one-dimensional reaction-diffusion equations,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 607–609, 2006.

[28] D. del-Castillo-Negrete, B. A. Carreras, and V. E. Lynch, “Front dynamics in reaction-diffusionsystems with Levy flights: a fractional diffusion approach,” Physical Review Letters, vol. 91, no. 1,Article ID 018302, 4 pages, 2003.

[29] V. E. Lynch, B. A. Carreras, D. del-Castillo-Negrete, K. M. Ferreira-Mejias, and H. R. Hicks,“Numerical methods for the solution of partial differential equations of fractional order,” Journal ofComputational Physics, vol. 192, no. 2, pp. 406–421, 2003.

[30] B. Baeumer, M. Kovacs, and M. M. Meerschaert, “Fractional reproduction-dispersal equations andheavy tail dispersal kernels,” Bulletin of Mathematical Biology, vol. 69, no. 7, pp. 2281–2297, 2007.


Research ArticleHe’s Variational Iteration Method for SolvingFractional Riccati Differential Equation

H. Jafari and H. Tajadodi

Department of Mathematics and Computer Science, University of Mazandaran,P.O. Box 47416-95447, Babolsar, Iran

Correspondence should be addressed to H. Jafari, [email protected]

Received 10 August 2009; Accepted 28 January 2010

Academic Editor: Shaher M. Momani

Copyright q 2010 H. Jafari and H. Tajadodi. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We will consider He’s variational iteration method for solving fractional Riccati differentialequation. This method is based on the use of Lagrange multipliers for identification of optimalvalue of a parameter in a functional. This technique provides a sequence of functions whichconverges to the exact solution of the problem. The present method performs extremely well interms of efficiency and simplicity.

1. Introduction

The fractional calculus has found diverse applications in various scientific and technologicalfields [1, 2], such as thermal engineering, acoustics, electromagnetism, control, robotics,viscoelasticity, diffusion, edge detection, turbulence, signal processing, and many otherphysical processes. Fractional differential equations (FDEs) have also been applied inmodeling many physical, engineering problems, and fractional differential equations innonlinear dynamics [3, 4].

The variational iteration method was proposed by He [5] and was successfullyapplied to autonomous ordinary differential equation [6], to nonlinear partial differentialequations with variable coefficients [7], to Schrodinger-KdV, generalized Kd and shallowwater equations [8], to linear Helmholtz partial differential equation [9], recently tononlinear fractional differential equations with Caputo differential derivative [10, 11], andto other fields, [12]. The variational iteration method gives rapidly convergent successiveapproximations of the exact solution if such a solution exists; otherwise a few approximationscan be used for numerical purposes. The method is effectively used in [6–8, 13–15] and thereferences therein. Jafari et al. applied the variational iteration method to the Gas Dynamics


Equation and Stefan problem [13, 14]. We consider here the following nonlinear fractionalRiccati differential equation:

Dα∗y(t) = A(t) + B(t)y + C(t)y2, (1.1)

subject to the initial conditions

y(k)(0) = ck, k = 0, 1, . . . , n − 1, (1.2)

where α is fractional derivative order, n is an integer, A(t), B(t), and C(t) are known realfunctions, and ck is a constant. There are several definitions of a fractional derivative of orderα > 0. The two most commonly used definitions are the Riemann-Liouville and Caputo.Each definition uses Riemann-Liouville fractional integration and derivatives of whole order.The difference between the two definitions is in the order of evaluation. Riemann-Liouvillefractional integration of order a is defined as

Iαf(x) =1

Γ(α)

∫x

0(x − t)α−1f(t)dt, α > 0, x > 0. (1.3)

The following two equations define Riemann-Liouville and Caputo fractional derivatives oforder α, respectively:

Dαf(x) =dm

dxm(Im−αf(x)

), (1.4)

Dα∗f(x) = I

m−α(dm

dxmf(x)

), (1.5)

where m − 1 < α � m and m ∈ N. We have chosen to use the Caputo fractionalderivative because it allows traditional initial and boundary conditions to be included inthe formulation of the problem, but for homogeneous initial condition assumption, thesetwo operators coincide. For more details on the geometric and physical interpretation forfractional derivatives of both the Riemann-Liouville and Caputo types, see [1].

2. Analysis of the Variational Iteration Method

We consider the fractional differential equation

Dα∗y(t) = A(t) + B(t)y + C(t)y2, 0 < α � 1, (2.1)

with initial condition y(0) = 0, where Dα = dα/dtα. According to the variational iterationmethod [5], we construct a correction functional for (2.1) which reads

yn+1 = yn + Iαλ(ξ)[dαyndξα

−A(t) − B(t)yn − C(t)y2n

]. (2.2)


To identify the multiplier, we approximately write (2.2) in the form

yn+1 = yn +∫ t

0λ(ξ)

[dαyndξα


]dξ, (2.3)

where λ is a general Lagrange multiplier, which can be identified optimally via the variationaltheory, and yn is a restricted variation, that is, δyn = 0.

The successive approximation yn+1, n � 0 of the solution y(t) will be readily obtainedupon using Lagrange’s multiplier, and by using any selective function y0. The initial valuey(0) and yt(0) are usually used for selecting the zeroth approximation y0. To calculate theoptimal value of λ, we have

δyn+1 = δyn + δ∫ t

0λ(ξ)

dyndξ

dξ = 0. (2.4)

This yields the stationary conditions λ′(ξ) = 0, and 1 + λ(ξ) = 0, which gives

λ = −1. (2.5)

Substituting this value of Lagrangian multiplier in (2.3), we get the following iterationformula

yn+1 = yn − Iα[dαyndξα


], (2.6)

and finally the exact solution is obtained by

y(t) = limn→∞

yn(t). (2.7)

3. Applications and Numerical Results

To give a clear overview of this method, we present the following illustrative examples.

Example 3.1. Consider the following fractional Riccati differential equation:

dαy

dtα= −y2(t) + 1, 0 < α � 1, (3.1)

subject to the initial condition y(0) = 0.

The exact solution of (3.1) is y(t) = (e2t − 1)/(e2t + 1), when α = 1.In view of (2.6) the correction functional for (3.1) turns out to be

yn+1 = yn − Iα(dαyndξα

+ y2n − 1

)dξ. (3.2)


0.2

0.4

0.6

0.8

y

0.2 0.4 0.6 0.8 1 1.2 1.4

t

Figure 1: Dashed line: Approximate solution.

Beginning with y0(t) = tα/Γ(1 + α), by the iteration formulation (3.2), we can obtain directlythe other components as

y1(t) =tα

Γ(1 + α)− Γ(1 + 2α)t3α

(α + 1)2Γ(1 + 3α),

y2(t) =tα

Γ(1 + α)− Γ(1 + 2α)t3α

Γ(1 + α)2Γ(1 + 3α)+

23+2αΓ(4α)Γ(1/2 + α)t5α√πΓ(α)Γ(1 + α)Γ(1 + 3α)Γ(1 + 5α)

− 64αΓ(1 + 2α)2Γ(1/2 + 3α)t7α√πΓ(1 + α)4Γ(1 + 3α)Γ(1 + 7α)

,

...

(3.3)

and so on. The nth Approximate solution of the variational iteration method converges to theexact series solution. So, we approximate the solution y(t) = limn→∞yn(t).

In Figure 1, Approximate solution (y(t) ∼= y3(t)) of (3.4) using VIM and the exactsolution have been plotted for α = 1. In Figure 2, Approximate solution (y(t) ∼= y3(t)) of (3.4)using VIM and the exact solution have been plotted for α = 0.98.

Comment. This example has been solved using HAM, ADM, and HPM in [16–18]. It shouldbe noted that these methods have given same result after applying the Pade approximants ony(t).


0.2

0.4

0.6

0.8

y

0.2 0.4 0.6 0.8 1 1.2 1.4

t


Example 3.2. Consider the following fractional Riccati differential equation:

dαy

dtα= 2y(t) − y2(t) + 1, 0 < α � 1, (3.4)

subject to the initial condition y(0) = 0.

The exact solution of (3.4) is y(t) = 1 +√

2tanh(√

2t + (1/2) log((√

2 − 1)/(√

2 + 1))),when α = 1.

Expanding y(t) using Taylor expansion about t = 0 gives

y(t) = t + t2 +t3

3− t4

3− 7t5

15− 7t6

45+

53t7

315+

71t8

315+ · · · . (3.5)

The correction functional for (3.4) turns out to be

yn+1 = yn − Iα(dαyndξα

− 2yn + y2n − 1

)dξ. (3.6)


0.5

1

1.5

2

y

0.2 0.4 0.6 0.8 1 1.2 1.4

t


Beginning with y0(t) = tα/Γ(1 + α), by the iteration formulation (3.6), we can obtain directlythe other components as

y1(t) =tα

Γ(α + 1)+

√π21−2αt2α

Γ(α + 1)Γ(α + 1/2)− 4αt3αΓ(α + 1/2)√

πΓ(α + 1)Γ(3α + 1),

y2(t) =tα

Γ(α + 1)+

√π21−2αt2α

Γ(α + 1)Γ(α + 1/2)− 2−1+2αΓ(α + 1/2)t4α√

παΓ(α + 1)Γ(4α)+

43α

Γ(3α + 1)

− Γ(1 + 2α)t3α

Γ(α + 1)2Γ(3α + 1)− 12Γ(3α)t4αΓ(α)Γ(2α + 1)Γ(4α + 1)

−√π24−2αΓ(4α)t5α

Γ(α)Γ(α + 1/2)Γ(2α + 1)Γ(5α + 1)+

23+2αΓ(4α)Γ(α + 1/2)t5α√πΓ(α)Γ(α + 1)Γ(3α + 1)Γ(5α + 1)

+20Γ(5α)t6α

Γ(α)Γ(α + 1)Γ(3α + 1)Γ(6α + 1)− 1024αΓ(α + 1/2)2Γ(3α + 1/2)t7α√

π3Γ(α + 1)2Γ(3α + 1)Γ(7α + 1)

...

(3.7)

and so on. In Figure 3, Approximate solution (y(t) ∼= y3(t)) of (3.4) using VIM and the exactsolution have been plotted for α = 1. In Figure 4, Approximate solution (y(t) ∼= y3(t)) of (3.4)using VIM and the exact solution have been plotted for α = 0.98.


0.5

1

1.5

2

y

0.2 0.4 0.6 0.8 1 1.2 1.4

t


4. Conclusion

In this paper the variational iteration method is used to solve the fractional Riccati differentialequations. We described the method, used it on two test problems, and compared the resultswith their exact solutions in order to demonstrate the validity and applicability of the method.

Acknowledgments

The authors express their gratitude to the referees for their valuable suggestions andcorrections for improvement of this paper. Mathematica has been used for computations inthis paper.

References


[2] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.[3] H. Jafari and V. Daftardar-Gejji, “Solving a system of nonlinear fractional differential equations using

Adomian decomposition,” Journal of Computational and Applied Mathematics, vol. 196, no. 2, pp. 644–651, 2006.

[4] J. G. Lu and G. Chen, “A note on the fractional-order Chen system,” Chaos, Solitons & Fractals, vol. 27,no. 3, pp. 685–688, 2006.

[5] J. He, “A new approach to nonlinear partial differential equations,” Communications in NonlinearScience and Numerical Simulation, vol. 2, no. 4, pp. 230–235, 1997.

[6] J.-H. He, “Variational iteration method for autonomous ordinary differential systems,” AppliedMathematics and Computation, vol. 114, no. 2-3, pp. 115–123, 2000.


[7] J.-H. He, “Variational principles for some nonlinear partial differential equations with variablecoefficients,” Chaos, Solitons & Fractals, vol. 19, no. 4, pp. 847–851, 2004.

[8] M. A. Abdou and A. A. Soliman, “New applications of variational iteration method,” Physica D, vol.211, no. 1-2, pp. 1–8, 2005.

[9] S. Momani and S. Abuasad, “Application of He’s variational iteration method to Helmholtzequation,” Chaos, Solitons & Fractals, vol. 27, no. 5, pp. 1119–1123, 2006.

[10] Z. M. Odibat and S. Momani, “Application of variational iteration method to nonlinear differentialequations of fractional order,” International Journal of Nonlinear Sciences and Numerical Simulation, vol.7, no. 1, pp. 27–34, 2006.

[11] S. Momani and Z. Odibat, “Numerical approach to differential equations of fractional order,” Journalof Computational and Applied Mathematics, vol. 207, no. 1, pp. 96–110, 2007.

[12] J.-H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of ModernPhysics B, vol. 20, no. 10, pp. 1141–1199, 2006.

[13] H. Jafari, H. Hosseinzadeh, and E. Salehpoor, “A new approach to the gas dynamics equation: anapplication of the variational iteration method,” Applied Mathematical Sciences, vol. 2, no. 48, pp. 2397–2400, 2008.

[14] H. Jafari, A. Golbabai, E. Salehpoor, and Kh. Sayehvand, “Application of variational iteration methodfor Stefan problem,” Applied Mathematical Sciences, vol. 2, no. 60, pp. 3001–3004, 2008.

[15] H. Jafari and A. Alipoor, “A new method for calculating General Lagrange’s multiplier in thevariational iteration method,” Numerical Method for Partial Differential Equations, In press, 2010.

[16] J. Cang, Y. Tan, H. Xu, and S.-J. Liao, “Series solutions of non-linear Riccati differential equations withfractional order,” Chaos, Solitons & Fractals, vol. 40, no. 1, pp. 1–9, 2009.

[17] S. Momani and N. Shawagfeh, “Decomposition method for solving fractional Riccati differentialequations,” Applied Mathematics and Computation, vol. 182, no. 2, pp. 1083–1092, 2006.

[18] Z. Odibat and S. Momani, “Modified homotopy perturbation method: application to quadratic Riccatidifferential equation of fractional order,” Chaos, Solitons & Fractals, vol. 36, no. 1, pp. 167–174, 2008.


Research ArticleSolitary Wave Solutions for a Time-FractionGeneralized Hirota-Satsuma Coupled KdVEquation by a New Analytical Technique

Majid Shateri and D. D. Ganji

Department of Mechanical Engineering, Babol University of Technology,P.O. Box 484, 47148 71167 Babol, Iran

Correspondence should be addressed to D. D. Ganji, ddg [email protected]

Received 17 May 2009; Accepted 7 July 2009

Academic Editor: Shaher Momani

Copyright q 2010 M. Shateri and D. D. Ganji. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

A new iterative technique is employed to solve a system of nonlinear fractional partial differentialequations. This new approach requires neither Lagrange multiplier like variational iterationmethod (VIM) nor polynomials like Adomian’s decomposition method (ADM) so that can be moreeasily and effectively established for solving nonlinear fractional differential equations, and willovercome the limitations of these methods. The obtained numerical results show good agreementwith those of analytical solutions. The fractional derivatives are described in Caputo sense.

1. Introduction

In recent years, it has been turned out that fractional differential equations can beused successfully to model many phenomena in various fields such as fluid mechanics,viscoelasticity, physics, chemistry, and engineering. For instance, the fluid−dynamics trafficmodel with fractional derivatives [1] is able to eliminate the deficiency arising from theassumption of continuum traffic flow, and the nonlinear oscillation of earthquakes can bemodeled by fractional derivatives [2]. Fractional differentiation and integration operators canalso be used for extending the diffusion and wave equations [3]. Most of fractional differentialequations do not have exact analytical solutions, hence considerable heed has been focusedon the approximate and numerical solutions of these equations. Although variationaliteration method [4−8] and Adomian’s decomposition method [9−14] are approaches thathave been utilized extensively to provide analytical approximations of linear and nonlinearproblems, they have limitations due to complicated algorithms of calculating Adomianpolynomials for nonlinear fractional problems, and an inherent inaccuracy in determining


the Lagrange multiplier for fractional equations. In this study, a new alternative procedurethat needs no Lagrange multiplier or Adomian polynomials is used to obtain an analyticalapproximate solution of a system of nonlinear fractional partial differential equations (1.1) toillustrate the effectiveness, accuracy, and convenience of this method.

In this work, we consider the solution of generalized Hirota−Satsuma coupled KdVof time−fractional order which is presented by a system of nonlinear partial differentialequations, of the form:

Dαt u =

12uxxx − 3uux + 3(vw)x,

Dαt v = −vxxx + 3uvx,

Dαt w = −wxxx + 3uwx,

0 < α < 1, (1.1)

subject to the following initial conditions:

u(x, 0) =β − 2k2

3+ 2k2tanh2(kx),

v(x, 0) = −4k2c0

(β + k2)

3c21

+4k2(β + k2)tanh(kx)

3c1,

w(x, 0) = c0 + c1tanh(kx),

(1.2)

where k, c0, c1 /= 0, and β are arbitrary constants. The Hirota−Satsuma system of equations[15] was introduced to describe the interaction of two long waves with different dispersionrelations. The case of α = 1 in system (1.1) was solved by Wu et al. [16]. If c = −β, then in onecase the u(x, t) = ((β − 2k2)/3) + 2k2tanh2(k(x − ct)), v(x, t) = −(4k2c0(β + k2)/3c2

1) + (4k2(β +k2)/3c1)tanh(k(x − ct)) and w(x, t) = c0 + c1tanh(k(x − ct)) are travelling−wave solutions ofsystem (1.1) when α = 1.

2. Basic Definitions

In this section, there are some basic definitions and properties of the fractional calculus theorywhich are used in this paper.

Definition 2.1. A real function f(x), x > 0, is mentioned to be in the space Cμ, μ ∈ R if thereexists a real number p(> μ) such that f(x) = xpf1(x), where f1(x) ∈ C[0,∞), and it is said tobe in the space Cm

μ if f (m) ∈ Cμ, m ∈N.

Definition 2.2. The left−sided Riemann−Liouville fractional integral operator of order α ≥ 0,of a function f ∈ Cμ, μ ≥ −1 is defined as

D−αf(x) =1

Γ(α)

∫x

0(x − t)α−1f(t)dt, α > 0, x > 0,

D0f(x) = f(x).

(2.1)


The properties of the operator D−α can be found in [1, 17], and we only mention thefollowing (in this case, f ∈ Cμ, μ ≥ −1, α, β ≥ 0 and γ > −1):

D−αD−βf(x) = D−(α+β)f(x),

D−αD−βf(x) = D−βD−αf(x),

D−αxγ =Γ(γ + 1

)

Γ(α + γ + 1

)xα+γ .

(2.2)

The Riemann−Liouville derivative has certain disadvantages, in trying way to modelreal−world phenomena with fractional differential equations. Therefore, we will employ amodification of fractional differential operator Dα

∗ proposed by Caputo, in his work [18] onthe theory of viscoelasticity.

Definition 2.3. The (left−sided) Caputo fractional derivative of f(x) is defined as

Dα∗f(x) = D

α−mDmf(x)

=1

Γ(m − α)

∫x

0(x − t)m−α−1f (m)(t)dt,

(2.3)

for m − 1 < α ≤ m, m ∈N, x > 0, f ∈ Cm−1

Also, we need two of its basic properties.

Lemma 2.4. If m − 1 < α ≤ m, m ∈N, and f ∈ Cmμ , μ ≥ −1, then one has:

Dα∗D

−αf(x) = f(x),

D−αDα∗f(x) = f(x) −

m−1∑

k=0

f (k)(0+)xk

k!, x > 0.

(2.4)

The Caputo fractional derivative is considered here, because it allows traditional initialand boundary conditions to be included in the formulation of the problem.

In this work, we consider the one−dimensional linear nonhomogeneous fractionalpartial differential equations in fluid mechanics, where the unknown function u(x, t) isassumed to be a causal function of time, that is, vanishing for t < 0.

Definition 2.5. For m as the smallest integer that exceeds α, the Caputo time−fractionalderivative operator of order α > 0 is defined as

Dα∗tu(x, t) =

∂αu(x, t)∂tα

=

⎧⎪⎪⎨

⎪⎪⎩

1Γ(m − α)

∫ t

0(t − τ)m−α−1 ∂

mu(x, τ)∂τm

dτ, m − 1 < α < m,

∂mu(x, t)∂tm

, α = m ∈N.

(2.5)

For more information on the mathematical properties of fractional derivatives andintegrals, one can consult the mentioned references.


−0.4

−0.3

−0.2

−0.1

0

u(1, 0.6)v(1, 0.6)w(1, 0.6)

−3 −2 −1 0 1

h

Figure 1: h−curves of u(x, t): dash, v(x, t): dash dot, and w(x, t): solid when k = 0.1, α = 1, β = 1.5, c0 = 1.5,c1 = 0.1 at point (1,0.6).

3. Basic Ideas of Fractional Iteration Method (FIM)

As pointed in [19], to illustrate fractional iteration method, we consider the followingnonlinear fractional differential equation (more general form can be considered without lossof generality):

Dα∗y(x) − f

(x, y(x)

)= 0, y(k)(0) = ak, k = 0, 1, . . . , m − 1, (3.1)

where the fractional differential operator Dα∗ is dened as in (2.3), m − 1 < α ≤ m, m ∈ N, f is

a nonlinear function of y, and y is an unknown function to be determined later. We want tofind a solution y of (3.1) having the form

y(x) = limn→∞

yn(x). (3.2)

Let H(x)/= 0 denote the so−called auxiliary function. Multiplying (3.1) by H(x) andthen applying D−γ , the Riemann−Liouville fractional integral operator, of order γ ≥ 0 definedby (2.1), on both sides of the resulted term yields

D−γ(H(x)

[Dα∗y(x) − f

(x, y(x)

)])= 0. (3.3)

Let h/= 0 denote the so−called auxiliary parameter. Multiplying (3.3) by h and then adding y,the solution of (3.1), on both sides of the resulted term yields

y(x) = y(x) + hD−γ(H(x)

[Dα∗y(x) − f

(x, y(x)

)]). (3.4)


0.493

0.498

0.503

0.508

0.513

−40−20

020

404

32

10

t

x

(a)

0.493

0.498

0.503

0.508

0.513

−40−20

020

404

32

10

t

x

(b)

Figure 2: The solitary wave solution of u(x, t), FIM result (a) and exact solution (b), when k = 0.1, α = 1,β = 1.5, c0 = 1.5, c1 = 0.1.

−3.22

−3.12

−3.02

−2.92

−2.82

−40−20

020

40 4

3

2

1

0

t

x

(a)

−3.22

−3.12

−3.02

−2.92

−2.82

−40−20

020

40 4

3

2

1

0

t

x

(b)

Figure 3: The solitary wave solution of v(x, t), FIM result (a) and exact solution (b), when k = 0.1, α = 1,β = 1.5, c0 = 1.5, c1 = 0.1.

However (3.4) can be solved iteratively as follows:

yn+1(x) = yn(x) + hD−γ(H(x)

[Dα∗yn(x) − f

(x, yn(x)

)]). (3.5)

In (3.5), the subscript n denotes the nth iteration, and provided that the right hand ofit, that is, y(x)+hD−γ(H(x)[Dα

∗y(x)−f(x, y(x))]), is a contractive mapping. The convergenceof (3.5) is ensured by Banach’s fixed point theorem [20], as is shown in [19].

Now, we introduce a new convenient technique for controlling the convergence regionand rate of solution series for this method. Assume that we gain a family of solution series in


1.4

1.45

1.5

1.55

1.6

−40−20

020

40 4

3

2

1

0

t

x

(a)

1.4

1.45

1.5

1.55

1.6

−40−20

020

40 4

3

2

1

0

t

x

(b)

Figure 4: The solitary wave solution of w(x, t), FIM result (a) and exact solution (b), when k = 0.1, α = 1,β = 1.5, c0 = 1.5, c1 = 0.1.

Table 1: Numerical values when α = 0.5, 0.75, 1.0 and k = 0.1, β = 1.5, c0 = 1.5, c1 = 0.1 for u(x, t).

t xα = 0.5 α = 0.75 α = 1.0

uFIM uHPM uFIM uHPM uFIM uHPM uExact

0.2

0 0.4935113355 0.4933513333 0.4933937249 0.4933513333 0.4933513226 0.4933513333 0.49335132250.25 0.4935979177 0.4934136490 0.4934546200 0.4934060408 0.4933937118 0.4933937581 0.49339371150.5 0.4937081918 0.4935005536 0.4935399023 0.4934853752 0.4934607890 0.4934608711 0.49346078910.75 0.4938416235 0.4936116158 0.4936491516 0.4935889426 0.4935522228 0.4935523392 0.4935522227

1 0.4939975728 0.4937462882 0.4937818335 0.4937162326 0.4936675611 0.4936677111 0.4936675613

0.4

0 0.4936853330 0.4934053333 0.4935033001 0.4934053333 0.4934051602 0.4934053333 0.49340516090.25 0.4938008948 0.4935090262 0.4935963610 0.4934954686 0.4934771397 0.4934775983 0.49347714010.5 0.4939391903 0.4936368323 0.4937131147 0.4936097847 0.4935733947 0.4935741334 0.49357339450.75 0.4940995646 0.4937881198 0.4938529939 0.4937477171 0.4936934501 0.4936944620 0.4936934499

1 0.4942812668 0.4939621476 0.4940153259 0.4939085897 0.4938367205 0.4938379949 0.4938367203

0.6

0 0.4938553137 0.4934953333 0.4936435681 0.4934953333 0.4934944586 0.4934953333 0.49349446250.25 0.4939921388 0.4936348749 0.4937639292 0.4936174202 0.4935955155 0.4935973485 0.49359551860.5 0.4941508292 0.4937979026 0.4939071763 0.4937630802 0.4937202663 0.4937230371 0.49372026750.75 0.4943306471 0.4939836133 0.4940726248 0.4939315968 0.4938681006 0.4938717813 0.4938681010

1 0.4945307679 0.4941911034 0.4942594934 0.4941221502 0.4940383061 0.4940428589 0.4940383061

the auxiliary parameter h by the means of fractional iteration method. Like HAM, by plottingthe amount of the function or one of its derivatives at a particular point with respect to theauxiliary parameter h (which is the so−called h−curve), we can obtain a proper value of hthat ensures the convergence of the solution series. This proper value of h corresponds to thecurve segment nearly parallel to the horizontal axis in the h−curve plot. Therefore, if we seth any value in this region, which is so−called the valid region of h, we are quite sure that thecorresponding solution series converge.

Having freedom for choosing the auxiliary function H(x), the auxiliary parameter h,the initial approximation y0(x), and the fractional integral order γ, that is, fundamental to


Table 2: Numerical values when α = 0.5, 0.75, 1.0 and k = 0.1, β = 1.5, c0 = 1.5, c1 = 0.1 for v(x, t).

t x α = 0.5 α = 0.75 α = 1.0vFIM vHPM vFIM vHPM vFIM vHPM vExact

0.2

0 −3.004851885 −3.009951963 −3.010186962 −3.011485031 −3.013961813 −3.013960000 −3.0139618110.25 −2.999873282 −3.004930481 −3.005175951 −3.006462592 −3.008937820 −3.008936014 −3.0089378190.5 −2.994919279 −2.999927784 −3.000183326 −3.001457026 −3.003927607 −3.003925818 −3.0039276060.75 −2.989995777 −2.994950010 −2.995215173 −2.996474487 −2.998937350 −2.998935586 −2.998937348

1 −2.985108534 −2.990003168 −2.990277460 −2.991521007 −2.993973118 −2.993971391 −2.993973120

0.4

0 −2.998707345 −3.001587258 −3.003540427 −3.004319121 −3.007934500 −3.007920000 −3.0079344750.25 −2.993774914 −2.996584581 −2.998560913 −2.999314737 −3.002927784 −3.002913367 −3.0029277630.5 −2.988873896 −2.991611047 −2.993607731 −2.994336092 −2.997942249 −2.997927984 −2.9979422280.75 −2.984009951 −2.986672646 −2.988686777 −2.989389199 −2.992983946 −2.992969899 −2.992983924

1 −2.979188572 −2.981775197 −2.983803790 −2.984479922 −2.988058789 −2.988045031 −2.988058768

0.6

0 −2.994081735 −2.994318388 −2.997767879 −2.997835535 −3.001928965 −3.001880000 −3.0019287660.25 −2.989192682 −2.989342884 −2.992826759 −2.992857834 −2.996948389 −2.996899772 −2.9969481970.5 −2.984339775 −2.984405468 −2.987918608 −2.987913837 −2.991996239 −2.991948208 −2.9919960540.75 −2.979528475 −2.979511953 −2.983049116 −2.983009390 −2.987078404 −2.987031190 −2.987078224

1 −2.974764060 −2.974667936 −2.978223778 −2.978150144 −2.982200600 −2.982154433 −2.982200431

Table 3: Numerical values when α = 0.5, 0.75, 1.0 and k = 0.1, β = 1.5, c0 = 1.5, c1 = 0.1 for w(x, t).

t xα = 0.5 α = 0.75 α = 1.0

wFIM wHPM wFIM wHPM wFIM wHPM wExact

0.2

0 1.507523900 1.504990747 1.504874024 1.504229289 1.502999100 1.503000000 1.5029991000.25 1.509996714 1.507484860 1.507362938 1.506723878 1.505494461 1.505495357 1.5054944610.5 1.512457311 1.509969643 1.509842719 1.509210085 1.507982979 1.507983864 1.5079829770.75 1.514902760 1.512442049 1.512310343 1.511684858 1.510461580 1.510462458 1.510461582

1 1.517330199 1.514899088 1.514762850 1.514145194 1.512927259 1.512928117 1.512927258

0.4

0 1.510575823 1.509145401 1.508175285 1.507788516 1.505992798 1.506000000 1.5059928100.25 1.513025704 1.511630175 1.510648553 1.510274137 1.508479570 1.508486739 1.5084795880.5 1.515459985 1.514100473 1.513108742 1.512746974 1.510955834 1.510962922 1.5109558470.75 1.517875852 1.516553321 1.515552926 1.515204040 1.513418571 1.513425547 1.513418581

1 1.520270576 1.518985828 1.517978251 1.517642422 1.515864843 1.515871673 1.515864850

0.6

0 1.512873311 1.512755768 1.511042443 1.511008840 1.508975680 1.509000000 1.5089757780.25 1.515301647 1.515227046 1.513496642 1.513481209 1.511449479 1.511473624 1.5114495710.5 1.517712032 1.517679404 1.515934464 1.515936836 1.513909155 1.513933009 1.5139092450.75 1.520101750 1.520109958 1.518353088 1.518372820 1.516351785 1.516375237 1.516351875

1 1.522468182 1.522515926 1.520749779 1.520786351 1.518774537 1.518797467 1.518774621

the validity and flexibility of the FIM, we can suppose that all of them are properly chosen,therefore, the iterative scheme (3.5) will converge to the exact solution. Accordingly, thesuccessive approximations yn(x), n ≥ 0, of the solution y(x) will be obtained by choosingy0(x) that at least satises the initial and/or boundary conditions. Consequently, the exactsolution may be obtained by using y(x) = limn→∞yn(x).

4. Applications

In this section, we implement fractional iteration method to generalized Hirota−Satsumacoupled KdV of time−fractional order when 0 < α ≤ 1. For convenience in applying FIMmethod, we choose the initial conditions given in (1.2) as the initial approximations:


u0(x, t) =β − 2k2

3+ 2k2tanh2(kx),

v0(x, t) = −4k2c0

(β + k2)

3c21

+4k2(β + k2)tanh(kx)

3c1,

w0(x, t) = c0 + c1tanh(kx).

(4.1)

Choosing γ = α and H(x, t) = 1, we can construct the iterative scheme (3.5) forinvestigation of the traveling wave solution of (1.1) as follows:

un+1(x, t) = un(x, t) + hDt−α

×[

D∗tαun(x, t) −

12∂3

∂x3un(x, t) + 3un(x, t)

∂

∂xun(x, t) − 3

∂

∂x(vn(x, t)wn(x, t))

]

,

vn+1(x, t) = vn(x, t) + hDt−α

×[

D∗tαvn(x, t) +

∂3

∂x3vn(x, t) − 3un(x, t)

∂

∂xvn(x, t)

]

,

wn+1(x, t) = wn(x, t) + hDt−α

×[

D∗tαwn(x, t) +

∂3

∂x3wn(x, t) − 3un(x, t)

∂

∂xwn(x, t)

]

.

(4.2)

Substituting the initial approximations, (4.1), into (4.2), for the case α = 1, yields

u1 = 0.493 + 0.02 tanh(0.1x)2 + h(

0.016(

0.1 − 0.1 tanh(0.1x)2)2

tanh(0.1x)t

− 0.0008 tanh(0.1x)3(

0.1 − 0.1 tanh(0.1x)2)t + 0.12

(0.493 + 0.02 tanh(0.1x)2

)

× tanh(0.1x)(

0.1 − 0.1 tanh(0.1x)2)t − 3

(0.001342 − 0.001342 tanh(0.1x)2

)

×(1.5 + 1.5 tanh(0.1x))t−3(−0.001342 + 0.001342 tanh(0.1x))(

0.15 − 0.15 tanh(0.1x)2)t

),

v1 = −0.001342 + 0.001342 tanh(0.1x) + h(−0.002684

(0.1 − 0.1 tanh(0.1x)2

)2t

+ 0.0005368 tanh(0.1x)2(

0.1 − 0.1 tanh(0.1x)2)t − 3

(0.493 + 0.02 tanh(0.1x)2

)

×(

0.001342 − 0.001342 tanh(0.1x)2)t

),

w1 = 1.5 + 1.5 tanh(0.1x) + h(−0.3

(0.1 − 0.1 tanh(0.1x)2

)2t

+ 0.06 tanh(0.1x)2(

0.1 − 0.1 tanh(0.1x)2)− 3

(0.493 + 0.02 tanh(0.1x)2

)

×(

0.15 − 0.15 tanh(0.1x)2)t

).

(4.3)


5. Result and Discussion

In this section, four figures are presented corresponding to FIM results and exact solutionsfor the solitary wave solutions u(x, t), v(x, t), and w(x, t) with the initial conditions (1.2),when k = 0.1, α = 1, β = 1.5, c0 = 1.5, c1 = 0.1. Furthermore, numerical values for the caseα = 0.5, 0.75, 1.0, and k = 0.1, β = 1.5, c0 = 1.5, c1 = 0.1 are obtained for u(x, t), v(x, t), andw(x, t).

Demonstrating the exactness of FIM, the numerical results are presented and onlyfew iterations are required to achieve accurate solutions. The convergence of FIM for thegeneralized fractional−order Hirota−Satsuma−coupled KdV equation is controllable, usingthe so−called h−curves presented in Figure 1 which are obtained based on the fourth−orderFIM approximate solutions. In general, by the means of the so−called h−curve, it is straightforward to choose a proper value of h which ensures that the solution series is convergent.This proper value of h corresponds to the curve segment nearly parallel to the horizontal axis.Both exact results and approximate solutions obtained for the first four approximations areplotted in Figures 2, 3, and 4. There are no visible differences in two solutions of each pair ofdiagrams.

Tables 1, 2, and 3 show the numerical values by FIM when α = 0.5, 0.75, 1.0 andk = 0.1, β = 1.5, c0 = 1.5, c1 = 0.1 for u(x, t), v(x, t), and w(x, t) respectively.

6. Conclusion

In this paper, the fractional iteration method (FIM) has been successfully applied to studyHirota−Satsuma−coupled KdV of time−fractional−order equation. FIM results are comparedwith the exact solutions and those obtained by Homotopy perturbation method [21].

The results show that fractional iteration method is a powerful and efficient techniquein finding exact and approximate solutions for nonlinear partial differential equations offractional order. The method provides the user with more realistic series solutions thatconverge very rapidly in real physical problems.

Compared with the ADM and VIM, the FIM has following advantages, [19].

(1) The auxiliary parameter h provides us with a convenient way to modify and controlthe convergence region of the solution.

(2) The solution of a given nonlinear problem can be expressed by an infinite number ofsolution series and thus can be more efficiently approximated by a better selectionof the auxiliary parameter values.

(3) Unlike the ADM, the FIM method is free from the need to use Adomianpolynomials.

(4) This method has no need for the Lagrange multiplier, correction functional,stationary conditions, the variational theory, and so forth, which eliminates thecomplications that exist in the VIM.

(5) The fractional iteration method can be easily comprehended with only a basicknowledge of fractional calculus.

(6) Compared to the ADM and VIM, the presented method proves simpler in itsprinciples and more convenient for computer algorithms.

In this work, we used Maple Package to calculate the series obtained by fractionaliteration method.


References

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New York, NY, USA, 1997.[3] W. R. Schneider and W. Wyss, “Fractional diffusion and wave equations,” Journal of Mathematical

Physics, vol. 30, pp. 134–144, 1989.[4] J. H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous

media,” Computer Methods in Applied Mechanics and Engineering, vol. 167, pp. 57–68, 1998.[5] J. H. He, “Variational iteration method—some recent results and new interpretations,” Journal of

Computational and Applied Mathematics, vol. 207, no. 1, pp. 3–17, 2007.[6] J. H. He and X. H. Wu, “Variational iteration method: new development and applications,” Computers

and Mathematics with Applications, vol. 54, no. 7-8, pp. 881–894, 2007.[7] H. Tari, D. D. Ganji, and H. Babazadeh, “The application of He’s variational iteration method to

nonlinear equations arising in heat transfer,” Physics Letters, Section A, vol. 363, no. 3, pp. 213–217,2007.

[8] S. Abbasbandy and A. Shirzadi, “The variational iteration method for a family of fifth-order boundaryvalue differential equations,” International Journal of Nonlinear Dynamics in Engineering and Sciences,vol. 1, no. 1, pp. 39–46, 2009.

[9] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer AcademicPublishers, Boston, Mass, USA, 1994.

[10] G. Adomian, “A review of the decomposition method in applied mathematics,” Journal ofMathematical Analysis and Applications, vol. 135, no. 2, pp. 501–544, 1988.

[11] G. Adomian, “Solutions of nonlinear P.D.E,” Applied Mathematics Letters, vol. 11, no. 3, pp. 121–123,1998.

[12] Q. Esmaili, A. Ramiar, E. Alizadeh, and D. D. Ganji, “An approximation of the analytical solution ofthe Jeffery-Hamel flow by decomposition method,” Physics Letters A, vol. 372, no. 19, pp. 3434–3439,2008.

[13] A. M. Wazwaz, “A new algorithm for calculating adomian polynomials for nonlinear operators,”Applied Mathematics and Computation, vol. 111, no. 1, pp. 53–69, 2000.

[14] S. Momani and Z. Odibat, “Analytical solution of a time-fractional Navier-Stokes equation byAdomian decomposition method,” Applied Mathematics and Computation, vol. 177, pp. 488–494, 2006.

[15] R. Hirota and J. Satsuma, “Soliton solutions of a coupled Korteweg-de Vries equation,” Physics LettersA, vol. 85, no. 8-9, pp. 407–408, 1981.

[16] Y. Wu, X. Geng, X. Hu, and S. Zhu, “A generalized Hirota-Satsuma coupled Korteweg-de vriesequation and Miura transformations,” Physics Letters A, vol. 255, pp. 259–264, 1999.

[17] I. Podlubny, “Numerical solution of ordinary fractional differential equations by the fractionaldifference method,” in Advances in Difference Equations, S. Elaydi, I. Gyori, and G. Ladas, Eds., Gordonand Breach, Amsterdam, The Netherlands, 1997.

[18] M. Caputo, “Linear models of dissipation whose Q is almost frequency independent,” Journal of theRoyal Astronomical Society, vol. 13, pp. 529–539, 1967.

[19] A. Ghorbani, “Toward a new analytical method for solving nonlinear fractional differentialequations,” Computer Methods in Applied Mechanics and Engineering, vol. 197, pp. 4173–4179, 2008.

[20] M. Reed and B. Simon, Methods of Modern Mathematical Physics, I: Functional Analysis, Academic Press,New York, NY, USA, 1980.

[21] Z. Z. Ganji, D. D. Ganji, and Y. Rostamiyan, “Solitary wave solutions for a time-fraction generalizedHirota-Satsuma coupled KdV equation by an analytical technique,” Applied Mathematical Modelling,vol. 33, no. 7, pp. 3107–3113, 2009.


Research ArticleTime-Optimal Control of Systems withFractional Dynamics

Christophe Tricaud and YangQuan Chen

Center for Self-Organizing and Intelligent Systems (CSOIS), Department of Electrical andComputer Engineering, Utah State University, 41260 Old Main Hill, Logan, UT 84322-4160, USA

Correspondence should be addressed to Christophe Tricaud, [email protected]

Received 1 August 2009; Accepted 5 December 2009

Academic Editor: Wen Chen

Copyright q 2010 C. Tricaud and Y. Chen. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We introduce a formulation for the time-optimal control problems of systems displaying fractionaldynamics in the sense of the Riemann-Liouville fractional derivatives operator. To propose asolution to the general time-optimal problem, a rational approximation based on the Hankeldata matrix of the impulse response is considered to emulate the behavior of the fractionaldifferentiation operator. The original problem is then reformulated according to the new modelwhich can be solved by traditional optimal control problem solvers. The time-optimal problem isextensively investigated for a double fractional integrator and its solution is obtained using eithernumerical optimization or time-domain analysis.

1. Introduction

In the world surrounding us, the physical laws of dynamics are not always followed by allsystems. When the considered systems are complex or can only be studied on a macroscopicscale, they sometimes divert from the traditional integer order dynamic laws. In some cases,their dynamics follow fractional-order laws meaning that their behavior is governed byfractional-order differential equations [1]. As an illustration, it can be found in the literature[2] that materials with memory and hereditary effects, and dynamical processes, such asgas diffusion and heat conduction, in fractal porous media can be more precisely modeledusing fractional-order models than using integer-order models. Another vein of research hasidentified the dynamics of a frog’s muscle to display fractional-order behavior [3].

Optimal Control Problems (OCPs) or Integer-Order Optimal Controls (IOOCs) canbe found in a wide variety of research topics such as engineering and science of course, buteconomics as well. The field of IOOCs has been investigated for a long time and a largecollection of numerical techniques has been developed to solve this category of problems [4].


The main objective of an OCP is to obtain control input signals that will make a givensystem or process satisfy a given set of physical constraints (either on the system’s statesor control inputs) while extremizing a performance criterion or a cost function. FractionalOptimal Control Problems (FOCPs) are OCPs in which the criterion and/or the differentialequations governing the dynamics of the system display at least one fractional derivativeoperator. The first record of a formulation of the FOCP was given in [5]. This formulationwas general but includes constraints on the system’s states or control inputs. Later, a generaldefinition of FOCP was formulated in [6] that is similar to the general definition of OCPs. Thenumber of publications linked to FOCPs is limited since that problem has only been recentlyconsidered.

Over the last decade, the framework of FOCPs has hatched and grown. In [5], Agrawalgives a general formulation of FOCPs in the Riemann-Liouville (RL) sense and proposesa numerical method to solve FOCPs based on variational virtual work coupled with theLagrange multiplier technique. In [7], the fractional derivatives (FDs) of the system areapproximated using the Grunwald-Letnikov definition, providing a set of algebraic equationsthat can be solved using numerical techniques. The problem is defined in terms of theCaputo fractional derivatives in [8] and an iterative numerical scheme is introduced to solvethe problem numerically. Distributed systems are considered in [9] and an eigenfunctiondecomposition is used to solve the problem. Ozdemir et al. [10] also use eigenfunctionexpansion approach to formulate an FOCP of a 2-dimensional distributed system. Cylindricalcoordinates for the distributed system are considered in [11]. A modified Grunwald-Letnikovapproach is introduced in [12] which leads to a central difference scheme. Frederico andTorres [13–15], using similar definitions of the FOCPs, formulated a Noether-type theoremin the general context of the fractional optimal control in the sense of Caputo and studiedfractional conservation laws in FOCPs. In [6], a rational approximation of the fractionalderivatives operator is used to link FOCPs and the traditional IOOCs. A new solution schemeis proposed in [16], based on a different expansion formula for fractional derivatives.

In this article, we introduce a formulation to a special class of FOCP: the FractionalTime-Optimal Control Problem (FTOCP). Time-optimal control problems are also referredto in the literature as minimum-time control problems, free final time-optimal control, orbang-bang control problems. These different denominations define the same kind of optimalcontrol problem in which the purpose is to transfer a system from a given initial stateto a specified final state in minimum time. So far, this special class of FOCPs has beendisregarded in the literature. In [6], such a problem was solved as an example to demonstratethe capability and generality of the proposed method but no thorough studies were done.

The article is organized as follows. In Section 2, we give the definitions of fractionalderivatives in the RL sense and FOCP and introduce the formulation of FTOCP. In Section 3,we consider the problem of the time-optimal control of a fractional double integrator andpropose different schemes to solve the problem. In Section 4, the solution to the problemis obtained for each approach for a given system. Finally, we give our conclusions inSection 5.

2. Formulation of the Fractional Time-Optimal Control Problem

2.1. The Fractional Derivative Operator

There exist several definitions of the fractional derivative operator: Riemann-Liouville,Caputo, Grunwald-Letnikov, Weyl, as well as Marchaud and Riesz [17–20]. Here, we are


interested in the Riemann-Liouville definition of the fractional derivatives for the formulationof the FOCP.

The Left Riemann-Liouville Fractional Derivative (LRLFD) of a function f(t) isdefined as

aDαt f(t) =

1Γ(n − α)

(d

dt

)n∫ ta

(t − τ)n−α−1f(τ)dτ, (2.1)

where Γ(·) is the Gamma function defined for any complex number z as

Γ(z) =∫0

∞tz−1e−tdt (2.2)

and where the order of the derivative α satisfies n − 1 ≤ α < n. The Right Riemann-LiouvilleFractional Derivative (RRLFD) is defined as

tDαb f(t) =

1Γ(n − α)

(− ddt

)n∫bt

(τ − t)n−α−1f(τ)dτ. (2.3)

2.2. Fractional Optimal Control Problem Formulation

With the RL definition of the fractional derivatives operator given in (2.1) and (2.3), we canspecify a general FOCP: find the optimal control u(t) for a fractional dynamical system thatminimizes the following performance criterion:

J(u) = Go(x(a), x(b)) +∫ba

Lo(x, u, t)dt (2.4)

subject to the following system dynamics:

aDαt x(t) = H(x, u, t) (2.5)

with initial condition

x(a) = xa (2.6)

and with the following constraints:

umin(t) ≤ u(t) ≤ umax(t),

xmin(a) ≤ x(a) ≤ xmax(a),

Lνti(t, x(t), u(t)) ≤ 0,

Gνei(x(a), x(b)) ≤ 0,

Gνee(x(a), x(b)) = 0,

(2.7)


where x is the state variable, t ∈ [a, b] stands for the time, and L, G, and H are arbitrary givennonlinear functions. The subscripts o, ti, ei, and ee on the functions G(·, ·) and L(·, ·, ·) standfor, respectively, objective function, trajectory constraint, endpoint inequality constraint andendpoint equality constraint.

2.3. Fractional Time-Optimal Control Problem Formulation

A FTOCP is defined by a performance index of the form

J(u) =∫ba

1dt = b − a, (2.8)

which appears when we desire to minimize the time required to reach a given target xb givensome initial conditions xa.

Under such conditions, the problem is to transfer the system whose dynamics aregiven by

aDαt x(t) = H(x, u, t) (2.9)

from a given initial state x(a) = xa to the desired state x(b) = xb. The minimum time requiredto reach the target is defined as t∗.

To ensure that the problem has a solution, the control variables are required to beconstrained in the following way:

umin ≤ u(t) ≤ umax. (2.10)

In the following, we will make use of the minimum principle to determine the optimal controllaw u∗(t) for the previously defined problem. We define the state associated with the optimalcontrol law as x∗(t).

We define the Hamiltonian H for the problem described by the dynamic system (2.9)and the criterion (2.8) as

H(x, u, t) = 1 + λ(H(x, u, t) − aD

αt x), (2.11)

where λ(t) stands for the costate variable. The optimal costate variable is defined as λ∗(t).In the case of constrained systems, the results given in [5] do not apply as the control

function u(t) is constrained and does not have arbitrary variations. Indeed, if the control u(t)lies on the boundary in (2.10), then the variations are not arbitrary. Instead, we need to usePontryagin’s minimum principle [4]. According to the proof of the theorem in [21], we usefor demonstration arbitrary variations in the control signal u(t) = u∗(t) + δu(t). We defineboth the increment ΔJ and the (first) variation δJ of the performance index J as

ΔJ(u∗, δu) = J(u) − J(u∗) ≥ 0 for minimum

= δJ(u∗, δu) +O2,(2.12)


where the first variation is defined as

δJ =∂J

∂uδu(t). (2.13)

With the constraints (2.10) and making the assumption that all the admissible variationson the control ‖δu(t)‖ are small enough to ensure that the sign of the increment ΔJ canbe determined by the sign of the variation δJ , the necessary condition on u∗ to minimizeJ becomes

δJ(u∗(t), δu(t)) ≥ 0. (2.14)

According to [21, Chapter 2], the first variation can be defined as

δJ(u∗(t), δu(t)) =∫ba

([∂H∂x

+ λ(t)]∗δx(t) +

[∂H∂u

]′∗δu(t) +

[∂H∂λ

− x(t)]′∗δλ(t)

)dt

+[∂S

∂x+ λ(t)

]′∗bδxb +

[H +

∂S

∂t

]∗bδb.

(2.15)

In the previous equation,

(1) if the optimal state x∗ equations are satisfied, then we obtain the state relation,

(2) if the costate λ∗ is chosen so that the coefficient of the dependent variation δx in theintegrand is identically zero, then we obtain the costate equation,

(3) the boundary condition is chosen so that it results in the auxiliary boundarycondition.

When all of the previous enumerated items are satisfied, the first variation can bereformulated as

δJ(u∗, δu) =∫ [

∂H∂u

]′δudt. (2.16)

The integrand in the previous relation is the first-order approximation to change in theHamiltonian H due to a change in u alone. This means that by definition

[∂H∂u

(x∗, u∗, λ∗, t)]δu ≡ H(x∗, u∗ + δu, λ∗, t) −H(x∗, u∗, λ∗, t) (2.17)

combining the previous two equations leads us to

δJ(u∗, δu) =∫ba

[H(x∗, u∗ + δu, λ∗, t) −H(x∗, u∗, λ∗, t)]dt. (2.18)


Now, using the above, the necessary condition becomes

∫ba

[H(x∗, u∗ + δu, λ∗, t) −H(x∗, u∗, λ∗, t)]dt ≥ 0, (2.19)

for all admissible δu less than a small value. The relation becomes

H(x∗, u∗ + δu, λ∗, t) ≥ H(x∗, u∗, λ∗, t). (2.20)

Replacing u∗ + δu by u, the necessary condition becomes

H(x∗, u∗, λ∗, t) ≤ H(x∗, u, λ∗, t). (2.21)

When applied to our problem, we obtain

1 + λ∗(H(x∗, u∗, t) − aD

αt x

∗) ≤ 1 + λ∗(H(x∗, u, t) − aD

αt x

∗), (2.22)

which can be simplified to

λ∗(H(x∗, u∗, t) − aD

αt x

∗) ≤ λ∗(H(x∗, u, t) − aDαt x

∗),H(x∗, u∗, t) ≤ H(x∗, u, t).

(2.23)

The state and costate equations can be retrieved from [5] and give

aDαt x = H(x, u, t),

tDαbλ =

∂H(x, u, t)∂x

λ,(2.24)

with

x(a) = xa, x(b) = xb, (2.25)

where we again note that b is free. We can notice that u∗ is the control signal that causesG(x∗, u, t) to take its minimum value.

Let us consider the simple case of a fractional system with the following: dynamics

aDαt x = Ax + Bu. (2.26)

The state and costate equations become

aDαt x = Ax + Bu,

tDαbλ = Aλ,

(2.27)


with

x(a) = xa, x(b) = xb. (2.28)

Using Pontryagin’s minimum principle, we have

u∗Bλ∗ ≤ uBλ∗. (2.29)

Defining q∗ = bλ∗ gives us

u∗q∗ ≤ uq∗, (2.30)

u∗q∗ = minumin≤u≤umax

{uq∗}. (2.31)

We can now derive the optimal control sequence u∗(t). Given (2.31),

(1) if q∗(t) is positive, then the optimal control u∗(t) must be the smallest admissiblecontrol umin value so that

minumin≤u≤umax

{uq∗}= −q∗ = − ∣∣q∗∣∣, (2.32)

(2) and if q∗(t) is negative, then the optimal control u∗(t) must be the largest admissiblecontrol umax value so that

minumin≤u≤umax

{uq∗}= q∗ = −∣∣q∗∣∣. (2.33)

Combining (2.32) and (2.33) gives us the following control law:

u∗(t) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

umax if q∗(t) < 0,

umin if q∗(t) > 0,

undetermined if q∗(t) = 0.

(2.34)

In Section 3, we provide several numerical methods to obtain the control u, state x, andcostate λ for a specific problem from the literature.

3. Solution of the Time-Optimal Control ofa Fractional Double Integrator

In this section, we consider the following FTOCP:

minu,T

J(u, T) .= T, (3.1)


subject to

x1 = x2, x1(0) = 0, x1(T) = A,

0Dαx2 = u, x2(0) = 0, x2(T) = 0,

umin ≤ u(t) ≤ umax, ∀t ∈ [0, T].

(3.2)

3.1. Solution Using Rational Approximation of the Fractional Operator

It is possible for time-optimal control problems to be reformulated into traditional optimalcontrol problems by augmenting the system dynamics with additional states (one additionalstate for autonomous problems). For that purpose, the first step is to specify a nominal timeinterval, [a b], for the problem and to define a scaling factor, adjustable by the optimizationprocedure, to scale the system dynamics, and hence, in effect, scale the duration of the timeinterval accordingly. This scale factor and the scaled time are represented by the extra states.

The problem defined by (3.1)-(3.2) can accordingly be reformulated as follows: findthe control u(t) (satisfying umin ≤ u(t) ≤ umax) over the time interval [01], which minimizesthe quadratic performance index

J(u) = T, (3.3)

subject to the following dynamics:

x1 = Tx2,

0Dαt x2 = Tu,

T = 0,

(3.4)

where the initial conditions are

x1(0) = 0,

x2(0) = 0,

T(0) = n,

(3.5)

where T(0) is the initial value chosen by the user. Final state constraints are

x1(1) = A,

x2(1) = 0.(3.6)

According to [22], we can approximate the operator aDαt using a state space definition

aDαt x = u⇐⇒

{z = Az + bu

x = cz

}. (3.7)


Such approximation is called rational approximation of the fractional operator. To ensure theapplicability of our method, we need to define a new state vector y(t) such that

y(t) =

⎡⎢⎢⎣x1(t)

z(t)

T

⎤⎥⎥⎦, (3.8)

where z(t) is the state vector of the rational approximation for the fractional-order systemdescribed by 0D

αt x2 = u.

Using the methodology proposed in [6], we reformulate the problem defined by (3.1)-(3.2). Find the control u(t) (satisfying umin ≤ u(t) ≤ umax), which minimizes the quadraticperformance index

J(u) = T, (3.9)

subjected to the following dynamics:

y =

⎡⎢⎢⎣

c[y2(t) · · ·yN+1(t)

]TA[y2(t) · · ·yN+1(t)

]T + bu(t)0

⎤⎥⎥⎦, (3.10)

the initial condition

y(0) =[0 0 · · · 0 T

]T, (3.11)

and the final state constraints given by

y1(T) = 300,

c[y2(T) · · ·yN+1(T)

]T = 0.(3.12)

Such a definition allows the problem to be solved by any traditional optimal controlproblem solver.


3.2. Solution Using Bang-Bang Control Theory

The solution to the problem defined by (3.1)–(3.2) can also be solved using bang-bang controltheory. In the integer case (α = 1), the solution in the time domain is well documented and isgiven by

T1 =

√− 2Aumin

umax(umax − umin),

T = T1 +

√− 2Aumax

umin(umax − umin),

(3.13)

where T1 is the switching time.Such a solution is obtained by deriving the system’s dynamic equations in the time

domain: for t ∈ [0, T1] (acceleration)

u(t) = umax,

x2(t) = umaxt,

x1(t) = umaxt2

2,

(3.14)

and for t ∈ [T1, T] (deceleration)

u(t) = umin,

x2(t) = umin(t − T1) + x2(T1),

x1(t) = umin(t − T1)2

2+ x2(T1)(t − T1) + x1(T1).

(3.15)

Finding the switching time T1 and the final time T can be done by solving the system of finaltime conditions:

x1(T2) = A,

x2(T2) = 0.(3.16)

We can apply a similar technique to solve the fractional problem. For t ∈ [0, T1] (acceleration)

u(t) = umax,

x2(t) = umax · tα

Γ(α + 1),

x1(t) = umax · tα+1

Γ(α + 2),

(3.17)


and for t ∈ [T1, T2] (deceleration)

u(t) = umin,

x2(t) = umin · (t − T1)α

Γ(α + 1)+ x2(T1)

(t − T1)α−1

Γ(α),

x1(t) = umin · (t − T1)α+1

Γ(α + 2)+ x2(T1) · (t − T1)α

Γ(α + 1)+ x1(T1).

(3.18)

Finding the switching time T1 and the final time T can be achieved by solving the system offinal time conditions:

x1(T2) = A,

x2(T2) = 0.(3.19)

When expanded, it becomes

(T2 − T1)α+1

Γ(α + 2)umin +

Tα1Γ(α + 1)

(T2 − T1)α

Γ(α + 1)umax +

Tα+11

Γ(α + 2)umax = A,

(T2 − T1)α

Γ(α + 1)umin +

Tα1Γ(α + 1)

(T2 − T1)α−1

Γ(α)umax = 0.

(3.20)

Renaming T21 = T − T1, we get

Tα+121

Γ(α + 2)umin +

Tα1 Tα21

Γ(α + 1)2umax +

Tα+11

Γ(α + 2)umax = A,

Tα21

Γ(α + 1)umin +

Tα1 Tα−121

Γ(α + 1)Γ(α)umax = 0.

(3.21)

The solution to the system of equations is as follows: solve for T21

− umin

αΓ(α + 2)Tα+1

21 − uminΓ(α)Γ(α + 2)

(Γ(α)

umin

umax

)1/α

T(α+1)/α21 = A,

T1 =(−Γ(α)umin

umaxT21

)1/α

,

T = T21 + T1 .

(3.22)


0 5 10 15 20 25

Time (s)

0

50

100

150

200

250

300

Posi

tion

Figure 1: States x(t) as functions of time t for the Bang-Bang control problem for α = 1.

4. Results

In this section, we find the solution to the problem defined in Section 3 for the followingparameter values:

(i) umin = −2,

(ii) umax = 1,

(iii) A = 300.

The analytical solution for this system for α = 1 (the traditional double integrator) isgiven in [23] by T ∗ = 30 as

u(t) =

⎧⎪⎪⎨⎪⎪⎩

1 for 0 ≤ t < 20,

−2 for 20 ≤ t ≤ 30,

x(t) =

⎧⎪⎪⎨⎪⎪⎩

t2

2for 0 ≤ t < 20,

−t2 + 60t − 600 for 20 ≤ t ≤ 30,

x(t) =

⎧⎪⎪⎨⎪⎪⎩t for 0 ≤ t < 20,

60 − 2t for 20 ≤ t ≤ 30.

(4.1)


0 5 10 15 20 25 30

Time (s)

0

50

100

150

200

250

300

Posi

tion

Figure 2: States x(t) as functions of time t for the Bang-Bang control problem for α = 0.9.

0 10 20 30

Time (s)

0

50

100

150

200

250

300

Posi

tion


To solve the problem, we use the RIOTS 95 Matlab Toolbox [24]. The acronym RIOTS means“recursive integration optimal trajectory solver.” It is a Matlab toolbox developed to solvea wide variety of optimal control problems. For more information about the toolbox, pleaserefer to [25].

Figure 1 shows the states x(t) as functions of t for α = 1. Figures 2, 3, 4, 5 and 6 showthe state x(t) as functions of t for different values of α (0.9, 0.8, 0.7, 0.6, and 0.5, respectively).We can observe that when the order α approaches 1, the optimal duration nears its value forthe double integrator case.

Since it is possible to obtain the analytical solution of the problem from (3.22), wegive in Figure 7 the plot of the duration of the control T versus the order of the fractionalderivative. As we can see, for α = 1, the solution matches the results obtained for a doubleintegrator.


0 10 20 30

Time (s)

0

50

100

150

200

250

300

Posi

tion


0 10 20 30 40

Time (s)

0

50

100

150

200

250

300

Posi

tion


5. Conclusions

We developed a formulation for fractional time-optimal control problems. Such problemsoccur when the dynamics of the system can be modeled using the fractional derivativesoperator. The formulation made use of the the Lagrange multiplier technique, Pontryagin’sminimum principle, and the state and costate equations. Considering a specific set ofdynamical equations, we were able to demonstrate the bang-bang nature of the solution offractional time-optimal control problems, just like in the integer-order case. We were ableto solve a special case using both optimal control theory and bang-bang control theory. Theoptimal control solution can be obtained using a rational approximation of the fractionalderivative operator whereas the bang-bang control solution is derived from the time-domainsolution for the final time constraints. Both methods showed similar results, and in both cases


0 10 20 30 40 50

Time (s)

0

50

100

150

200

250

300

Posi

tion


0.5 0.6 0.7 0.8 0.9 1

Order α

30

35

40

45

50

55

Dur

atio

n(s)

Figure 7: Duration of the control T as a function of the order α.

as the order α approaches the integer value 1, the numerical solutions for both the state andthe control variables approach the analytical solutions for α = 1.

References

[1] A. Oustaloup, F. Levron, B. Mathieu, and F. M. Nanot, “Frequency-band complex nonintegerdifferentiator: characterization and synthesis,” IEEE Transactions on Circuits and Systems I, vol. 47,no. 1, pp. 25–39, 2000.

[2] M. Zamani, M. Karimi-Ghartemani, and N. Sadati, “FOPID controller design for robust performanceusing particle swarm optimization,” Fractional Calculus & Applied Analysis, vol. 10, no. 2, pp. 169–187,2007.

[3] L. Sommacal, P. Melchior, A. Oustaloup, J.-M. Cabelguen, and A. J. Ijspeert, “Fractional multi-modelsof the frog gastrocnemius muscle,” Journal of Vibration and Control, vol. 14, no. 9-10, pp. 1415–1430,2008.


[4] D. E. Kirk, Optimal Control Theory: An Introduction, Dover, New York, NY, USA, 2004.[5] O. P. Agrawal, “A general formulation and solution scheme for fractional optimal control problems,”

Nonlinear Dynamics, vol. 38, no. 1–4, pp. 323–337, 2004.[6] C. Tricaud and Y. Q. Chen, “Solving fractional order optimal control problems in riots 95—a general-

purpose optimal control problem solver,” in Proceedings of the 3rd IFAC Workshop on FractionalDifferentiation and Its Applications, Ankara, Turkey, November 2008.

[7] O. P. Agrawal and D. Baleanu, “A Hamiltonian formulation and a direct numerical scheme forfractional optimal control problems,” Journal of Vibration and Control, vol. 13, no. 9-10, pp. 1269–1281,2007.

[8] O. P. Agrawal, “A quadratic numerical scheme for fractional optimal control problems,” ASME Journalof Dynamic Systems, Measurement and Control, vol. 130, no. 1, Article ID 011010, 6 pages, 2008.

[9] O. P. Agrawal, “Fractional optimal control of a distributed system using eigenfunctions,” Journal ofComputational and Nonlinear Dynamics, vol. 3, no. 2, Article ID 021204, 6 pages, 2008.

[10] N. Ozdemir, D. Karadeniz, and B. B. Iskender, “Fractional optimal control problem of a distributedsystem in cylindrical coordinates,” Physics Letters A, vol. 373, no. 2, pp. 221–226, 2009.

[11] N. Ozdemir, O. P. Agrawal, B. B. Iskender, and D. Karadeniz, “Fractional optimal control of a 2-dimensional distributed system using eigenfunctions,” Nonlinear Dynamics, vol. 55, no. 3, pp. 251–260,2009.

[12] D. Baleanu, O. Defterli, and O. P. Agrawal, “A central difference numerical scheme for fractionaloptimal control problems,” Journal of Vibration and Control, vol. 15, no. 4, pp. 583–597, 2009.

[13] G. S. F. Frederico and D. F. M. Torres, “Noether’s theorem for fractional optimal control problems,”in Proceedings of the 2nd IFACWorkshop on Fractional Differentiation and Its Applications, Porto, Portugal,July 2006.

[14] G. S. F. Frederico and D. F. M. Torres, “Fractional conservation laws in optimal control theory,”Nonlinear Dynamics, vol. 53, no. 3, pp. 215–222, 2008.

[15] G. S. F. Frederico and D. F. M. Torres, “Fractional optimal control in the sense of Caputo and thefractional Noether’s theorem,” International Mathematical Forum, vol. 3, no. 10, pp. 479–493, 2008.

[16] Z. D. Jelicic and N. Petrovacki, “Optimality conditions and a solution scheme for fractional optimalcontrol problems,” Structural and Multidisciplinary Optimization, vol. 38, no. 6, pp. 571–581, 2009.

[17] P. L. Butzer and U. Westphal, “An introduction to fractional calculus,” in Applications of FractionalCalculus in Physics, pp. 1–85, World Scientific, River Edge, NJ, USA, 2000.

[18] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations,John Wiley & Sons, New York, NY, USA, 1993.


[20] K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation andIntegration to Arbitrary Order, vol. 11 of Mathematics in Science and Engineering, Academic Press,London, UK, 1974.

[21] D. S. Naidu, Optimal Control Systems, vol. 2 of Electrical Engineering Series, CRC Press, Boca Raton, Fla,USA, 2002.

[22] C. Tricaud and Y. Q. Chen, “Solution of fractional order optimal control problems using SVD-basedrational approximations,” in Proceedings of the American Control Conference (ACC ’09), pp. 1430–1435,St. Louis, Mo, USA, June 2009.

[23] A. E. Bryson and Y. C. Ho, Applied Optimal Control, Hemisphere, New York, NY, USA, 1975.[24] A. L. Schwartz, Theory and implementation of numerical methods based on runge-kutta integration for solving

optimal control problems, Ph.D. thesis, University of California at Berkeley, Berkeley, Calif, USA, 1989.[25] A. L. Schwartz, E. Polak, and Y. Q. Chen, “RIOTS—A MATLAB Toolbox for Solving Optimal Control

Problems,” 1997, http://www.accesscom.com/∼adam/RIOTS/.


Research ArticleOn the Speed of Spread for FractionalReaction-Diffusion Equations

Hans Engler

Department of Mathematics, Georgetown University, Box 571233, Washington, DC 20057, USA

Correspondence should be addressed to Hans Engler, [email protected]

Received 12 August 2009; Revised 12 October 2009; Accepted 25 October 2009

Academic Editor: Om Agrawal

Copyright q 2010 Hans Engler. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

The fractional reaction diffusion equation ∂tu + Au = g(u) is discussed, where A is a fractionaldifferential operator on R of order α ∈ (0, 2), the C1 function g vanishes at ζ = 0 and ζ = 1, andeither g ≥ 0 on (0, 1) or g < 0 near ζ = 0. In the case of nonnegative g, it is shown that solutionswith initial support on the positive half axis spread into the left half axis with unbounded speed ifg(ζ) satisfies some weak growth condition near ζ = 0 in the case α > 1, or if g is merely positive ona sufficiently large interval near ζ = 1 in the case α < 1. On the other hand, it shown that solutionsspread with finite speed if g ′(0) < 0. The proofs use comparison arguments and a suitable familyof travelling wave solutions.

1. Introduction

The scalar reaction-diffusion equation

∂tu(x, t) − ∂2xu(x, t) = g(u(x, t)) (1.1)

has been the subject of much study, beginning with the celebrated paper [1]. The authorsof [1] proposed this equation, with g being positive and concave on (0, 1) such that g(0) =g(1) = 0, as a model for a population that undergoes logistic growth and Brownian diffusion.If u(x, 0) = H(x), the Heaviside function, and g(u) = u − u2, the equation in fact hasan exact probabilistic interpretation, given in [2]. Consider a population of particles thatundergo independent Brownian motion and branching processes, with each child particleagain following the same behavior. Then u(x, t) is the probability that there is a particle tothe left of position x at time t, assuming that there was exactly one particle at position x = 0at time t = 0. Equation (1.1) also can be derived heuristically for the mean behavior of an


interacting particle process in which two types of particles (call them A- and B-particles)simultaneously undergo Brownian diffusion and conversion reactions

A + B −→ 2A, A + B −→ 2B (1.2)

with suitable reaction rates. Then the volume density fraction u(x, t) of A-particles in thehydrodynamic limit of large particle numbers per unit volume formally satisfies (1.1) withg(u) = cu(1 − u) where c depends on the reaction rates in (1.2); see [3] for a discussion of theunderlying limit procedure and an exact connection to a stochastic version of (1.1). If insteadthe conversion reactions are

2A + B −→ 3A, 2A + B −→ A + 2B, (1.3)

then the equation for u becomes (1.1) with g(u) = cu2(1−u). Other polynomial reaction termsg occur in similar ways.

For (1.1) with g(u) = u(1 − u), it is known that solutions approach a wave profile ψ inthe sense that

u(x +m(t), t) −→ ψ(x) (t −→ ∞), (1.4)

where m(t) is the median, u(m(t), t) = 1/2. It turns out that m(t) = c∗t+O(log t) for a suitableasymptotic finite wave speed c∗. Larger asymptotic speeds are only possible if the initial dataare supported on R. A more general result, given in [4], implies that there is a critical speedc∗ such that for fairly general initial data u(·, 0) that are nonnegative and supported on (0,∞),

lim supt→∞

supx<−ct

u(x, t) = 0 (1.5)

whenever c > c∗ and

lim inft→∞

infx<−ct

u(x, t) = 1 (1.6)

whenever c < c∗. If u is interpreted as the density of a quantity whose spread is governed by(1.1), a runner may escape from it by running to −∞ at a speed c > c∗, but this quantity willcatch up with and engulf it if its speed is c < c∗. In this sense, solutions of the (1.1) exhibitfinite speed of spread.

Equation (1.1) was also derived in [5] to describe antiphase domain coarsening inalloys. In this situation, g(0) = g(1) = g(u∗) = 0 for some u∗ ∈ (0, 1), and g < 0 on (0, u∗),and g > 0 on (u∗, 1). In this case there exists exactly one wave speed c∗ with associated waveprofile. In particular, (1.5) and (1.6) still hold for this c∗. The first of the two cases (the KPPcase) corresponds to “pulled” fronts (the state u = 0 is unstable) while the second case (theAllen-Cahn case) results in a “pushed” front (the state u = 0 is stable). More on these twofundamentally different situations may be found in [6] and the references given there. A vastrange of applications leading to related models is discussed in [7].


The purpose of this note is a study of the fractional reaction-diffusion equation:

∂tu(x, t) +Au(x, t) = g(u(x, t)). (1.7)

Here A = Aαρ is a pseudodifferential operator with symbol p = pαρ that is homogeneous ofdegree α ∈ (0, 2], such that p(−λ) = p(λ) and |p(1)| = 1. Following the presentation in [8], wewrite p in the Riesz-Feller form:

p(λ) = e−i(π/2) sign(λ)ρ|λ|α, (1.8)

where the skewness parameter ρ must also satisfy |ρ| ≤ min{α, 2 − α}. The operator A is theinfinitesimal generator of a stable Levy process, that is, a continuous time stochastic processthat has cadlag paths and independent stationary increments with stable distributions; see[9]. Paths of such a process must have jumps, and the variance of the displacement must beinfinite.

For ρ = 0 we obtain fractional powers of the usual negative one-dimensionalLaplacian, abbreviated often by (−Δ)α/2. There are various real variable representations ofsuch operators, for example, as singular integral operators or as limits of suitable differenceoperators; see [8]. In the special case where α = 1 and −1 ≤ ρ ≤ 1, there is the representation

A1,ρu(x) =d

dx

(cos

ρπ

2Hu(x) + sin

ρπ

2u(x)

), (1.9)

where Hu is the Hilbert transform of u. In particular, for ρ = ±1, this is an ordinary first-orderderivative, not a fractional derivative.

The function g is always assumed to satisfy g(0) = g(1) = 0. We are interested in boththe KPP-case, that is, g(ζ) ≥ 0 for 0 < ζ < 1, and the Allen-Cahn case, that is, g(ζ) < 0 for ζnear 0 and g(ζ) > 0 for ζ near 1.

The equation occurs in the heuristic hydrodynamic limit of interacting particlepopulations in which conversion reactions such as (1.2) or (1.3) occur together with motionby a stable Levy process. It has been proposed in, for example, [10–15]. It should be notedthat the term “anomalous diffusion” is also used for situations in which the first-order timederivative is replaced by a fractional order derivative; see [8] where the present case of afirst-order time derivative is called space-fractional diffusion. Another generalization of (1.1)consists in allowing time delays; see [16]. These further generalizations will not be discussedhere.

There is strong evidence that (1.7) does not admit traveling wave solutions if 0 < α < 2and g is positive and concave on (0, 1). Rather, numerical results in [11, 17] suggest that forinitial data that are supported on the positive half axis and increase there from 0 to 1, themedian satisfies m(t) ∼ −ect for some c > 0. In [18], the estimates

lim supt→∞

supx<−ect

u(x, t) = 0, lim inft→∞

infx>−edt

u(x, t) = 1 (1.10)

are shown to hold for such initial data whenever c > c∗ > d > 0, where c∗ = g ′(0)/α. Thus thesupport of a solution grows asymptotically like an exponential. For the case where g < 0 on


some interval (0, u∗), the results in [14, 19] suggest on the other hand that there exist waveprofile solutions that move with constant speed, although no rigorous proofs are given there.

The main results of this note are concerned with the existence and nonexistence of afinite speed of spread and take the form (1.5) and (1.6). It is shown that for a large class ofright-hand sides g that are nonnegative on (0, 1), the speed of spread is infinite; that is, theestimate (1.6) holds for all positive speeds c. It is not necessary to assume that g ′(0) > 0, andif α < 1, one does not even have to assume that g is strictly positive on (0, 1). On the otherhand, if g ′(0) < 0 and therefore g is negative near 0, then it will be shown that there exists afinite speed of spread; that is, (1.5) holds for some finite positive c. These results are statedand proven in Section 3. Some basic existence and comparison results for (1.7) are sketchedin Section 4.

To prove these results, comparison arguments are employed which follow fromintegral representations of solutions of (1.7) and which are therefore extensions of similararguments for the study of (1.1). The challenge then is to come up with suitable comparisonsolutions. Since (1.7) is nonlocal in nature, techniques from ordinary differential equationscannot be employed to construct such solutions. Instead, in this paper a set of travelling wavesolutions is used that comes directly from the fundamental solution of the linear problem (see(2.1) below). These solutions are discussed in Section 2 and may be of independent interest.

2. A Class of Travelling Wave Solutions

In this section, it will be shown that fundamental solutions of fractional diffusion equations(without reaction terms) lead to traveling wave solutions u(x, t) = U(x + ct) of (1.7), forsuitable functions g. A two-parameter family will be constructed for each possible choiceof α and ρ, one parameter being the speed c. The main contribution of this section is thecharacterization of the nonlinear function g that is required to make the equation hold.Throughout this section, let α ∈ (0, 2) and |ρ| < min{α, 2 − α}.

Consider the “free” equation

∂tu(x, t) +Aαρu(x, t) = 0, (2.1)

where Aαρ is the pseudodifferential operator with symbol pαρ defined in (1.8). It is knownthat (2.1) has the fundamental solution

(x, t) −→Wαρ(x, t) = t−1/αfαρ(xt−1/α

)(2.2)

with initial data Wαρ(x, 0) = δ0(x), the delta distribution; see [8]. In particular, Wαρ(x, 1) =fαρ(x). Here fαρ is the probability density function of a stable distribution; see [20, 21].There are many ways to parametrize such distributions. For α/∈ {0, 1, 2} the version used herecorresponds to form (B) in [21] with the same choice of α, scale parameter γ = 1, locationparameter δ = 0, and skewness parameter:

β = ± ρ

min{α, 2 − α} , (2.3)


where the sign is positive if 0 < α < 1 and negative if 1 < α < 2. For α = 1, the form used herecorresponds to form (C) in [21] with β = ρ, γ = 1, δ = 0.

There is also the special solution

(x, t) −→ Vαρ(x, t) = Fαρ(xt−1/α

), (2.4)

where Fαρ(x) =∫x−∞fαρ(s)ds is a cumulative distribution function. Then Vαρ solves (2.1) with

initial data Vαρ(x, 0) = H(x), the Heaviside function. The equations hold in the sense ofdistributions, and the initial data are attained in this sense. It is known that fαρ is positive,infinitely differentiable, and unimodal. Also, as x → ∞, there are expansions

1 − Fαρ(x) ∼∑j≥1

cjαρx−jα,

fαρ(x) ∼∑j≥1

cjαρx−1−jα,

(2.5)

and as x → −∞

Fαρ(x) ∼∑j≥1

djαρ(−x)−jα,

fαρ(x) ∼∑j≥1

djαρ(−x)−1−jα.(2.6)

These are convergent expansions if 0 < α < 1 and asymptotic expansions if 1 < α < 2; see [8].For the remainder of this section, we suppress the subscripts α and ρ in most formulae

that involve Aαρ, fαρ, and Fαρ. For fixed c ∈ R and τ > 0 we consider the function

Uτ(ξ) = F(ξτ−1/α

). (2.7)

Set ucτ(x, t) = Uτ(x + ct), then

Aucτ(x, t) = −∂τF((x + ct)τ−1/α

)

=1α

((x + ct)τ−1/α−1

)f((x + ct)τ−1/α

)

∂tucτ(x, t) = cτ−1/αf((x + ct)τ−1/α

),

(2.8)

and therefore

∂tucτ(x, t) +Aucτ(x, t) =(

1α

((x + ct)τ−1/α−1

)+ cτ−1/α

)f((x + ct)τ−1/α

). (2.9)

Now (x + ct)τ−1/α = F−1(ucτ(x, t)) and consequently

∂tucτ(x, t) +Aucτ(x, t) = cτ−1/αg0(ucτ(x, t)) +1ατ

g1(ucτ(x, t)) (2.10)


with

g0(ζ) = f(F−1(ζ)

), g1(ζ) = F−1(ζ)f

(F−1(ζ)

). (2.11)

Equation (2.10) is of the form (1.7), with g(ζ) = cτ−1/αg0(ζ) + 1/(ατ)g1(ζ).In the case α = 1 and −1 < ρ < 1, everything is explicit. Let κ = cos(πρ)/2 and

σ = sin(πρ)/2. Then by results in [21],

F(x) =12+

1π

arctanx − σκ

,

F−1(ζ) = σ − κ cot(πζ),

f(x) =1

πκ(

1 + (x − σ)2/κ2) ,

g0(ζ) =1κπ

sin2(πζ),

g1(ζ) =σ

πκsin2(πζ) − 1

πcos(πζ) sin(πζ).

(2.12)

It remains to characterize the functions g0, g1 in the general case.

Proposition 2.1. Let 0 < α < 2, |ρ| < min{α, 2 − α}. The functions g0, g1 have the followingproperties.

(a) g0 and g1 are infinitely differentiable on (0, 1).

(b) The function g0 is positive on (0, 1). The function g1 is negative on (0, Fαρ(0)) and positiveon (Fαρ(0), 1). The function

ζ −→ cτ−1/αg0(ζ) +1ατ

g1(ζ) (2.13)

is negative on (0, u∗) and positive on (u∗, 1), where u∗ = Fαρ(−cατ−1/α+1).

(c) As ζ ↓ 0, g0(ζ) = O(ζ1+1/α) and g0(1 − ζ) = O(ζ1+1/α).

(d) As ζ ↓ 0, g1(ζ) = −αζ +O(ζ1+1/α). As ζ ↑ 1, g1(ζ) = α(1 − ζ) +O((1 − ζ)1+1/α).

(e) The functions g0 and g1 can be represented as

g0(ζ) =d

dζ

∫F−1(ζ)

−∞f2(s)ds,

g1(ζ) =d

dζ

∫F−1(ζ)

−∞sf2(s)ds.

(2.14)

Proof. Property (a) follows since f and F together with its inverse are infinitely differentiable.Property (b) is obvious. Properties (c) and (d) follow from the asymptotic expansions (2.5)and (2.6). Finally (e) can be checked by differentiation.


Property (e) will not be used in what follows. It should be noted that g0 and g1 are ofclass C1 on [0, 1] but are not infinitely differentiable at the interval endpoints, except if α = 1.Clearly g0 and g1 do not depend on c or τ . We are therefore free to form fairly arbitrary linearcombinations of g0 and g1 by choosing c and τ .

The construction provides travelling wave solutions for (1.7) for a special class offunctions for which g ∈ C1([0, 1]), g(0) = g(u∗) = g(1) for some u∗ ∈ (0, 1), and g ′(0) <0, g ′(u∗) > 0, g ′(1) < 0. This suggests that (1.7) possesses travelling wave solutions for moregeneral functions g with these properties.

If the same construction is attempted for the case α = 2, it turns out that g0 and g1 aremerely continuous on [0, 1], with derivatives that have logarithmic singularities near ζ = 0and ζ = 1. Therefore the arguments in the next section cannot be extended to the case α = 2,and indeed the results of the next section do not hold in that case.

3. Results on the Speed of Spread

This section contains the main results of this paper. As before, the operator A has symbol(1.8) with 0 < α < 2 and |ρ| < min{α, 2 − α}. We always assume that u is a solution of (1.7)and that g ∈ C1([0, 1],R) with g(0) = g(1) = 0. Initial data u0 will be assumed to satisfy

u0 ∈ C(R,R), 0 ≤ u0(x) ≤ 1, limx→∞

u0(x) = 1, supp(u0) ⊂ [0,∞). (3.1)

The results in Section 4 then imply that (1.7) has a unique mild solution u that exists for allx ∈ R, t > 0, and this solution satisfies 0 ≤ u(x, t) ≤ 1 for all (x, t). The notation of that sectionwill also be used here.

We first discuss the case where g ≥ 0 on (0, 1). The main result in this case is thefollowing.

Theorem 3.1. Let u be the solution of (1.7) with u0 satisfying (3.1).

(a) Let α > 1. Assume that g > 0 on (0, 1) and that there are 0 < γ < α/(α − 1) , c0 > 0 suchthat for all ζ ∈ [0, 1/2]

g(ζ) ≥ c0ζγ . (3.2)

Then for all c > 0

lim inft→∞

infx≥−ct

u(x, t) = 1. (3.3)

(b) Let α = 1. Assume that g > 0 on (0, 1). Then for all c > 0

lim inft→∞

infx≥−ct

u(x, t) = 1. (3.4)

(c) Let α < 1. Assume that g ≥ 0 on (0, 1) and g(ζ) > 0 for ζ ∈ [(α − ρ)/2α, 1). Then for allc > 0

lim inft→∞

infx≥−ct

u(x, t) = 1. (3.5)


The result shows that the speed of spread is unbounded (i.e., (1.6) holds for all c > 0),and it exhibits different mechanisms for this phenomenon. Recall that in the interpretationof [1], the function g is responsible for the growth of a substance whose density is given byu, while A describes the spread of this substance. If α ∈ (0, 2), the substance spreads with ajump process, not with Brownian diffusion, and jumps of magnitude exceedingM occur witha probability that is O(M−α) for large M. If α > 1, the mean jump distance is still finite. Inthis case, the growth rate g(u) at small densities (small u) is responsible for the unboundedspeed of spread. If α is close to 1, this growth can be very weak (g(ζ) ∼ ζγ with large γ) yetthe speed of spread is still unbounded. If on the other hand α < 1, then large jumps tend tobe more frequent, and jump sizes have unbounded mean. In this case, the growth rate forsmall densities is no longer the reason why the speed of spread is unbounded; in fact theremay be no growth at all for small densities (g(u) = 0 for small u) for this to occur. Rather,the unbounded speed of spread results from growth that occurs solely for large densities(g(ζ) > 0 only for ζ ≥ (α−ρ)/2α). The substance is transported towards −∞ due large (α < 1)negative jumps, resulting in an unbounded speed of spread. It is known that in this case,(α − ρ)/2α is the probability that a jump is negative. If this fraction is large, then growththat occurs only for large densities, that is, g(u) > 0 on [(α − ρ)/2α, 1), already leads to anunbounded speed of spread. The case α = 1 is intermediate: any growth for small densities(g(ζ) > 0 for ζ > 0) results in an unbounded speed of spread.

In the case α > 1, it would be interesting to know if the speed of spread is stillunbounded if γ ≥ α/(α − 1) or if a finite speed of spread occurs ((1.5) holds for large c) ifγ becomes sufficiently large, that is, if growth is extremely weak for small densities u. In thecase α < 1, it would be interesting to know if a finite speed of spread is possible at all if g ≥ 0and g is not identically equal to 0.

The main result in the case where g is negative near ζ = 0 is the following.

Theorem 3.2. Let u be the solution of (1.7) with initial data u0 satisfying (3.1). Assume that g ′(0) <0. Then there exists c > 0 such that

lim supt→∞

supx≤−ct

u(x, t) = 0. (3.6)

The result shows that negative proportional growth at small densities (g ′(0) < 0)always limits the speed of spread of a substance whose growth and spread are governed by(1.7), even for processes whose jump sizes tend to be very large (α < 1). I am not aware of aninterpretation of this result in the context of material science, similar to the use of (1.1) in [5].

The proofs will be given below. The main tools in the proofs are the comparisonarguments given in the next section, together with the following crucial auxiliary result.

Lemma 3.3. Let g ∈ C1([0, 1],R) and let g0, g1 be defined as in (2.11), depending on α and ρ.

(a) Let α ∈ (1, 2). Suppose that g(ζ) > 0 for all ζ ∈ (0, 1] and that there exist c0 > 0 andγ < α/(α − 1) such that g(ζ) ≥ c0ζ

γ for all ζ ∈ [0, 1/2]. Then given any c > 0 there existsτ > 0 such that for all ζ ∈ [0, 1]

g(ζ) ≥ cτ−1/αg0(ζ) + (ατ)−1g1(ζ). (3.7)


(b) Let α = 1. Suppose that g > 0 on (0, 1]. Then given any c > 0 there exists τ > 0 such thatfor all ζ ∈ [0, 1]

g(ζ) ≥ cτ−1g0(ζ) + τ−1g1(ζ). (3.8)

(c) Let α ∈ (0, 1). Suppose that g ≥ 0 on [0, 1] and g(ζ) > 0 for all ζ ∈ [(α − ρ)/2α, 1). Thengiven any c > 0 there exists τ > 0 such that for all ζ ∈ [0, 1]

g(ζ) ≥ cτ−1/αg0(ζ) + (ατ)−1g1(ζ). (3.9)

(d) Suppose that g ′(0) < 0 and g(ζ) = 0 for ζ ∈ [1 − ε, 1] for some ε. Then there exist c ∈ R

and τ > 0 such that for all ζ ∈ [0, 1]

g(ζ) ≤ cτ−1/αg0(ζ) + (ατ)−1g1(ζ). (3.10)

Proof. Consider first statement (a). Let us write F = Fαρ and f = fαρ. Let α > 1 and let M > 0be large enough such that for some c1, c2 > 0 and all x ≤ −M

F(x) ≥ c1|x|−α, f(x) ≤ c2|x|−1−α. (3.11)

This is possible by (2.6). Let c > 0 be given, then we may increase M further such that also

c0cγ

1Mr ≥ c2

α − 1α

cα/(α−1), (3.12)

where r = (α/(α− 1))− γ > 0. Now set δ = F(−M). Then for 0 < ζ = F(x) ≤ δ, that is, x < −M,and for all τ ≥ (M/c)α/(α−1)

cτ−1/αg0(F(x)) + (ατ)−1g1(F(x)) =(cτ−1/α + (ατ)−1x

)f(x)

≤ α − 1α

cα/(α−1)|x|1/(1−α)f(x)

≤ α − 1α

cα/(α−1)|x|1/(1−α)c2|x|−1−α

=α − 1α

cα/(α−1)c2|x|−α2/(α−1),

(3.13)

where a standard calculus argument has been used to see that the expression (cτ−1/α +(ατ)−1x) is maximal for τ = (|x|/c)α/(α−1). We estimate further, using the choice of M,

cτ−1/αg0(F(x)) + (ατ)−1g1(F(x)) ≤ c0cγ

1Mr |x|−α2/(α−1)

≤ c0cγ

1 |x|r |x|−α2/(α−1) = c0

(c1|x|α

)γ

≤ c0(F(x))γ ≤ g(F(x)).

(3.14)


Therefore, for all τ ≥ (M/c)α/(α−1) and all ζ < δ = F(−M),

cτ−1/αg0(ζ) + (ατ)−1g1(ζ) ≤ g(ζ). (3.15)

Since g > 0 on [δ, 1] by assumption, this inequality can be achieved also on [δ, 1] by increasingτ even further. This proves part (a).

The proof of part (b) is straight forward: given c > 0, note that cg0(ζ)+g1(ζ) ≤ 0 on theinterval [0, F(−c)]. Then (cg0(ζ) + g1(ζ))/τ ≤ g(ζ) is true if τ is sufficiently large.

To prove part (c), let again c > 0 be given. Let u∗ = inf{u ∈ [0, 1] | g(u) > 0}. Thenu∗ < (α−ρ)/2α = F(0). Pick τ large enough such that F(−cατ1−1/α) > u∗. This is possible sinceα < 1. Then on [0, u∗],

cτ−1/αg0(ζ) + (ατ)−1g1(ζ) ≤ g(ζ) (3.16)

since the left-hand side is nonpositive there by Proposition 2.1. By increasing τ further, wecan obtain this estimate also for ζ ∈ [u∗, 1], using again that g is assumed to be positive on[(1 − β)/2, 1].

To prove part (d), note first that

cτ−1/αg0(ζ) + (ατ)−1g1(ζ) ≥ g(ζ) (3.17)

on an interval [0, δ] as soon as τ−1 + g ′(0) > 0, that is, for sufficiently small τ . Increasing csufficiently and noting that g = 0 near ζ = 1 extends this inequality to the entire interval[0, 1].

Proof of Theorem 3.1. The proof uses the same argument for all three parts; so we give detailsonly in part (a). Let ε > 0. We replace u with u = (1 + ε)u and g with g, where g(ζ) =(1 + ε)g((1 + ε)−1ζ). Then

∂tu +Au = g(u). (3.18)

Then g(ζ) ≥ c0ζγ for ζ ∈ [0, 1/2], possibly with a changed c0, and additionally g > 0 on (0, 1].

Let c > 0 be given, then there exists τ > 0 such that

g(ζ) ≥ (c + 1)τ−1/αg0(ζ) + (ατ)−1g1(ζ) (3.19)

for all ζ ∈ [0, 1] by Lemma 3.3. By extending g0 and g1 to be zero on [1, 1 + ε], this inequalityis true on [0, 1 + ε]. Now find a constant d such that v0(x) = H(x − d) ≤ u(x, 0) for all x. Thisis possible since limx→∞u(x, 0) = 1 + ε. By Proposition 4.3, we see that

u(x, t) ≥ F((x − d)t−1/α

)(3.20)


for all x ∈ R, t > 0. This is in particular true for t = τ . Now use Proposition 4.1 and (3.19) toinfer that

u(x, t) ≥ F((x − d + (c + 1)t)τ−1/α

)(3.21)

for all x ∈ R, t ≥ τ . Therefore for t ≥ τ and x ≥ −ct,

u(x, t) ≥ F((x − d + (c + 1)t)τ−1/α

)

≥ F((−ct − d + (c + 1)t)τ−1/α

)

= F((t − d)τ−1/α

).

(3.22)

As t → ∞, the right-hand side goes to 1. Rewriting this in terms of u, we see that

lim inft→∞

infx≥−ct

u(x, t) ≥ (1 + ε)−1. (3.23)

Since ε > 0 was arbitrary, the desired result follows.In case of part (b), the same argument can be used without changes, appealing to part

(b) of Lemma 3.3.In case of part (c), we have to restrict ε such that g > 0 on [(α − ρ)/2α, 1], that is,

g > 0 on [(α − ρ)/(2α(1 + ε)), 1]. The rest of the proof is again unchanged, using part (c) ofLemma 3.3.

Proof of Theorem 3.2. We replace u with u = (1/2)u and g with g, where g(ζ) = (1/2)g(2ζ) for0 ≤ ζ ≤ 1/2 and g(ζ) = 0 for ζ ∈ (1/2, 1]. Then g ′(0) = g ′(0) < 0 and

∂tu +Au = g(u). (3.24)

By Lemma 3.3, part (d), there exist c > 0 and τ > 0 such that

g(ζ) ≤ (c − 1)τ−1/αg0(ζ) + (ατ)−1g1(ζ). (3.25)

Since limx→∞u(x, 0) = 1/2 and u(x, 0) = 0 for x < 0, we can find d > 0 such that F(x +d)τ−1/α) ≥ u(x, 0) for all x ∈ R. Using Proposition 4.1, one sees that

F((x + d + (c − 1)t)τ−1/α

)≥ u(x, t) (3.26)

for all x ∈ R, t > 0. Therefore for t > 0 and x ≤ −ct,

u(x, t) ≤ F((x + d + (c − 1)t)τ−1/α

)

≤ F((−ct + d + (c − 1)t)τ−1/α

)

= F((−t + d)τ−1/α

).

(3.27)


The right-hand side tends to 0 as t → ∞. In terms of u, this implies

lim supt→∞

supx≤−ct

u(x, t) = 0. (3.28)

This concludes the proof.

4. Facts about Fractional Reaction-Diffusion Equations

In this section we summarize some basic theory about (1.7) that is needed in this note. Abroader and deeper discussion may be found in [10].

We work in the Banach space

Clim ={w ∈ C(R) | lim

x→∞w(x) and lim

x→−∞w(x) exist

}(4.1)

equipped with the supremum norm ‖ · ‖. Let A = Aαρ be the pseudodifferential operatorwith symbol (1.8) and parameters α, ρ, with 0 < α < 2 and |ρ| < min{α, 2 − α}. Subscriptsα, ρ will again frequently be suppressed. Solutions of the free equation (2.1) with initial datau(·, 0) = ϕ ∈ Clim then can be written in terms of the fundamental solution given in (2.2),namely,

u(x, t) =∫

R

Wαρ

(x − y, t)ϕ(y)dy. (4.2)

For fixed α and ρ and ϕ ∈ Clim, we therefore define

S(t)ϕ(x) = u(x, t), (4.3)

where u is given by (4.2). Then S(t)t≥0 is a positive C0 semigroup on Clim and a Fellersemigroup on the subspace of functions in Clim that vanish at ±∞. If ψ is a continuous functionfrom [0, T] to Clim, then solutions of the inhomogeneous equation

∂tu(x, t) +Au(x, t) = ψ(x, t), u(·, 0) = ϕ (4.4)

can be written with the variation-of-constants formula:

u(·, t) = S(t)ϕ +∫ t

0S(t − s)ψ(·, s) ds. (4.5)

A continuous curve u : [0, T] → Clim that satisfies (4.5) is commonly called a mild solutionof (4.4). Next let g : [0,∞) × R → R be locally Lipschitz continuous in both variables and letϕ ∈ Clim. Then the equation ∂tu(x, t)+Au(x, t) = g(t, u(x, t)) (for which (1.7) is a special case)has a unique mild solution u ∈ C([0, T),Clim), where 0 < T ≤ ∞ is maximal. Either T = ∞, or


‖u(·, t)‖ → ∞ as t ↑ T . The solution can be obtained as the locally in time uniform limit of theiteration scheme:

un+1(·, t) = S(t)ϕ +∫ t

0S(t − s)g(s, un(·, s)) ds (n = 0, 1, . . .) (4.6)

with u0 being arbitrary, for example, u0(·, t) = S(t)ϕ. It is possible to set up a more generalsolution theory, but this is not needed for the purposes of this paper.

Solutions of (1.7) satisfy comparison theorems. Results of this type are true for allFeller semigroup. A systematic study of such semigroups and their generators was carriedout in [22], following the seminal work on this topic in [23]. For the sake of completeness, acomparison result is stated here, and its proof is sketched.

Proposition 4.1. Let u, v ∈ C([0, T],Clim) be mild solutions of the equations

∂tu +Au = g(u), ∂tv +Av = h(v), (4.7)

where g, h : R → R are locally Lipschitz continuous. If

g(ζ) ≤ h(ζ), ∀ζ ∈ R,

u(·, 0) ≤ v(·, 0), (4.8)

then

u(x, t) ≤ v(x, t) ∀(x, t) ∈ R × [0, T]. (4.9)

Proof. Let M = max[0,T](‖u(·, t)‖ + ‖v(·, t)‖ + 1). Let λ > |g ′(ζ)| + |h′(ζ)| for all |ζ| ≤M. Withoutloss of generality we may assume that g and h are constant outside [−M,M]. Set

U(x, t) = eλtu(x, t), V (x, t) = eλtv(x, t), (4.10)

and observe that U and V satisfy

∂tU +AU = g(t,U),

∂tV +AV = h(t, V )(4.11)

with g(t, ζ) = λζ + eλtg(e−λtζ) and h(t, ζ) defined similarly. Clearly, g(t, ζ) ≤ h(t, ζ) for all ζ.The function g is nondecreasing in its second argument, since for almost all ζ

∂ζg(t, ζ) = λ + g ′(e−λζ

)≥ 0. (4.12)

Consider the iteration scheme:

Un+1(·, t) = S(t)u(·, 0) +∫ t

0S(t − s)g(s,Un(·, s))ds (4.13)


and similarly for Vn and h. The scheme for the Un converges to the limit U, and the schemefor the Vn converges to the limit V .

We now employ a standard induction argument to show that Un ≤ Vn on R× [0, T] forall n. Let U0(·, t) = S(t)u(·, 0) and V0(·, t) = S(t)v(·, 0), then U0 ≤ V0 on R × [0, T] since S is apositive semigroup and u(·, 0) ≤ v(·, 0). Suppose Un ≤ Vn on R × [0, T], then

Un+1(·, t) = S(t)u(·, 0) +∫ t

0S(t − s)g(s,Un(·, s))ds

≤ S(t)v(·, 0) +∫ t

0S(t − s)g(s, Vn(·, s))ds

≤ S(t)v(·, 0) +∫ t

0S(t − s)h(s, Vn(·, s))ds

= Vn+1(·, t),

(4.14)

which completes the induction step. Taking the limit, this implies that U ≤ V and thereforealso u ≤ v on R × [0, T]. This proves the proposition.

Corollary 4.2. Consider a mild solution u ∈ C([0, T),Clim) of (1.7) and assume that g is locallyLipschitz continuous. If g(γ) ≥ 0 for some γ and u(·, 0) ≥ γ , then u(·, t) ≥ γ for all t. If g(γ) ≥0 ≥ g(δ) for some γ < δ and γ ≤ u(·, 0) ≤ δ, then γ ≤ u(·, t) ≤ δ for all t, and the solution can becontinued to R × [0,∞).

The proof consists in observing that the constant functions v(x, t) = γ and w(x, t) =δ solve (1.7) with right-hand sides 0 and therefore must be pointwise bounds for thesolution, by Proposition 4.1. If the solution remains bounded between two constants, thenits supremum norm remains bounded and it can be continued to R × [0,∞).

Also required is a comparison result for solutions whose initial data are step functions.Since such initial data are not in Clim, a separate argument is required.

Proposition 4.3. Let u ∈ C([0, T],Clim) be a mild solution of (1.7), with locally Lipschitzcontinuous g. Assume that

u(x, 0) ≥ v0(x) = a0 +N∑j=1

ajH(x − cj

) ∀x ∈ R, (4.15)

where aj ∈ R, c1 < c2 · · · < cN , and H is the Heaviside function. Let γ = minRv0(x) and δ =maxRv0(x). Assume also that g ≥ 0 on [γ, δ]. Then

u(x, t) ≥ a0 +N∑j=1

ajVαρ(x − cj , t

) ∀x ∈ R, 0 < t ≤ T, (4.16)

where Vαρ is defined in (2.4).


Proof. We know that u(x, t) ≥ γ and thus may assume that g(ζ) ≥ 0 also for ζ < γ . For arbitraryε, σ > 0, we set

vεσ(x) =1σ

∫σ

0v0(x − z)dz − ε. (4.17)

Then vεσ is piecewise linear and constant outside the interval [c1, cN + σ]; in particular, vεσ ∈Clim. Solving (2.1) with initial data vε,σ gives the solution

Gεσ(x, t) =1σ

∫σ

0

N∑j=1

ajVαρ(x − z − cj , t

)+ a0 − ε. (4.18)

Given ε > 0, it is possible to find σ > 0 such that vεσ < u0(x) on R, since u0 is uniformlycontinuous. Clearly, γ − ε ≤ Gεσ ≤ δ. We may therefore view Gεσ as a solution of (1.7) with aright-hand side h that satisfies h(ζ) = 0 ≤ g(ζ) for ζ ≤ δ and h(ζ) ≤ g(ζ) also for ζ > δ. Thenby Proposition 4.1

u(x, t) ≥ Gε,σ(x, t) ∀x ∈ R, 0 < t ≤ T. (4.19)

Send δ to 0, then since Vαρ is uniformly continuous, (4.16) is obtained with a0 replaced bya0 − ε on the right-hand side. Now send ε to 0 and (4.16) follows.

5. Conclusion

In this note, conditions for the speed of spread of solutions of fractional scalar reaction-diffusion equations to be finite or infinite have been derived. If the reaction term is positivefor all positive arguments, then this speed is shown to be infinite as soon as the reaction termdescribes some very weak growth for low densities. This is in contrast to the correspondingproblem for standard diffusion, where the speed of spread is always finite for such reactionterms. On the other hand, if the reaction term is negative for small positive arguments, thenthe speed of spread is finite, just as it is for the case of standard diffusion.

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[20] J. Nolan, Stable Distributions—Models for Heavy Tailed Data, chapter 1, Birkhauser, Boston, Mass, USA,2010.

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[22] J.-M. Bony, P. Courrege, and P. Priouret, “Semi-groupes de Feller sur une variete a bord compacte etproblemes aux limites integro-differentiels du second ordre donnant lieu au principe du maximum,”Annales de l’Institut Fourier, vol. 18, pp. 369–521, 1968.

[23] W. von Waldenfels, “Positive Halbgruppen auf einem n-dimensionalen Torus,” Archiv fur Mathema-tische Logik und Grundlagenforschung, vol. 15, pp. 191–203, 1964.


Research ArticleRiesz Potentials for Korteweg-de Vries Solitonsand Sturm-Liouville Problems

Vladimir Varlamov

Department of Mathematics, The University of Texas-Pan American, Edinburg, TX 78539-2999, USA

Correspondence should be addressed to Vladimir Varlamov, [email protected]

Received 9 August 2009; Accepted 9 November 2009

Academic Editor: Mark M. Meerschaert

Copyright q 2010 Vladimir Varlamov. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

Riesz potentials (also called Riesz fractional derivatives) and their Hilbert transforms arecomputed for the Korteweg-de Vries soliton. They are expressed in terms of the full-rangeHurwitz Zeta functions ζ+(s, a) and ζ−(s, a). It is proved that these Riesz potentials and theirHilbert transforms are linearly independent solutions of a Sturm-Liouville problem. Various newproperties are established for this family of functions. The fact that the Wronskian of the system ispositive leads to a new inequality for the Hurwitz Zeta functions.

1. Introduction

In recent years the theory of fractional derivatives and integrals called Fractional Calculushas been steadily gaining importance for applications. Ordinary and partial differentialequations of fractional order have been widely used for modeling various processes inphysics, chemistry, and engineering (see, e.g., [1–3] and the references therein). Recenttheoretical developments shed new light on the interpretation and properties of fractionalderivatives. Having written the latter in the form of Stieltjes integrals, Podlubny [4] foundnew physical and geometric interpretation of these structures relating them to inhomogeneityof time. Extension of the classical maximum principle to the case of a time-fractional diffusionequation appeared in the recent work of Luchko [5]. In the present paper we are concernedwith Riesz fractional derivatives (also called Riesz potentials; see [6, page 88], and [7, page117]) that are defined as fractional powers of the Laplacian Dα = (−Δ)α/2 with α ∈ R. Theyare well known for their role in investigating the solvability of nonlinear partial differentialequations, and the Korteweg-de Vries equation (KdV henceforth) in particular (see, e.g., [7–11] and the references therein). In the current work, Riesz potentials of KdV solitons arecomputed and their relation to ordinary differential equations is established.


We continue the study of Riesz fractional derivatives of solutions to Korteweg-de-Vries-type equations started in [12]. After appropriate rescaling, KdV can be written in theform

ut + uxxx + 3(u2)x= 0, x ∈ R, t > 0. (1.1)

It is well known that the fundamental solution of the Cauchy problem for the linearizedKdV is expressed in terms of the Airy function of the first kind Ai(x) and its Hilberttransform (conjugate) in terms of the Scorer functionGi(x). The papers [12–14] were devotedto the study of fractional properties of the Airy functions and their conjugates and to theestablishing of related properties for KdV-type equations.

Although there exists extensive literature on solitons, as far as we know, a study oftheir fractional properties is still missing. A preliminary investigation of Riesz potentials fora KdV soliton was carried out in [15]. In this paper the emphasis was put on the issue ofwhether solitons inherit fractional properties of fundamental solutions. Riesz potentials of asoliton, uα(X) = Dαu0(X), where u0(X) = 2sech2 X, X = x − 4t, and Dα = (−∂2

x)α/2, and their

Hilbert transforms, vα(X) = −Huα(X), were obtained in terms of the Hurwitz Zeta functionof a complex argument, ζ(s, z) with s = 2 + α and z = 1/2 + iX/π . It was proved in [15]that the zero mean properties hold for both uα(X) and vα(X) with α > 0. This confirmedthe predictions based on the properties of fundamental solutions of the linearized Cauchyproblems.

The goal of the current paper is to go further and to study Riesz potentials of solitonsas solutions of differential equations. We intend to show that these functions and their Hilberttransforms form linearly independent systems of solutions for a second-order ordinarydifferential equation in a self-adjoint form. This fact may be helpful in understanding theissue of using these structures as intrinsic mode functions in signal processing (see [16, 17]and the references therein), that is, in using Riesz potentials for expansions. In this context itis interesting to point out that the graphs of the functions uα(X) reveal a striking similarityto those of the Airy wavelets generated by the function Ai′(x)Ai′(−x) (see [18, page 34], andFigure 3 below).

For the analysis to follow; we employ the full-range Hurwitz Zeta functions: ζ+(s, a) =ζ(s, a)+ζ(s, 1−a) and ζ−(s, a) = ζ(s, a)−ζ(s, 1−a) (symmetric and antisymmetric combinationsof ζ(s, a) and ζ(s, 1 − a)), recently introduced in [19] for a ∈ R. We prove that the functionswα(X) = uα(X) + ivα(X), α > −1 are solutions of the Sturm-Liouville problem

− d

dX

(Pα(X)

dw

dX

)+Qα(X)w = λ ρα(X)w, X ∈ R, (1.2)

limX→±∞

w(X) = 0, (1.3)

for λ = 1. Here Pα(X) > 0, ρα(X) > 0, and Qα(X) is a real function. The essential point consistsin proving that the Wronskian of uα(X) and vα(X) is positive for all α > −1 and x ∈ R. Itallows one to prove that Pα(X) and ρα(X) are positive and to estimate the number of zeros ofuα(X) and vα(X) on any bounded interval.


The fact that this Wronskian is positive also leads to a new inequality for the HurwitzZeta functions

ζ(s, z)ζ(s + 1, z) + ζ(s, z)ζ(s + 1, z) > 0, (1.4)

where s > 1, z = 1/2+iX/π , and the bar over the letter denotes complex conjugation. As far aswe know, there are no results in the literature on the arguments of the Hurwitz Zeta functions.However, (1.4) provides some information on this issue. Indeed, setting ζ(s, z) = rse

iϕs andζ(s + 1, z) = rs+1e

iϕs+1 in (1.4) allows one to deduce the relation cos(ϕs+1(z) − ϕs(z)) > 0 fors > 1, z = 1/2 + iX/π , and X ∈ R.

The paper is organized as follows. In Section 2, we provide the necessary informationon the special functions involved. Section 3 is devoted to the study of Riesz potentials for KdVsolitons and their Hilbert transforms. In Section 3.1, the main properties of these functions aresummarized. Sturm-Liouville problem (1.2) is derived in Section 3.2. Section 3.3 deals withthe properties of the Wronskian W[uα, vα]. Zeros of the functions uα and vα are studied inSection 3.4. In Section 4, the inequality (1.4) is discussed.

2. Preliminaries

Introduce the Fourier transform of the function f : R → R by

f(ξ) = F{f}(ξ) =

∫∞−∞e−iξxf(x)dx, (2.1)

and the inverse Fourier transform by

f(x) = F−1{f}(x) =

12π

∫∞−∞eiξxf(ξ)dξ. (2.2)

For real α and x ∈ R, define Riesz potentials of a function f(x) by the formula (see [7,page 117])

Dαf(x) =1

2π

∫∞−∞|ξ|αf(ξ)eiξxdξ, (2.3)

provided that the integral in the right-hand side exists. Define derivatives of Dαxf(x) with

respect to α by

∂nαDαxf(x) =

12π

∫∞−∞|ξ|αlnn |ξ|f(ξ)eiξxdξ, n ∈ N, (2.4)

provided that these integrals exist.Introduce the Hilbert transform of the function f by (see [20, page 120])

Hf(x) =1πP.V.

∫∞−∞

f(y)

y − xdy,(2.5)


where x ∈ R and P.V. denotes the Cauchy principal value of the integral. According to ourchoice of the Fourier transform (Hf)(ξ) = i sgn(ξ)f(ξ), one can see thatH2 = −I on Lp(R), p ≥1, where I is the identity operator. Also, ∂x = H ◦ D and ∂2

x = −D2, where the operator D isdefined by (2.3).

Next, introduce the Trigamma function by (see [21, page 260, 6.4.1])

ψ ′(z) =d2

dz2ln Γ(z) =

∫∞0

te−zt

1 − e−t dt, R(z) > 0. (2.6)

Notice that (see [21], page 260, 6.4.7)

ψ ′(z) + ψ ′(1 − z) = −π d

dz{cot(πz)}. (2.7)

Also, the following asymptotic expansion holds:

ψ ′(z) ∼ 1z+

12z2

+1

6z3+O(

1z5

)for z −→ ∞,

∣∣arg z∣∣ < π. (2.8)

The Hurwitz (generalized) Zeta function is defined by (see [22, page 88])

ζ(s, a) =∞∑n=0

1(n + a)s

for R(s) > 1, a ∈ C, a /= 0,−1,−2, . . . . (2.9)

This implies that

∂aζ(s, a) = −sζ(s + 1, a). (2.10)

This function has the integral representation

ζ(s, a) =1

Γ(s)

∫∞0

xs−1e−ax

1 − e−x dx for R(s) > 1, R(a) > 0, (2.11)

where Γ(x) is the Gamma function. In two particular cases, we have (see [22]) that

ζ

(s,

12

)= (2s − 1)ζ(s), (2.12)

ζ(2, a) = ψ ′(a), (2.13)

where ζ(s) is the Riemann Zeta function.The singularity of ζ(s, a) as s → 1 is given by the relation

lims→ 1

{ζ(s, a) − 1

s − 1

}= −Γ

′(a)Γ(a)

= −ψ(a). (2.14)


The asymptotic expansion of ζ(s, a) for large a is (see [23])

ζ(s, a) ∼ 12a−s +

a1−s

s − 1+

1Γ(s)

∞∑k=1

B2k

(2k)!Γ(2k + s − 1)

a2k+s−1, |a| <∞,

∣∣arg a∣∣ < π, (2.15)

where Bn are the Bernoulli numbers.Introduce the full-range forms of Hurwitz Zeta functions (see [19])

ζ+(s, a) =∞∑

n=−∞

1[(n + a)2

]s/2 , (2.16)

ζ−(s, a) =∞∑n=0

1[(n + a)2

]s/2−

−1∑n=−∞

1[(n + a)2

]s/2. (2.17)

Representations (2.16) and (2.17) imply that

ζ+(s, a) = ζ(s, a) + ζ(s, 1 − a), ζ−(s, a) = ζ(s, a) − ζ(s, 1 − a). (2.18)

Hence, follow the symmetric and antisymmetric properties of the functions,

ζ+(s, a) = ζ+(s, 1 − a), ζ−(s, a) = −ζ−(s, 1 − a). (2.19)

It follows from (2.16) that ζ+(s, a) is a periodic function of a, with the unit period. It is evenwith respect to a = 0 and a = 1/2. The function ζ−(s, a) is odd about a = 1/2 and ζ−(s, 1/2) =0. These functions satisfy the functional differential equations

∂aζ+(s, a) = −sζ−(s + 1, a), ∂aζ−(s, a) = −sζ+(s + 1, a). (2.20)

Consider now that s ∈ R and denote by a the complex conjugate of a. Then ζ(s, a) =ζ(s, a). It implies that, for a = a1 + ia2,

ζ+(s, a) = 2R{ζ(s, a)} = 2Γ(s)

∫∞0

xs−1e−a1x cosa2x

1 − e−x dx. (2.21)

In a similar way,

ζ−(s, a) = 2iI{ζ(s, a)} = − 2iΓ(s)

∫∞0

xs−1e−a1x sin(a2x)1 − e−x dx. (2.22)

Denote by W[u, v] the Wronskian of the functions u(x) and v(x), that is, W[u, v] =∣∣∣ u v

u′ v′

∣∣∣. For reader’s convenience, we present here [24, Theorem 5.3].


Theorem 2.1. Let p(t) > 0, q(t) be real valued and continuous for 0 ≤ t ≤ T . Let u and v be realvalued solutions of the equation

(pu′)′ + qu = 0, (2.23)

satisfying the condition

p(t)W[u(t), v(t)] = C0 > 0. (2.24)

LetN be the number of zeros of u(t) on [0, T]. Then

∣∣∣∣∣πN − C0

∫T0

dt

p(t)[v2(t) + u2(t)]

∣∣∣∣∣ ≤ π. (2.25)

In conclusion of this section, we would like to quote an interesting result concerningintegrals over the real axis (see [25]).

Theorem 2.2. For any integrable function f(x) and g(x) = x − c2/x with c = const ∈ R,

P.V.

∫∞−∞f(g(x)

)dx = P.V.

∫∞−∞f(x)dx. (2.26)

Moreover, the above formula holds true if

g(x) = x −∞∑j=1

bj

x − cj, (2.27)

where {bj} is any sequence of positive constants, cj are any real constants, and the series is convergent.

3. Fractional Derivatives of A KdV Soliton and Their Conjugates

In this section, we consider Riesz fractional derivatives of a KdV soliton and their Hilberttransforms and establish their properties. We notice that all the graphs were obtained withthe Mathematica 6 software.

We take the soliton solution of (1.1) in the form u0(X) = 2sech2 X with X = x − 4t (seeFigure 1) and introduce the function

wα(X) = uα(X) + ivα(X), (3.1)

where

uα(X) = Dαu0(X), vα = −DαHu0(X). (3.2)


−4 −2 2 4

0.5

1

1.5

2

2.5

Figure 1: Graph of the soliton u0(X).

Notice that the functions uα(X) and vα(X) form a conjugate pair (see [20, page 120]) since

uα(X) = Hvα(X), vα(X) = −Huα(X). (3.3)

The next statement was proved in [15]. Using the functions (2.16) and (2.17), werewrite it in a more convenient form.

Theorem 3.1. The functions uα(X) and vα(X) have the following representations for α > −1 andX ∈ R:

uα(X) =2Γ(2 + α)π2+α

ζ+

(2 + α,

12+ i

X

π

), (3.4)

vα(X) = i2Γ(2 + α)π2+α

ζ−

(2 + α,

12+ i

X

π

), (3.5)

where ζ+(s, a) and ζ−(s, a) are the full-range Hurwitz Zeta functions (see (2.16) and (2.17)) and Γ(s)is the Gamma function.

3.1. Properties of the Functions uα and vα

In this subsection, we collect the properties of the functions uα and vα. Some of them wereestablished in [15] and some are given for the first time as follows.

Properties of the Functions uα and vα

(1) The functions uα(X) and vα(X) satisfy the functional differential equations

u′α(X) = −vα+1(X), v′α(X) = uα+1(X), (3.6)

where α > −1, X ∈ R, and the prime denotes differentiation with respect to X. This followsfrom (3.3) and the relation d/dX = H ◦D.


(2) The functions uα(X) are even and the functions vα(X) are odd on R (see [15] andFigures 1–4).

(3) The function uα(X) is periodic with the period X = iπ . It follows from theperiodicity of ζ+(s, a) with the unit period.

(4) For all α > 0 (see [15]),

∫∞−∞uα(X)dX = 0,

∫∞−∞vα(X)dX = 0. (3.7)

These properties are reflected on the graphs (see Figures 3 and 4).

(5) For all α > 0 and n ∈ N,

∂nα

∫∞−∞uα(X)dX = 0, ∂nα

∫∞−∞vα(X)dX = 0. (3.8)

These relations follow from the differentiation of the identities in (3.7).

(6) For all α > 0 and c ∈ R,

P.V.

∫∞−∞uα

(X − c

2

X

)dX = 0, P.V.

∫∞−∞vα

(X − c

2

X

)dX = 0. (3.9)

Moreover,

P.V.

∫∞−∞uα

⎛⎝X −

∞∑j=1

bj

X − cj

⎞⎠dX = 0,

P.V.

∫∞−∞vα

⎛⎝X −

∞∑j=1

bj

X − cj

⎞⎠dX = 0,

(3.10)

where bj is any sequence of positive constants, cj are any real constants, and the seriesconverges. These relations follow from (3.7) and Theorem 2.2 .

(7) The functional sequence of Riesz potentials {uα(X)} converges pointwise to thesoliton u0(X), and the functional sequence {vα(X)} converges pointwise to the


−4 −2 2 4

−2

−1.5

−1

−0.5

0.5

1

1.5

2

Figure 2: Graph of the conjugate soliton v0(X).

conjugate soliton v0(X) for α → 0+ (see [15]). Notice that

∫∞−∞u0(X)dX = 4, P.V.

∫∞−∞v0(X)dX = 0 (3.11)

(see Figures 1 and 2). Here the conjugate soliton v0(X) is given by

v0(X) =2iπ2

[ψ ′(

12+ i

X

π

)− ψ ′(

12− iX

π

)]. (3.12)

Equation (3.12) can be recovered from (3.5) with α = 0 thanks to (2.13).

Remark 3.2. The conjugate soliton (3.12) is an algebraic solitary wave for extended KdV:

vt + vxxx + 3[2v ·Hv −H

(v2)]

x= 0, (3.13)

obtained by applying the Hilbert transform to (1.1) and setting v = −Hu. The term “algebraicsolitary wave” is explained by the fact that v0(X) has a 1/X decay for large X. More precisely(see [15]),

v0(X) ∼ 2π· 1X

+O(

1X2

)for |X| −→ ∞. (3.14)

(8) For α ≥ 0, the functions uα and vα are the elements of L2(R). Moreover, they areorthogonal in the principal value sense, namely, for all α1, α2 ≥ 0,

P.V.

∫∞−∞uα1(X)vα2(X)dX = 0. (3.15)


−4 −2 2 4

−10

10

20

Figure 3: Graph of the fractional derivative u3.8(X).

Equations (3.12) and (3.14) imply that v0 ∈ L2(R). The fact that uα and vα with α > 0 are theelements of L2(R) follows from their asymptotics obtained in [15], namely,

uα(X) ∼ 4Γ(1 + α)π2+α

cos[(1 + α) arctan(2X/π)][1/4 + (X/π)2

](1+α)/2+O

(1

|X|2+α

),

vα(X) ∼ 4Γ(2 + α)π2+α

2 sin[(1 + α) arctan(2X/π)]

(1 + α)[1/4 + (X/π)2

](1+α)/2+O

(1

|X|2+α

).

(3.16)

Orthogonality of uα and vα follows from the fact that all uα(X) with α ≥ 0 are even functionsand all vα(X) with α ≥ 0 are odd functions of X (see Property 2 and Figures 1–4).

(9) At the point X = 0, one has for all α > −1

uα(0) =4Γ(2 + α)π2+α

(2α+2 − 1

)ζ(2 + α) > 0, (3.17)

vα(0) = 0, (3.18)

where ζ(s) is the Riemann Zeta function (see [15]).

3.2. Sturm-Liouville Problem

It is convenient to represent wα(X) in the exponential form, namely,

wα(X) = Rα(X) exp[iΘα(X)], (3.19)


−4 −2 2 4

−20

−10

10

20

Figure 4: Graph of the conjugate fractional derivative v3.8(X).

where

Rα(X) =√[uα(X)]2 + [vα(X)]2,

Θα(X) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

Φα(X), uα(X) > 0,

Φα(X) + π, uα(X) < 0, vα(X) ≥ 0,

Φα(X) − π, uα(X) < 0, vα(X) < 0,

Φα(X) = arctanvα(X)uα(X)

.

(3.20)

Exponential representation (3.19) allows one to deduce the boundary value problem for thefunctions wα(X). It turns out that for all α > −1 the functions wα(X) solve the equation

d

dX

(Pα(X)

dw

dX

)−Qα(X)w + ρα(X)w = 0, X ∈ R, (3.21)

where

Pα(X) =C

W[uα(X), vα(X)], C = const,

Qα(X) =[Pα(X)R′α(X)]′

Rα(X),

ρα(X) = [Θ′α(X)]2 ,

(3.22)

Θ′α(X) =W[uα(X), vα(X)]

R2α(X)

. (3.23)

Here W[uα(X), vα(X)] is the Wronskian of uα(X) and vα(X). Below we shall use a shorternotation Wα(X) =W[uα(X), vα(X)].


−4 −2 2 4

−2

−1.5

−1

−0.5

0.5

1

1.5

2

Figure 5: Graph of the arctangent function Φ3.8(X).

Taking into account the behavior of wα(X) for large X (see [15]), we can restate theobtained results in another form. Indeed, wα(X) are solutions of the Sturm-Liouville problem

− d

dX

(Pα(X)

dw

dX

)+Qα(X)w = λ ρα(X)w, X ∈ R, (3.24)

limX→±∞

w(X) = 0, (3.25)

corresponding to λ = 1. Without loss of generality, we can choose the constant in (3.22) to bepositive. In the next subsection, we shall prove that Wα(X) > 0. It implies that Pα(X) > 0 andρα(X) > 0 in (3.22).

It follows from (3.7) that for α > 0 the functions wα(X) also satisfy the zero meancondition

∫∞−∞wα(X)dX = 0. (3.26)

This reflects the oscillatory behavior of wα(X) for α > 0 (see Figures 3 and 4).The graph of the arctangent function Φα(X) is shown in Figure 5. Φα(X) conveniently

serves as a zero counter for both functions: uα(X) and vα(X). It possesses zeros at the pointswhere vα(X) has zeros and has jumps at the points where uα(X) has zeros.

Remark 3.3. Observe that a general solution of (3.24) can be written in the form

w(g)(X) = C1uα(X) + C2uα(X)∫X

0

Wα

(y)

u2α

(y) dy, (3.27)

where C1, C2 = const.


Remark 3.4. We would like to point out that the differential equation given by (3.21) can befactored in the following way (see [26, page 269]):

d

dX

(Pα(X)

dw

dX

)−Qα(X)w + ρα(X)w =

Wα(X)uα(X)

d

dX

[u2α(X)

Wα(X)d

dX

(w(X)uα(X)

)]. (3.28)

3.3. Wronskian of uα and vα

Lemma 3.5. The following properties hold for the WronskianWα(X) for α > −1 and all x ∈ R:

Wα(−X) =Wα(X), W ′α(0) = 0, (3.29)

Wα(X) > 0. (3.30)

Proof. We start with (3.29). Taking into account relations in (3.3) and the fact that ∂X = H ◦D,we can write

Wα(X) = uα(X)v′α(X) − vα(X)u′α(X) = uα(X)uα+1(X) + vα(X)vα+1(X). (3.31)

Since the functions uα(X) are even and vα(X) are odd with respect to X ∈ R, Wα(X) is evenfor X ∈ R. Differentiation of (3.31) with the help of (3.6) and (3.18) yields W ′

α(0) = 0. Next,we turn to the proof of (3.30). Since the functions uα(X) and vα(X) are linearly independentsolutions of the equation (3.24), Wα(X)/= 0 for all α > −1 and x ∈ R. It remains to establishthe sign of the Wronskian. In view of (3.17) and (3.18),

Wα(0) = uα(0)uα+1(0) =16Γ(2 + α)Γ(3 + α)

π5+2α

(22+α − 1

)(23+α − 1

)ζ(2 + α)ζ(3 + α) > 0. (3.32)

By Abel’s formula, for all X ∈ R,

Wα(X) =Wα(0) exp

[−∫X

0

P ′α(η)

Pα(η) dη], (3.33)

where the function

P ′α(X)Pα(X)

= − i(s + 1)π

· ζ−(s, z)ζ+(s + 2, z) − ζ+(s, z)ζ−(s + 2, z)ζ+(s, z)ζ+(s + 1, z) − ζ−(s, z)ζ−(s + 1, z)

,

s = 2 + α, z = 1/2 + iX/π,

(3.34)


0

20

40

0

1

2

−4

−2

0

2

4

Figure 6: Three-dimensional graph of the Wronskian Wα(X).

is continuous on R. After some simplification, we have that

Wα(X) =Wα(0) exp{−s + 1

π

I[ζ(s, z)]R[ζ(s + 2, z)] −R[ζ(s, z)]I[ζ(s + 2, z)]R[ζ(s, z)]R[ζ(s + 1, z)] + I[ζ(s, z)]I[ζ(s + 1, z)]

}. (3.35)

This representation yields (3.30). The lemma is proved.

Three-dimensional graph of Wα(X) is given in Figure 6.

Remark 3.6. What does the positivity of the Wronskian yield for the soliton and its conjugate?For α = 0, (3.30) simplifies to read

W0(X) = 2sech2(X)[v′0(X) + tanhX · v0(X)

]> 0. (3.36)

We notice that v0(X) > 0 for X > 0, v0(−X) = −v0(X) for X ∈ R, and v0(0) = 0 (see Figure 2).Integrating the inequality v′0(X)/v0(X) + tanhX > 0 over the interval [ε,X] for X ≥ ε > 0 andthe inequality v′0(X)/v0(X) + tanhX < 0 over [−ε,X] for X ≤ −ε < 0 yields the estimate

|v0(X)| > |v0(ε)|cosh εcoshX

for |X| ≥ ε > 0. (3.37)

3.4. Zeros of the Functions uα(X) and vα(X)

This subsection is devoted to the estimates of the number of zeros for the functions inquestion. By a strictly monotone change of variable

y =∫Xb

dη

Pα(η) , b = const ∈ R, (3.38)


Equation (3.21) can be reduced to the equation

d2Wdy2

+ Qα

(y)W = 0, (3.39)

where W(y) = w(X(y)) and Qα(y) = Pα(X(y))Qα(X(y)). Therefore, any nontrivial solutionof (3.39) can have not more than a finite number of zeros on any bounded interval ([24], page323).

Theorem 3.7. Let Nα be the number of zeros of the function vα(X) on the interval [0, X0], whereX0 <∞. Then the following inequality holds:

Iα − 1 ≤Nα ≤ Iα + 1, (3.40)

where

Iα = arctaniζ−(2 + α, 1/2 + iX0/π)ζ+(2 + α, 1/2 + iX0/π)

. (3.41)

Proof. Since we chose C = 1 in (3.22),

Pα(X)W[uα(X), vα(X)] = 1. (3.42)

Therefore,

Θ′α(X) =v′αuα − u′αvα

v2α

> 0. (3.43)

Therefore, by Theorem 2.1 of Section 2, for the interval [0, X0] we have the estimate

−1 + Iα ≤Nα ≤ 1 + Iα, (3.44)

where

Iα =∫X0

0

Wα

(η)dη

v2α

(η)+ u2

α

(η) =∫X0

0Θ′α(η)dη = arctan

iζ−(2 + α, 1/2 + iX0/π)ζ+(2 + α, 1/2 + iX0/π)

. (3.45)

Here we have used the fact that ζ−(s, 1/2) = 0 for s > 1.

Theorem 3.8. The zeros of uα(X) separate and are separated by those of vα(X).

Proof. This follows from Sturm’s Separation Theorem (see [24], page 335).


4. Inequality for Hurwitz Zeta Functions

Here we discuss a new inequality for the Hurwitz Zeta functions which follows fromLemma 3.5. The next statement is a corollary of this lemma.

Corollary 4.1. For s > 1 and z = 1/2 + iX/π with X ∈ R, the following inequality holds:

K(s, z) = ζ(s, z)ζ(s + 1, z) + ζ(s, z)ζ(s + 1, z) > 0. (4.1)

This inequality can also be written in another form:

K(s, z) = R{ζ(s, z)}R{ζ(s + 1, z)} + I{ζ(s, z)}I{ζ(s + 1, z)} > 0. (4.2)

Proof. Dropping positive terms in front of the full-range Hurwitz Zeta functions in (3.4) and(3.5) and using (3.30) lead to

ζ+(s, z)ζ+(s + 1, z) − ζ−(s, z)ζ−(s + 1, z)

= 2[ζ(s, z)ζ(s + 1, z) + ζ(s, z)ζ(s + 1, z)]

= 4[R{ζ(s, z)}R{ζ(s + 1, z)} + I{ζ(s, z)}I{ζ(s + 1, z)}] > 0.

(4.3)

Remark 4.2. Setting ζ(s, z) = rs(z)eiϕs(z) and ζ(s + 1, z) = rs+1eiϕs+1(z), we can rewrite (4.1) in

the form

K = 2rsrs+1{

exp[i(ϕs+1 − ϕs

)]+ exp

[−i(ϕs+1 − ϕs

)]}= 4rsrs+1 cos

(ϕs+1 − ϕs

). (4.4)

It shows that, for z = 1/2 + iX/π , X ∈ R,

cos(ϕs+1(z) − ϕs(z)

)> 0. (4.5)

Introduce the scalar product for the complex-valued functions f = f1 + if2 and g = g1 + ig2 bythe formula

⟨f · g⟩= f · g = f1f2 + f2g2 + i

(f2g1 − f1g2

). (4.6)

Then for s > 1 and z = 1/2 + iX/π ,

K(s, z) = 2R{〈ζ(s, z) · ζ(s + 1, z)〉} > 0. (4.7)

Remark 4.3. The proof of (4.1) becomes quite difficult when one approaches it from the pointof view of special functions. For example, the use of integral representations (2.21) and (2.22)yields

K =1

2Γ(s)Γ(s + 1)

∫∫∞0

ts−1τs cos[X/π(t − τ)]sinh(t/2) sinh(τ/2)

dtdτ. (4.8)


The change of variables ξ = (t − τ)/2, η = (t + τ)/2 leads to

K =2

Γ(s)Γ(s + 1)

∫∞0

cos(

2X

πξ

)dξ

∫ ξ−ξ

(ξ + η

)s−1(η − ξ

)scoshη − cosh ξ

dη. (4.9)

It is not clear at all that the integral (4.9) is positive for all s > 1 and X ∈ R. However, (4.1)shows that it is. Multiplication of the series representations (2.16) and (2.17) does not makethe proof any easier.

References

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tions, vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands,2006.


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no. 30, Princeton University Press, Princeton, NJ, USA, 1970.[8] T. Kato, “On the Cauchy problem for the (generalized) Korteweg-de Vries equation,” in Studies in

Applied Mathematics, vol. 8 of Advances in Mathematics. Supplementary Studies, pp. 93–128, AcademicPress, New York, NY, USA, 1983.

[9] C. E. Kenig, G. Ponce, and L. Vega, “On the (generalized) Korteweg-de Vries equation,” DukeMathematical Journal, vol. 59, no. 3, pp. 585–610, 1989.

[10] C. E. Kenig, G. Ponce, and L. Vega, “Well-posedness and scattering results for the generalizedKorteweg-de Vries equation via the contraction principle,” Communications on Pure and AppliedMathematics, vol. 46, no. 4, pp. 527–620, 1993.

[11] J. C. Saut and R. Temam, “Remarks on the Korteweg-de Vries equation,” Israel Journal of Mathematics,vol. 24, no. 1, pp. 78–87, 1976.

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Research ArticleLinear Fractionally Damped Oscillator

Mark Naber

Department of Mathematics, Monroe County Community College, Monroe, MI 48161-9746, USA

Correspondence should be addressed to Mark Naber, [email protected]

Received 8 July 2009; Accepted 11 August 2009

Academic Editor: Mark M. Meerschaert

Copyright q 2010 Mark Naber. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

The linearly damped oscillator equation is considered with the damping term generalized to aCaputo fractional derivative. The order of the derivative being considered is 0 ≤ v ≤ 1. At thelower end (v = 0) the equation represents an undamped oscillator and at the upper end (v = 1) theordinary linearly damped oscillator equation is recovered. A solution is found analytically, and acomparison with the ordinary linearly damped oscillator is made. It is found that there are ninedistinct cases as opposed to the usual three for the ordinary equation (damped, over-damped, andcritically damped). For three of these cases it is shown that the frequency of oscillation actuallyincreases with increasing damping order before eventually falling to the limiting value given bythe ordinary damped oscillator equation. For the other six cases the behavior is as expected, thefrequency of oscillation decreases with increasing order of the derivative (damping term).

1. Introduction

In this paper the linearly damped oscillator equation is considered with the damping termreplaced by a fractional derivative [1] whose order, ν, will be restricted to, 0 ≤ ν ≤ 1,

D2t x + λ0D

νt x +ω2x = 0. (1.1)

Burov and Barkai [2, 3] examined such an equation in connection with critical behavior. Theywere able to determine a solution in terms of generalized Mittag-Leffler functions. Nonlinearfractional oscillators have been studied numerically by Zaslavsky et al. [4]. He was primarilyinterested in chaotic behavior. It is hoped that a careful study of the analytic solution tothe linear fractionally damped equation will help shed light on properties of the nonlinearequation and be of use for direct applications of fractionally damped oscillations (see, e.g.,[5, 6]).


In this paper the Caputo formulation of the fractional derivative will be used. TheCaputo derivative is preferred over the Riemann-Liouville derivative for physical reasons.Consider the Laplace transform of the two formulations of the fractional derivative for 0 <ν < 1

L(RL0 Dν

t f(t))= sνF(s) − RL

0 Dν−1t f(t)

∣∣0,

L(C0 D

νt f(t)

)= sνF(s) − f(t)

∣∣0.

(1.2)

The constant term arising from the Laplace transform of the Caputo derivative is merelythe initial value of the function. For the Riemann-Liouville derivative this is not the case.The constant term arising from the Laplace transform currently has no simple physicalinterpretation. Hence the Caputo fractional derivative seems to be more useful for modelingphysical systems.

If the order of the fractional damping term is allowed to become 3/2 (outside therange of values considered in this paper) the equation is usually referred to as the Bagley-Torvik equation (see, e.g., [1, 7, 8]). The solution of this equation exhibits damped oscillatorybehavior similar to what we expect to find for the equation studied in this paper. The Bagley-Torvik equation was originally derived to study the motion of a rigid plate in a Newtonianfluid [7].

The analytic solution to the fractionally damped equation is found by means ofLaplace transform. For the sake of clarity, and for pointing out some unique difficulties withthe factional equation, a comparison with the Laplace transform method as applied to thenonfractional case is made. It is found that there are nine distinct cases for the fractionallydamped equation as opposed to the usual three cases for the Nonfractional equation. In sixof the nine cases the results are as expected; increasing the order of the factional derivativeincreases the effects of the damping (i.e., the frequency of the damping slows as the orderof the derivative increases). However, in three cases, the frequency of the damping actuallyincreases as the order of the fractional derivative increases until a peak value is reached afterwhich the frequency falls to its Nonfractional limit. The physical reason for this increase inthe oscillation frequency is not yet clear.

2. The Nonfractional Case

Before a solution to the linear fractionally damped oscillator equation is constructed it will beuseful to review the Laplace transform method of solution for the linearly damped oscillatorequation

D2t x + λDtx +ω2x = 0. (2.1)

The constants λ and ω are taken to be real and positive. λ is the damping force per unitmass. ω2 is the restoring force per unit mass. In both cases (fractional and Nonfractional) thefollowing initial conditions will be used:

x(0) = x0, Dtx(0) = x1. (2.2)


Transforming (2.1) together with the initial conditions (2.2) gives the following:

s2X(s) − sx0 − x1 + λ(sX(s) − x0) +ω2X(s) = 0, (2.3)

or

X(s) =sx0

s2 + λs +ω2+

x1 + λx0

s2 + λs +ω2. (2.4)

Equation (2.4) can be inverted using tables, however, to shed light on a problem that willhappen later, (2.4) will be inverted via the complex inversion integral. The exponents on thes variable in both terms are whole numbers, hence there will not be a branch cut in the contourintegral and the Bromwich contour can be used

x(t) = Residue − 12πi

∫estX(s)ds. (2.5)

Recall that the Bromwich contour begins at γ − i∞ goes vertically up to γ + i∞ (where γ ischosen so that all poles will lie to the left of the vertical contour line and thus all poles will becaptured within the contour) and then travels in a half circle (to the left, counter clockwise)back to γ − i∞. For this problem there is no contribution from the contour integral. The onlycontribution comes from the residue. The residue is generated from the roots of the followingquadratic equation

s2 + λs +ω2 = 0. (2.6)

There are three different cases.(1) λ > 2ω; 2 unequal real roots that are negative

s1,2 =−λ ±

√λ2 − 4ω2

2. (2.7)

(2) λ = 2ω; 2 repeated real roots that are negative

s3 = −λ2= −ω. (2.8)

(3) λ < 2ω; 2 complex roots whose real parts are negative

s4,5 =−λ ± i

√4ω2 − λ2

2. (2.9)

See Figure 1 for a graphical representation of the location of the roots.Note that in cases one and three the poles will be of order one and in case two the

pole will be of order two. Note also that if there were no damping the poles would be on theimaginary axis at ± iω.


s2 s3 s1

s4

s5

iω

−iω

Im

Re

Figure 1

Computing the residue for case one gives

Residue = lims→ s1,2

(s − s1,2)est(

sx0

s2 + λs +ω2+

x1 + λx0

s2 + λs +ω2

), (2.10)

or

x(t) =es1t

2s1 + λ(s1x0 + x1 + λx0) +

es2t

2s2 + λ(s2x0 + x1 + λx0). (2.11)

As s1 and s2 are both negative this solution will decay exponentially. This is usually referredto as the over-damped case.

Case three is computed the same way as case one. Now the poles are complex so theexponential function can be expressed using sine and cosine with an over-all exponentialdamping factor

x(t) = e−αt(x0 cos

(ρt)+(2x1 + λx0)

2ρsin(ρt)), (2.12)

where ρ =√ω2 − λ2/4 and α = λ/2 Notice that the presence of damping causes the effective

angular frequency, ρ, to be smaller than the un-damped angular frequency; that is, theoscillations go slower, as one might expect if there were damping to impede the motion. Thisis usually referred to as the under-damped case. By comparison, cases one and two could beviewed as having a zero frequency or an infinite period.

For case two the pole is of order 2, and the residue is given by

lims→−ω

d

ds

((s +ω)2est

(sx0 + x1 + λx0

s2 + λs +ω2

)). (2.13)

Recall that λ = 2ω, this allows the denominator to be factored. The limit then becomes

lims→−ω

d

ds

(est(sx0 + x1 + 2ωx0)

), (2.14)

or

x(t) = e−ωt(t(ωx0 + x1) + x0). (2.15)


This is usually called the critically damped case. Graphs of sample solutions to these threecases can be found in any introductory book on differential equations.

3. The Fractional Case

Now consider (1.1). In this case, λ has units of time raised to the power ν − 2. Hence theover all units of the second term remain the same as in (2.1). There are two cases to consider,0 < ν < 1 and 1 < ν < 2. We will consider the former in this paper. The Laplace transform of(1.1) is

s2X − sx0 − x1 + λ(sνX − sν−1x0

)+ω2X = 0, (3.1)

or

X =sx0 + x1 + λsν−1x0

s2 + λsν +ω2. (3.2)

If (3.2) is inverted using a contour integral a branch cut is needed on the negative real axisdue to the fractional exponents on the complex variable s. Hence, a Hankel contour will beused. This contour starts at γ − i∞ and goes vertically up to γ + i∞ (where γ is again chosenso that all poles will lie to the left of the vertical contour line) then travels in a quarter circlearc (to the left) to just above the negative real axis (i.e., −∞). The contour then has a cut thatgoes into the origin (following the negative real axis), around the origin in a clockwise sense(to just below the negative real axis) and then back out to −∞. The contour then has anotherquarter circle arc to γ − i∞.

Now the question is, where are the poles? This is a somewhat more involved questionthan in the standard linearly damped model. To find the poles the following equation needsto be solved:

s2 + λsν +ω2 = 0, (3.3)

Which, for an arbitrary ν, is not a trivial problem. To determine if there are solutions, and ifso how many, let s = reiθ then (3.3) breaks into 2 equations, a real and an imaginary part

r2 cos(2θ) + λrν cos(νθ) +ω2 = 0,

r2 sin(2θ) + λrν sin(νθ) = 0.(3.4)

Could there be a solution on the positive real axis? No, in this case θ = 0 and the first equationof (3.4) would be the sum of three positive nonzero terms, which would never be zero. Couldthere be a solution on the negative real axis? No, in this case θ = π and the second termof the second equation of (3.4) would never be zero. Using similar arguments we can showthat there are no solutions on the positive or negative imaginary axes, recall 0 < ν < 1. Itcan also be shown that no solutions are in the right half plane (both terms of the secondequation would always be positive). If there are solutions, they should be in pairs, complexconjugates, with π/2 < θ < π and −π/2 > θ > −π . To attempt to find a solution first solve the


second equation of (3.4) for r and substitute this into the first equation (only look for positiveθ values first)

(−λsin(νθ)

sin(2θ)

)2/(2−ν)cos(2θ) + λ

(−λsin(νθ)

sin(2θ)

)ν/(2−ν)cos(νθ) +ω2 = 0. (3.5)

The reader may be worried about the negative sign and the fractional exponent in (3.5),however, for the restricted angular range being considered, π/2 < θ < π , sin(2θ) is alwaysnegative. So, the argument of the root will always be positive.

Given values for ν, λ, and ω it would appear to be impossible to solve (3.5) for θ.Equation (3.5) can be simplified to a more aesthetically pleasing form

((sin(νθ))ν

(sin(2θ))2

)1/(2−ν)

sin((2 − ν)θ) =(

ω

λ1/(2−ν)

)2

. (3.6)

Now it needs to be seen if there is a θ value that will satisfy (3.6). For this equationto be true sin((2 − ν)θ) needs to be positive. This will only happen on the restricted domainπ/2 < θ < π/(2 − ν). Now the question becomes, on this restricted domain can we pick a θvalue that will make the left-hand side of (3.6) as large or as small as we wish? Thus ensuringthat no matter what the values of ω, λ, and ν we are given we can always find a θ value thatwill satisfy (3.6). Consider the two limits

limθ→π/2+

((sin(νθ))ν

(sin(2θ))2

)1/(2−ν)

sin((2 − ν)θ) =∞,

limθ→π/(2−ν)−

((sin(νθ))ν

(sin(2θ))2

)1/(2−ν)

sin((2 − ν)θ) = 0.

(3.7)

Since the left-hand side of (3.6) is continuous in θ and we have the two limits above, (3.7),it is guaranteed that there will be at least one solution to (3.6) and hence there will be atleast two poles for the residue calculation. If we can show that the left hand side of (3.6)decreases monotonically in θ over the restricted domain then we know that there will be onlyone solution to (3.6), and thus only two poles in the residue calculation. To show that the lefthand side of (3.6) decreases monotonically in θ we need to show that the derivative of theleft hand side of (3.6) with respect to θ is always negative, that is,

∂

∂θ

⎧⎨⎩

((sin(νθ))ν

(sin(2θ))2

)1/(2−ν)

sin((2 − ν)θ)

⎫⎬⎭ < 0. (3.8)

Computing the derivative, doing some algebra, and throwing away over all factors that arealways positive we have

ν2sin2(2θ) − 4ν sin(2θ) sin(νθ) cos((2 − ν)θ) + 4sin2(νθ) > 0. (3.9)


θ =π

2 − ν

θ = − π

2 − ν

s6

s5

θ = −π2

θ =π

2Im

Re

Figure 2

On the restricted domain sin(νθ) > 0, sin(2θ) < 0, and cos((2 − ν)θ) ≤ 1. This reduces (3.9) to

ν2sin2(2θ) − 4ν sin(2θ) sin(νθ) + 4sin2(νθ) > 0. (3.10)

Equation (3.10) can now be factored into a perfect square and prove the assertion madein(3.8)

(ν sin(2θ) − 2 sin(νθ))2 > 0. (3.11)

Hence the left hand side of (3.6) will decrease monotonically on the restricted domain withthe upper bound being∞ and the lower bound being 0. To summarize, it has just been shownthat there is always one solution, with a positive angle, to (3.6) and this solution must be suchthat π/2 < θ < π/(2 − ν). Consequently there will be two poles for the residue calculationand they will be complex conjugates of each other. Notice that for the fractionally dampedequation repeated roots are not possible. Repeated roots can only happen when the order ofthe derivative becomes one. See Figure 2 for a graphical representation of the location of theroots.

Now the question of the poles that has been settled the solution to (1.1) can begenerated. Denote the two poles as

s6,7 = β ± iσ = re±iθ, (3.12)

where β and σ are determined from the r and θ values that satisfy (3.6) in the usual way,

r =√β2 + σ2 and tan(θ) = σ/β. Note that β is negative, the two solutions are in the second

and third quadrants and s7 is the complex conjugate of s6. Note also that β plays the role of−λ/2 from the Nonfractional case. The poles are of order one and the residue is given by

Residue = lims→ s6

(s − s6)est(sx0 + x1 + λsν−1x0

s2 + λsν +ω2

)+ lims→ s7

(s − s7)est(sx0 + x1 + λsν−1x0

s2 + λsν +ω2

)

= es6t

(s6x0 + x1 + x0λs

ν−16

2s6 + νλsν−16

)+ es6t

(s6x0 + x1 + x0λs

ν−16

2s6 + νλsν−16

),

(3.13)


where s7 has been replaced by s6. After some algebra this can be reduced to

2eβt cos(σt)x0(2r2 + νλ2r2ν−2 + λrν(ν + 2) cos(θ(ν − 2))

)+ A

4r2 + 4νλνrν cos((2 − ν)θ) + ν2λ2νr2ν−2

+2eβt sin(σt)x0(λrν(ν − 2) sin((ν − 2)θ)) + x1

(2r sin(θ) + νλrν−1 sin(θ(ν − 1))

)

4r2 + 4νλrν cos((2 − ν)θ) + ν2λ2r2ν−2,

(3.14)

where A denote to x1(2r cos(θ) + νλrν−1 cos(θ(ν − 1))).For the contour integral the only contributions come from the paths along the negative

real axis

λ

π

∫∞0

(Rx0 − x1) sin(νπ) + (x0/R)(R2 +ω2) sin(π(ν − 1))

(R2 +ω2)2 + 2λRν(R2 +ω2) cos(νπ) + (λRν)2e−RtRνdR. (3.15)

The solution to (1.1) is then (3.15) subtracted from (3.14). This may look overly complicatedbut the solution does have the general form of

x(t) = Aeβt cos(σt) + Beβt sin(σt) −Decay function. (3.16)

Notice that the decay function, (3.15), goes to zero if ν goes to zero or one; that is, if (1.1) goesto its Nonfractional limits the decay function goes away, as expected. The damping factoreβt is similar to the damping factor for the Nonfractional case, e−λt/2. Notice that since thepoles, for the residue calculation, have nonzero imaginary and nonzero real parts we will nothave the same three distinct cases as we did for the Nonfractional case (critically damped,over-damped, and under-damped).

4. The Oscillation Frequency

Consider the frequency of the oscillation component of the solution, σ = Im(s6). One questionwe might ask is: how does the frequency change as we change the order of the fractionaldamping? When ν is set to zero we have an un-damped oscillator with frequency

σ =√λ +ω2. (4.1)

When ν is set to one we have the three cases given in Section 2: over-damped, critically-damped, and under-damped. So, the frequency may be zero or nonzero, that is,

σ = 0 λ ≥ 2ω,

σ =

√4ω2 − λ2

2λ < 2ω.

(4.2)

For 0 < ν < 1 there will always be a nonzero frequency. Note that 0 ≤√

4ω2 − λ2/2 <√λ +ω2.


In the Nonfractional case increasing λ causes the frequency of oscillation to becomesmaller, monotonically, until the critical cases are reached and the oscillation period becomesinfinite (these are the critical and over-damped cases). In the fractional case the frequency ofoscillation, σ = Im(s6), now depends on the order of the derivative, ν, as well as λ and ω. Totry to determine how σ depends on these three parameters consider s to be a function of ν,on 0 ≤ ν ≤ 1, implicitly defined by

s2 + λsν +ω2 = 0, (4.3)

for fixed values of λ andω (both being positive). Let us restrict our attention to the upper halfplane for s. As such s will be one to one on 0 ≤ ν < 1. Due to the fractional exponent causinga branch cut on the negative real axis s will not be one to one at ν = 1.

Now the question arises, does σ fall monotonically with respect to ν? To get the answerto this consider the derivative of (4.3) with respect to ν and isolate ds/dν (remember, λ andω are being held fixed)

ds

dν= − λs

ν ln(s)s2s2 + λsνν

. (4.4)

The imaginary part of this equation is

dσ

dν= Im

(ds

dν

). (4.5)

Specifically, consider this equation at ν = 0

dσ

dν

∣∣∣∣ν=0

= Im

((s2 +ω2) ln(s)s

2s2 − ν(s2 +ω2)

)∣∣∣∣∣ν=0

=λ(ln(λ +ω2))

4√λ +ω2

. (4.6)

This gives three initial slopes for the rate of change of σ with respect to ν.

λ +ω2 > 1⇒ The frequency initially increases with increasing damping order.

λ+ω2 = 1⇒The frequency initially is not changing with increasing damping order.

λ +ω2 < 1⇒ The frequency initially decreases with increasing damping order.

This is not entirely what might have been expected. In the first case the oscillationfrequency actually increases before falling. Hence there will be some values of ν for whichthe fractional damping will actually cause the oscillations to go faster than the un-dampedoscillator (the damping will still cause the amplitude to decrease). Each of the above threecases can become any of the three Nonfractionally damped cases by letting ν → 1 ((2.7),(2.8), and (2.9)). Hence, there are nine cases for the linear fractionally damped oscillator.

There are some graphs of solutions to the imaginary part of (4.3) (the oscillationfrequency) for various values of ν, λ, and ω. In all three graphs the oscillation frequency is onthe vertical axis and the order of the derivative is on the horizontal axis. The three graphs foreach case correspond to what would be under-damped, critically damped, and over-dampedfor a damped oscillator with whole order derivatives.


0 0.2 0.4 0.6 0.8 1

Derivative order

0.9

1

1.1

1.2

1.3

1.4

Osc

illat

ion

freq

uenc

y

Figure 3

0 0.2 0.4 0.6 0.8 1

Derivative order

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Osc

illat

ion

freq

uenc

y

Figure 4

Figure 3 is a representative graph of case one, λ + ω2 > 1. The green graph is for λ =ω = 1, the black graph is for λ = 2 and ω = 1, and the red graph is for λ = 3 and ω = 1.

Figure 4 is a representative graph from case two, λ +ω2 = 1, a flat start. The red graphis for λ = 2(

√2− 1) and ω = λ/2, the green graph is for λ = 1/2 and ω = 1/

√2, and the black

graph is for λ = 15/16 and ω = 1/4.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Derivative order

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Osc

illat

ion

freq

uenc

y

Figure 5

Figure 5 is a representative graph for case three, a decreasing start. The green graph isfor λ = 1/2 and ω = 1/8, the black graph is for λ = 1/2 and ω = 1/4, and the red graph is forλ = ω = 1/2.

5. Conclusion

In this paper the linear fractionally damped oscillator equation was solved analytically. It wasfound that the solution is very similar to the Nonfractional case (decayed oscillations but withthe inclusion of an additional decay function). It was found that there are nine distinct cases,as opposed to the usual three for the ordinary damped oscillator. An unexpected result wasthat for three of the cases the oscillation frequency actually increases with increasing order ofderivative of the damping term (till a peak value is reached, then the frequency decreases asexpected). The physical reason for this increase in oscillation frequency is not yet clear.

References

[1] I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999.[2] S. Burov and E. Barkai, “Fractional langevin equation: over-damped, under-damped, and critical

behaviors,” http://arxiv.org/abs/0802.3777.[3] S. Burov and E. Barkai, “The critical exponent of the fractional langevin equation is αc ≈ 0.402,”

http://arxiv.org/abs/0712.3407.[4] G. M. Zaslavsky, A. A. Stanislavsky, and M. Edelman, “Chaotic and pseudochaotic attractors of

perturbed fractional oscillator,” Chaos, vol. 16, no. 1, Article ID 013102.[5] A. C. Galucio, J. FranCois, and F. Dubois, “On the use of fractional derivative operators to

describe viscoelastic damping in structural dynamics- FE formulation of sandwich beams and


approximation of fractional derivatives by using the scheme,” Derivation fractionaire enme-canique—etat-de-l’art et applications, CNAM Paris, France, November 2006, http://www.cnam.fr/lmssc/seminaires/derivfrac/galucio/.

[6] B. N. Narahari Achar, J. W. Hanneken, and T. Clarke, “Damping characteristics of a fractionaloscillator,” Physica A, vol. 339, no. 3-4, pp. 311–319, 2004.

[7] R. L. Bagley and P. J. Torvik, “On the appearance of the fractional derivative in the behavior of realmaterials,” Journal of Applied Mechanics, vol. 51, pp. 294–298, 1984.

[8] S. Saha Ray and R. K. Bera, “Analytical solution of the Bagley Torvik equation by Adomiandecomposition method,” Applied Mathematics and Computation, vol. 168, no. 1, pp. 398–410, 2005.


Research ArticlePositive Solution to Nonzero BoundaryValues Problem for a Coupled System of NonlinearFractional Differential Equations

Jinhua Wang, Hongjun Xiang, and Zhigang Liu

Department of Mathematics, Xiangnan University, Chenzhou 423000, China

Correspondence should be addressed to Hongjun Xiang, [email protected]

Received 13 April 2009; Accepted 9 June 2009

Academic Editor: Shaher Momani

Copyright q 2010 Jinhua Wang et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We consider the existence and uniqueness of positive solution to nonzero boundary valuesproblem for a coupled system of fractional differential equations. The differential operator is takenin the standard Riemann-Liouville sense. By using Banach fixed point theorem and nonlineardifferentiation of Leray-Schauder type, the existence and uniqueness of positive solution areobtained. Two examples are given to demonstrate the feasibility of the obtained results.

1. Introduction

Fractional differential equation can describe many phenomena in various fields of scienceand engineering such as control, porous media, electrochemistry, viscoelasticity, andelectromagnetic. There are many papers dealing with the existence and uniqueness ofsolution for nonlinear fractional differential equation; see, for example, [1–5]. In [1], theauthors investigated a singular coupled system with initial value problems of fractionalorder. In [2], Su discussed a boundary value problem of coupled system with zero boundaryvalues. By means of Schauder fixed point theorem, the existence of the solution is obtained.The nonzero boundary values problem of nonlinear fractional differential equations is moredifficult and complicated. No contributions exist, as far as we know, concerning the existenceof positive solution for coupled system of nonlinear fractional differential equations withnonzero boundary values.

In this paper, we consider the existence and uniqueness of positive solution tononzero boundary values problem for a coupled system of nonlinear fractional differentialequations:


Dαu(t) + f(t, v(t)) = 0, 0 < t < 1,

Dβv(t) + g(t, u(t)) = 0, 0 < t < 1,

u(0) = 0, u(1) = au(ξ),

v(0) = 0, v(1) = bv(ξ),

(1.1)

where 1 < α < 2, 1 < β < 2, 0 � a, b � 1, 0 < ξ < 1, f, g : [0, 1] × [0,+∞) → [0,+∞) aregiven functions, and D is the standard Riemann-Liouville differentiation. By using Banachfixed point theorem and nonlinear differentiation of Leray-Schauder type, some sufficientconditions for the existence and uniqueness of positive solution to the above coupledboundary values problem are obtained.

The rest of the paper is organized as follows. In Section 2, we introduce some basicdefinitions and preliminaries used in later. In Section 3, the existence and uniqueness ofpositive solution for the coupled boundary values problem (1.1) will be discussed, andexamples are given to demonstrate the feasibility of the obtained results.

2. Basic Definitions and Preliminaries

In this section, we introduce some basic definitions and lemmas which are used throughoutthis paper.

Definition 2.1 (see [6, 7]). The fractional integral of order α (α > 0) of a function y : (0,∞) →R is given by

Iαy(t) =1

Γ(α)

∫ t

0(t − s)α−1y(s)ds, (2.1)

provided that the right side is pointwise defined on (0,∞).

Definition 2.2 (see [6, 7]). The fractional derivative of order α > 0 of a continuous functiony : (0,∞) → R is given by

Dαy(t) =1

Γ(n − α)

(d

dt

)n∫ t

0(t − s)n−α−1y(s)ds, (2.2)

where n = [α] + 1 provided that the right side is pointwise defined on (0,∞).

Remark 2.3 (see [3]). The following properties are useful for our discussion:

(1) IαDαu(t) = u(t) −∑Nk=1 Ckt

α−k, Dαu(t) ∈ C(0, 1)⋂L(0, 1), Ck ∈ R, N = [α] + 1,

(2) DαIαu(t) = u(t),

(3) Dαtγ = Γ(γ + 1)/Γ(γ + 1 − α)tγ−α, α > 0, γ > −1, γ > α − 1, t > 0.


Lemma 2.4 (the nonlinear alternative of Leray and Schauder type [8]). Let E be a Banach spacewith C ⊆ E closed and convex. Let U be a relatively open subset of C with 0 ∈ U and let T : U → Cbe a continuous and compact mapping. Then either

(a) the mapping T has a fixed point inU, or

(b) there exist u ∈ ∂U and λ ∈ (0, 1) with u = λTu.

Consider

Dαu(t) + y(t) = 0, 0 < t < 1,

u(0) = 0, u(1) = au(ξ),(2.3)

then one has the following lemma.

Lemma 2.5. Let y ∈ C[0, 1] and 1 < α < 2, then u(t) is a solution of BVP (2.3) if and only if u(t) isa solution of the integral equation:

u(t) =∫1

0G1(t, s)y(s)ds, (2.4)

where

G1(t, s) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

[t(1 − s)]α−1 − atα−1(ξ − s)α−1 − (t − s)α−1(1 − aξα−1)(1 − aξα−1

)Γ(α)

, 0 � s � t � 1, s � ξ,

[t(1 − s)]α−1 − (t − s)α−1(1 − aξα−1)(1 − aξα−1

)Γ(α)

, 0 < ξ � s � t � 1,

[t(1 − s)]α−1 − atα−1(ξ − s)α−1(1 − aξα−1

)Γ(α)

, 0 � t � s � ξ � 1,

[t(1 − s)]α−1(1 − aξα−1

)Γ(α)

, 0 � t � s � 1, ξ � s.

(2.5)

Proof. Assume that u(t) is a solution of BVP (2.3), then by Remark 2.3, we have

u(t) = −Iαy(t) + C1tα−1 + C2t

α−2

= −∫ t

0

(t − s)α−1

Γ(α)y(s)ds + C1t

α−1 + C2tα−2.

(2.6)

By (2.3), we have

C2 = 0, C1 =∫1

0

(1 − s)α−1

Γ(α)(1 − aξα−1

)y(s)ds − a

∫ ξ

0

(ξ − s)α−1

Γ(α)(1 − aξα−1

)y(s)ds. (2.7)


Therefore, we obtain

u(t) = −∫ t

0

(t − s)α−1

Γ(α)y(s)ds +

∫1

0

tα−1(1 − s)α−1

Γ(α)(1 − aξα−1

)y(s)ds − a

∫ ξ

0

tα−1(ξ − s)α−1

Γ(α)(1 − aξα−1

)y(s)ds

=∫1

0G1(t, s)y(s)ds.

(2.8)

Conversely, if u(t) is a solution of integral equation (2.4), using the relation Dαtα−m =0, m = 1, 2, . . . ,N, where N is the smallest integer greater than or equal to α [3, Remark 2.1],we have

Dαu(t) = −Dα

(∫ t

0

(t − s)α−1

Γ(α)y(s)ds

)

+Dαtα−1

[∫1

0

(1 − s)α−1

Γ(α)(1 − aξα−1

)y(s)ds − a

∫ ξ

0

(ξ − s)α−1

Γ(α)(1 − aξα−1

)y(s)ds]

= −DαIαy(t) = −y(t).

(2.9)

A simple computation showed u(0) = 0, u(1) = au(ξ). The proof is complete.

Let

G2(t, s) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

[t(1 − s)]β−1 − btβ−1(ξ − s)β−1 − (t − s)β−1(1 − bξβ−1)(1 − bξβ−1

)Γ(β) , 0 � s � t � 1, s � ξ,

[t(1 − s)]β−1 − (t − s)β−1(1 − bξβ−1)(1 − bξβ−1

)Γ(β) , 0 < ξ � s � t � 1,

[t(1 − s)]β−1 − btβ−1(ξ − s)β−1(1 − bξβ−1

)Γ(β) , 0 � t � s � ξ � 1,

[t(1 − s)]β−1(1 − bξβ−1

)Γ(β) , 0 � t � s � 1, ξ � s,

(2.10)

we call G(t, s) = (G1(t, s), G2(t, s)) Green’s function of the boundary value problem (1.1).

Lemma 2.6. Let 0 � a, b � 1, then the function G(t, s) is continuous and satisfies

(1) G(t, s) > 0, for t, s ∈ (0, 1),

(2) G(t, s) � G(s, s), for t, s ∈ (0, 1).


Proof. It is easy to prove that G(t, s) is continuous on [0, 1] × [0, 1], here we omit it. Now weprove G1(t, s) > 0. Let

g1(t, s) =[t(1 − s)]α−1 − atα−1(ξ − s)α−1 − (t − s)α−1(1 − aξα−1)(

1 − aξα−1)Γ(α)

, 0 < s � t � 1, s � ξ,

g2(t, s) =[t(1 − s)]α−1 − (t − s)α−1(1 − aξα−1)(

1 − aξα−1)Γ(α)

, 0 < ξ � s � t � 1,

g3(t, s) =[t(1 − s)]α−1 − atα−1(ξ − s)α−1(

1 − aξα−1)Γ(α)

, 0 < t � s � ξ � 1,

g4(t, s) =[t(1 − s)]α−1(

1 − aξα−1)Γ(α)

, 0 < t � s � 1, ξ � s.

(2.11)

We only need to prove g1(t, s) > 0, 0 < s � t � 1, s � ξ. Since

[t(1 − s)]α−1 − atα−1(ξ − s)α−1 − (t − s)α−1(

1 − aξα−1)

= tα−1[(1 − s)]α−1 − a(ξ − s)α−1 −(

1 − s

t

)α−1(1 − aξα−1

),

(2.12)

set g(t) = (1 − s)α−1 − a(ξ − s)α−1 − (1 − s/t)α−1(1 − aξα−1), we have

g ′(t) = −(α − 1)(

1 − s

t

)α−2 s

t2

(1 − aξα−1

)� 0, for 0 < s < t � 1, s � ξ. (2.13)

Then g(t) is decreasing on (0, 1). Meanwhile,

g(1) = (1 − s)α−1 − a(ξ − s)α−1 − (1 − s)α−1(

1 − aξα−1)

= aξα−1

[(1 − s)α−1 −

(1 − s

ξ

)α−1]> 0, 0 < s < t � 1, s � ξ.

(2.14)

Therefore, g1(t, s) > 0, for 0 < s < t � 1, s � ξ. Clearly g1(t, s) > 0, t = s, so g1(t, s) >0, s, t ∈ (0, 1). It is easy to show that g2(t, s) > 0, g3(t, s) > 0, g4(t, s) > 0. Hence, G1(t, s) >0, s, t ∈ (0, 1).

Similarly, G2(t, s) > 0, s, t ∈ (0, 1). The proof of (1) is completed.Let

g2(t) =[t(1 − s)]α−1 − (t − s)α−1(1 − aξα−1)(

1 − aξα−1)Γ(α)

, 0 < ξ � s � t � 1, (2.15)


then,

g ′2(t) =

(α − 1)tα−2[(1 − s)]α−1 − (1 − s/t)α−2(1 − aξα−1)(1 − aξα−1

)Γ(α)

, 0 < ξ � s < t � 1,

[(1 − s)]α−1 − (1 − s/t)α−2(

1 − aξα−1)

� [(1 − s)]α−1 − (1 − s)α−2(

1 − aξα−1)

= [(1 − s)]α−2(aξα−1 − s

)� 0, 0 < ξ � s < t � 1,

(2.16)

therefore,

g ′2(t) � 0, 0 < ξ � s < t � 1. (2.17)

So, g2(t, s) is decreasing with respect to t. Similarly, g1(t, s) is decreasing with respect to t.Also g3(t, s) and g4(t, s) are increasing with respect to t. We obtain that G1(t, s) is decreasingwith respect to t for s � t and increasing with respect to t for t � s.

With the use of the monotonicity of G1(t, s), we have

max0�t�1

G1(t, s) = G1(s, s) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

[s(1 − s)]α−1 − a[s(ξ − s)]α−1

Γ(α)(1 − aξα−1

) , s ∈ (0, ξ],

[s(1 − s)]α−1

Γ(α)(1 − aξα−1

) , s ∈ [ξ, 1).

(2.18)

Similarly,

max0�t�1

G2(t, s) = G2(s, s) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

[s(1 − s)]β−1 − b[s(ξ − s)]β−1

Γ(β)(

1 − bξβ−1) , s ∈ (0, ξ],

[s(1 − s)]β−1

Γ(β)(

1 − bξβ−1) , s ∈ [ξ, 1).

(2.19)

The proof of (2) is completed.

3. Main Result

In this section, we will discuss the existence and uniqueness of positive solution for boundaryvalue problem (1.1).

We define the space X = {u(t) | u(t) ∈ C[0, 1]} endowed with ‖u‖X = max0�t�1|u(t)|,Y = {v(t) | v(t) ∈ C[0, 1]} endowed with ‖u‖Y = max0�t�1|v(t)|.

For (u, v) ∈ X × Y , let ‖(u, v)‖X×Y = max{‖u‖X, ‖v‖Y}.Define P = {(u, v) ∈ X × Y | u(t) � 0, v(t) � 0} , then the cone P ⊂ X × Y .


From Lemma 2.5 in Section 2, we can obtain the following lemma.

Lemma 3.1. Suppose that f(t, v) and g(t, u) are continuous, then (u, v) ∈ X × Y is a solution ofBVP (1.1) if and only if (u, v) ∈ X × Y is a solution of the integral equations

u(t) =∫1

0G1(t, s)f(s, v(s))ds,

v(t) =∫1

0G2(t, s)g(s, u(s))ds.

(3.1)

Let T : X × Y → X × Y be the operator defined as

T(u, v)(t) =

(∫1

0G1(t, s)f(s, v(s))ds,

∫1

0G2(t, s)g(s, u(s))ds

)

=: (T1v(t), T2u(t)),

(3.2)

then by Lemma 3.1, the fixed point of operator T coincides with the solution of system (1.1).

Lemma 3.2. Let f(t, v) and g(t, u) be continuous on [0, 1] × [0,∞) → [0,∞), then T : P → Pdefined by (3.2) is completely continuous.

Proof. Let (u, v) ∈ P , in view of nonnegativeness and continuity of functions G(t, s), f , and g,we conclude that T : P → P is continuous.

Let Ω ∈ P be bounded, that is, there exists a positive constant h > 0 such that ‖(u, v)‖ �h for all (u, v) ∈ Ω.

Let

M = max{∣∣f(t, v(t))∣∣ + 1 : 0 � t � 1, 0 � v � h

},

N = max{∣∣g(t, u(t))∣∣ + 1 : 0 � t � 1, 0 � u � h

},

(3.3)

then we have

|T1v(t)| =∣∣∣∣∣∫1

0G1(t, s)f(s, v(s))ds

∣∣∣∣∣ � M

∫1

0G1(s, s)ds,

|T2u(t)| =∣∣∣∣∣∫1

0G2(t, s)g(s, u(s))ds

∣∣∣∣∣ � N

∫1

0G2(s, s)ds.

(3.4)

Hence, ‖T(u, v)‖ � max{M∫10G1(s, s)ds,N

∫10G2(s, s)ds}. T(Ω) is uniformly bounded.

Since G1(t, s) is continuous on [0, 1]× [0, 1], it is uniformly continuous on [0, 1]× [0, 1].Thus, for fixed s ∈ [0, 1] and for any ε > 0, there exists a constant δ > 0, such that anyt1, t2 ∈ [0, 1] and |t1 − t2| < δ,

|G1(t1, s) −G1(t2, s)| < ε/M. (3.5)


Then

|T1(v)(t2) − T1(v)(t1)| � M

∫1

0|G1(t2, s) −G1(t1, s)|ds < ε. (3.6)

Similarly,

|T2(u)(t2) − T2(u)(t1)| � N

∫1

0|G2(t2, s) −G2(t1, s)|ds < ε. (3.7)

For the Euclidean distance d on R2, we have that if t1, t2 ∈ [0, 1] are such that |t2 − t1| < δ, then

d(T(u, v)(t2), T(u, v)(t1)) =√(T1v(t2) − T1v(t1))2 + (T2u(t2) − T2u(t1))2 <

√2ε. (3.8)

That is to say, T(P) is equicontinuous. By the means of the Arzela-Ascoli theorem, we haveT : P → P is completely continuous. The proof is completed.

Theorem 3.3. Assume that f(t, v) and g(t, u) are continuous on [0, 1]×[0,∞) → [0,∞), and thereexist two positive functions m(t), n(t) that satisfy

(H1) |f(t, v2) − f(t, v1)| � m(t)|v2 − v1|, for t ∈ [0, 1], v1, v2 ∈ [0,∞),

(H2) |g(t, u2) − g(t, u1)| � n(t)|u2 − u1|, for t ∈ [0, 1], u1, u2 ∈ [0,∞).

Then system (1.1) has a unique positive solution if

ρ =∫1

0G1(s, s)m(s)ds < 1, θ =

∫1

0G2(s, s)n(s)ds < 1. (3.9)

Proof. For all (u, v) ∈ P , by the nonegativeness of G(t, s) and f(t, v), g(t, u), we haveT(u, v)(t) � 0. Hence, T(P) ⊆ P.

‖T1v2 − T1v1‖ = maxt∈[0,1]

|T1v2 − T1v1|

= maxt∈[0,1]

∣∣∣∣∣∫1

0G1(t, s)

[f(s, v2(s)) − f(s, v1(s))

]ds

∣∣∣∣∣

�∫1

0G1(s, s)m(s)ds‖v2 − v1‖

� ρ‖v2 − v1‖.

(3.10)

Similarly,

‖T2u2 − T2u1‖ � θ‖u2 − u1‖. (3.11)


We have,

‖T(u2, v2) − T(u1, v1)‖ � max(ρ, θ

)‖(u2, v2) − (u1, v1)‖. (3.12)

From Lemma 3.2, T is completely continuous, by Banach fixed point theorem, the operatorT has a unique fixed point in P , which is the unique positive solution of system (1.1). Thiscompletes the proof.

Theorem 3.4. Assume that f(t, v) and g(t, u) are continuous on [0, 1] × [0,∞) → [0,∞) andsatisfy

(H3) |f(t, v(t))| � a1(t) + a2(t)|v(t)|,

(H4) |g(t, u(t))| � b1(t) + b2(t)|u(t)|,

(H5) A1 =∫1

0G1(s, s)a2(s)ds < 1, 0 < B1 =∫1

0G1(s, s)a1(s)ds < ∞,

(H6) A2 =∫1

0G2(s, s)b2(s)ds < 1, 0 < B2 =∫1

0G2(s, s)b1(s)ds < ∞.

Then the system (1.1) has at least one positive solution (u, v) in

C ={(u, v) ∈ P | ‖(u, v)‖ < min

(B1

1 −A1,

B2

1 −A2

)}. (3.13)

Proof. Let C = {(u, v) ∈ X × Y : ‖(u, v)‖ < r} with r = min(B1/(1 − A1), B2/(1 − A2)), definethe operator T : C → P as (3.2).

Let (u, v) ∈ C, that is, ‖(u, v)‖ < r. Then

‖T1v‖ = maxt∈[0,1]

∣∣∣∣∣∫1


∣∣∣∣∣

�∫1

0G1(s, s)(a1(s) + a2(s)|v(s)|)ds

≤∫1

0G1(s, s)a1(s)ds +

∫1

0G1(s, s)a2(s)ds‖v‖

= B1 +A1‖v‖ � r.

(3.14)

Similarly, ‖T2u‖ � r, so ‖T(u, v)‖ � r, T(u, v) ⊆ C. From Lemma 3.2 T : C → C iscompletely continuous.

Consider the eigenvalue problem

(u, v) = λT(u, v), λ ∈ (0, 1). (3.15)


Under the assumption that (u, v) is a solution of (3.15) for a λ ∈ (0, 1), one obtains

‖u‖ = ‖λT1v‖

= λmaxt∈[0,1]

∣∣∣∣∣∫1


∣∣∣∣∣

<

∫1

0G1(s, s)(a1(s) + a2(s)|v(s)|)ds

=∫1

0G1(s, s)a1(s)ds +

∫1

0G1(s, s)a2(s)ds‖v‖

= B1 +A1‖v‖ � r.

(3.16)

Similarly, ‖v‖ = ‖λT2u‖ < r, so ‖(u, v)‖ < r, which shows that (u, v)/∈ ∂C. By Lemma 2.4, Thas a fixed point in C. We complete the proof of Theorem 3.4.

Example 3.5. Consider the problem

D7/4u(t) + f(t, v(t)) = 0, 0 < t < 1,

D3/2v(t) + g(t, u(t)) = 0, 0 < t < 1,

u(0) = 0, u(1) =12u

(12

),

v(0) = 0, v(1) =34v

(12

),

(3.17)

where

f(t, v(t)) =tv(t)

(1 + t)(1 + v(t)), g(t, u(t)) = arctan

t

1 + t|sinu(t)|. (3.18)

Set v1(t), v2(t), u1(t), u2(t) ∈ [0,∞) and t ∈ [0, 1], then we have

∣∣f(t, v2(t)) − f(t, v1(t))∣∣ � t

1 + t|v2(t) − v1(t)|,

∣∣g(t, u2(t)) − g(t, u1(t))∣∣ � arctan

t

1 + t|u2(t) − u1(t)|.

(3.19)


Therefore,

ρ=∫1

0G1(s, s)m(s)ds �

∫1

0G1(s, s)ds

=1

Γ(7/4)(

1 − (1/2)7/4){∫1/2

0[s(1 − s)]3/4ds−

∫1/2

0

12

[s

(12−s

)]3/4

ds+∫1

1/2[s(1−s)]3/4ds

}

=2(

1 + (1/2)7/4)

5· Γ(3/4)Γ(1/2)

<45< 1,

θ=∫1

0G2(s, s)n(s)ds � π

4

∫1

0G2(s, s)ds

=π

41

Γ(3/2)(

1−(3/4)(1/2)1/2){∫1/2

0[s(1−s)]1/2ds−

∫1/2

0

34

[s

(12−s

)]1/2

ds+∫1

1/2[s(1−s)]1/2ds

}

=π

4

[1 − (3/4)(1/2)2

][1 − (3/4)(1/2)1/2

] · Γ(3/2)Γ(3)

≈0.6018 < 1.(3.20)

With the use of Theorem 3.3, BVP (3.17) has a unique positive solution.

Example 3.6. Consider the problem

D7/4u(t) + f(t, v(t)) = 0, 0 < t < 1,

D3/2v(t) + g(t, u(t)) = 0, 0 < t < 1,

u(0) = 0, u(1) =12u

(12

),

v(0) = 0, v(1) =34v

(12

),

(3.21)

where

f(t, v(t)) = t2 +t

1 + tln(1 + v(t)), g(t, u(t)) = 10 +

t2

20+ u(t). (3.22)

We have

∣∣f(t, v(t))∣∣t2 + t

1 + t· |v(t)|, ∣∣g(t, u(t))∣∣ �

(10 +

t2

20

)+ |u(t)|. (3.23)


Hence,

A1 =∫1

0G1(s, s)a2(s)ds �

∫1

0G1(s, s)ds =

2(

1 + (1/2)7/4)

5· Γ(3/4)Γ(1/2)

< 1,

B1 =∫1

0G1(s, s)a1(s)ds =

∫1

0G1(s, s) · s2ds < ∞,

A2 =∫1

0G2(s, s)b2(s)ds =

∫1

0G2(s, s)ds ≈ 0.7666 < 1,

B2 =∫1

0G2(s, s)b1(s)ds =

∫1

0G2(s, s)

(10 +

s2

20

)ds < ∞.

(3.24)

By Theorem 3.4, BVP (3.21) has at least one positive solution in

C ={(u, v) ∈ P | ‖u, v‖ < min

(B1

1 −A1,

B2

1 −A2

)}. (3.25)

Acknowledgments

This work was jointly supported by the Natural Science Foundation of Hunan ProvincialEducation Department under Grants 07A066 and 07C700, the Construct Program of the KeyDiscipline in Hunan Province, Aid Program for Science and Technology Innovative ResearchTeam in Higher Educational Institutions of Hunan Province, and the Foundation of XiangnanUniversity.

References

[1] C. Bai and J. Fang, “The existence of a positive solution for a singular coupled system of nonlinearfractional differential equations,” Applied Mathematics and Computation, vol. 150, no. 3, pp. 611–621,2004.

[2] X. Su, “Boundary value problem for a coupled system of nonlinear fractional differential equations,”Applied Mathematics Letters, vol. 22, no. 1, pp. 64–69, 2009.

[3] Z. Bai and H. Lu, “Positive solutions for boundary value problem of nonlinear fractional differentialequation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 495–505, 2005.

[4] S. Zhang, “Existence of solution for a boundary value problem of fractional order,” Acta MathematicaScientia, vol. 26, no. 2, pp. 220–228, 2006.

[5] S. Q. Zhang, “Positive of solution for boundary-value problems of nonlinear fractional differentialequations,” Electronic Journal of Differential Equations, vol. 36, pp. 1–12, 2006.


[7] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations,vol. 204 of North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006.

[8] E. Zeidler, Nonlinear Functional Analysis and Its Applications—I: Fixed-Point Theorems, Springer, NewYork, NY, USA, 1986.


Research ArticleThe Periodic Solutions of the Compound SingularFractional Differential System with Delay

XuTing Wei and XuanZhu Lu

School of Mathematics and Computer Science, Fuzhou Universiy, Fuzhou 350108, China

Correspondence should be addressed to XuTing Wei, [email protected]

Received 31 July 2009; Revised 16 November 2009; Accepted 1 December 2009


Copyright q 2010 X. Wei and X. Lu. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

The paper gives sufficient conditions on the existence of periodic solution for a class ofcompound singular fractional differential systems with delay, involving Nishimoto fractionalderivative. Furthermore, for the particular functions, the necessary conditions on the existence ofperiodic solution are also derived. Especially, for two-dimensional compound singular fractionaldifferential equation with delay, the criteria of existence of periodic solution are obtained. Finally,two examples are presented to verify the validity of criteria.

1. Introduction

In real life, there are many phenomena with time delay. The mathematical model derivedfrom engineering, physics, mechanics, control theory, chemical reactions, biology, andmedicine was made with a significant amount of delay, such as the limited signaltransmission speed human reaction time to the outside world. Therefore, the delay iswidespread in nature and society, in the introduction of time-delay differential equationsthere can be a more accurate description and explanation of various phenomena andprocesses.

Fractional calculus is the promotion of classical calculus. The study found thatfractional calculus was very suitable to describe long memory and hereditary propertiesof various materials and processes [1, 2]. In the recent years, fractional calculus becomes aresearch hotspot, its field of concern has become wide, such as the numerical method of theequation in [3], the existence and uniqueness of equations in [4], fractional Brownian motion,fractional reaction-diffusion equation and random walk [5, 6], fractional wavelet transform[7], and fractional control [8].

Most of the above-mentioned studies, utilize the Riemann-liouville fractional deriva-tive definition, which due to its nature of its definition is simple and relatively good. ButNishimoto definition of fractional calculus [9, 10], has not received a lot of attention, this may


be part of the naturalization due to the complexity of its definition, but compared to Riemann-liouville fractional calculus, it has a better nature, relevant results more concise useful.

The existence of periodic solutions of differential equations is one of the importantresearch directions of biomathematics [11–15], which has a wide range of applications, suchas the existence of periodic orbits of celestial movement and its stability.

In [12], the author discussed the following system:

Ex(t) = Ax(t) + Bx(t − τ1) + Cx(t − τ2), (1.1)

and obtained sufficient and necessary conditions for the existence of periodic solutions for thesystem. Taking into account the periodic solutions of the fractional time-delay system will bea very important practical significance; we are tried to generalize the corresponding resultsto the case of fractional order.

For the above reasons we consider the following compound singular fractionaldifferential system with delay:

HDαx(t) = Ax(t) + Bx(t − τ1) + Cx(t + τ2), (1.2)

where Dα denotes Nishimoto fractional derivative of order α, α > 0. H,A,B, and C areconstant system matrices of appropriate dimensions, and τ1 and τ2 are constants with τ1 >0, τ2 > 0, |H| = 0.

2. Definitions and Notations

In this section we introduce the definitions of fractional derivative/integral and related basicproperties used in the paper; more information can be obtained from [9, 10].

Definition 2.1 (see [9]). If the function f(z) is analytic (regular) inside and on C, here C :={C−, C+}, C− is a contour along the cut joining the points z and −∞ + iI(z), which starts fromthe point at −∞, encircles the point z once counter-clockwise, and returns to the point at −∞,C+ is a contour along the cut joining the points z and +∞ + iI(z), which starts from the pointat +∞, encircles the point z once counter-clockwise, and returns to the point at +∞,

fυ(z) =(f(z)

)υ :=

Γ(υ + 1)2πi

∫

c

f(ς)

(ς − z)υ+1dς

(υ ∈ R

Z− ;Z− := {−1,−2,−3, . . .}),

f−n(z) := limυ→−n

{fυ(z)

}(n ∈ N := {1, 2, 3, . . .}),

(2.1)

where ς, −π ≤ arg(ς − z) ≤ π , for C−, and 0 ≤ arg(ς − z) ≤ 2π , for C+.Then fυ(z) (υ > 0) is said to be the fractional derivative of f(z) of order υ and

fυ(z) (υ < 0) is said to be the fractional integral of f(z) of order −υ, provided that |fυ(z)| <∞ (υ ∈ R).

Let us recall the following useful properties associated with the definition introducedabove [9].


Property 1. For a constant λ,

(eλz)

υ= λυeλz (λ/= 0;υ ∈ R; z ∈ C). (2.2)


(e−λz)

υ= e−iπυλυe−λz (λ/= 0;υ ∈ R; z ∈ C). (2.3)

Property 3. If the function f(z) is singlevalued and analytic in some domain Ω ⊆ C, then

(fμ(z)

)υ= fμ+υ(z) =

(fυ(z)

)μ. (2.4)


(zλ)

υ= e−iπυ

Γ(υ − λ)Γ(−λ) zλ−υ

(υ ∈ R; z ∈ C;

∣∣∣∣Γ(υ − λ)Γ(−λ)

∣∣∣∣ < ∞). (2.5)

In the following section of this paper, we let Dα denote the α order Nishimoto derivative.

3. The Main Results

In this section, we discuss some problems to the system of the system (1.2).

Theorem 3.1. The sufficient condition for the existence of the nonconstant periodic solutions ofsystem (1.2) is that the following equation exists pure imaginary roots

det(λαH −A − Be−τ1λ − Ceτ2λ

)= 0. (3.1)

Proof. Assume that ηi is pure imaginary root of (3.1), let x(t) = Keηit(K ∈ Rn), substitutingx(t) in (1.2), then

((ηi)αH −A − Be−τ1(ηi) − Ceτ2(ηi)

)K = 0. (3.2)

As ηi is pure imaginary roots of (3.1), note that

det((

ηi)αH −A − Be−τ1(ηi) − Ceτ2(ηi)

)= 0. (3.3)

So (3.1) exists nonzero solution K, then, x(t) = Keηit is the nonconstant periodic solution of(1.2).

If the system have nonconstant periodic solutions, then we may wonder whetherthe solution satisfy (3.1), in fact, as you will see it holds when the function satisfy some


conditions. We know that if the function f(t) is a continuous smooth periodic function withperiod 2l, then it can be expressed as its fourier series form

f(t) =+∞∑

k=−∞Cke

ikπt/l, (3.4)

where Ck = (1/2l)∫ l−lf(ξ)e

−ikπt/ldξ.And as we also know that its fourier series expansion has the following properties:

f ′(t) =+∞∑

k=−∞Ck

ikπ

leikπt/l, (3.5)

we can even get the following relation if f(t) satisfy some more strictly condition:

f (k)(t) =+∞∑

k=−∞Ck

(ikπ

l

)k

eikπt/l. (3.6)

To obtain the similar property of our fractional derivative, what conditions the functionshould satisfy? We give the following function space.

Definition 3.2. If the periodic function f(t) is continuous and smooth on R, its α (α > 1) orderNishimoto derivative exists, then we let Ω(t) denote the corresponding function space whoseelements have the following property:

Dαf(t) = Dα

(+∞∑

k=−∞Cke

ikπt/l

)

=+∞∑

k=−∞CkD

α(eikπt/l

)=

+∞∑

k=−∞Ck

(ikπt

l

)α

eikπt/l, (3.7)

it is easy to know from the definition that eiλt ∈ Ω(t) (λ ∈ R), and so Ω(t) is nonempty.

Theorem 3.3. If x(t) is the non-constant periodic solution of (1.2), and further x(t) ∈ Ω(t), one canobtain the necessity of Theorem 3.1.

Proof. Suppose the period of x(t) is 2l, x(t) is continuous and differentiable because of α > 1,then we can denote it in the form of its fourier series:

x(t) =+∞∑

k=−∞Cke

ikπt/l. (3.8)

Since x(t) ∈ Ω(t), we have

Dαx(t) =+∞∑

k=−∞Ck

(ikπt

l

)α

eikπt/l, (3.9)

where Ck = (1/2l)∫ l−lx(ξ)e

−ikπt/ldξ.


We put (3.8) and (3.9) into (1.2), and obtain

∞∑

k=−∞

[(ikπ

l

)α

H −A − Be−ikπτ1/l − Ceikπτ2/l

]Cke

ikπt/l = 0, (3.10)

then multiply (1/2l)e−imπt/l (m = 0,±1,±2, . . .) on both sides of (3.10) and integrate it from −lto l, hence

∞∑

k=−∞

[(ikπ

l

)α


]Ck

12l

∫ l

−lei(k−m)πt/ldt = 0. (3.11)

It is easy to deduce that

12l

∫ l

−lei(k−m)πt/ldt =

⎧⎨

⎩

1, k = m,

0, k /=m,(3.12)

recalling (3.10), it reduces to

[(ikπ

l

)α


]Cm = 0 (m = 0,±1,±2, . . .). (3.13)

Thus, if there are no pure imaginary roots in (3.1), then for every k we have Ck = 0, accordingto (3.8), we conclude that x(t) = cons tan t vector which conflicts the suppose that x(t) is thenon-constant periodic solution of (1.2).

4. Two-Dimensional Case

For the case of the two-dimensional compound singular fractional differential system withdelay, there is

HDαx(t) = Ax(t) + Bx(t − τ1) + Cx(t + τ2), (4.1)

where α > 0, H =(

1 0

0 0

), A =

(a11 a12

a21 a22

), B =

(b11 b12

b21 b22

), C =

(c11 c12

c21 c22

), x(t) =

(x1(t)

x2(t)

),

and x1(t), x2(t) is scalar function.Using Theorem 3.1, we obtained the following theorem.


Theorem 4.1. If one of the following equations exists non-zero real root, then system(4.1) exists non-constant periodic solution, further more, if x(t) ∈ Ω(t), then the conclusion is sufficient and necessary

|A| + |B| cos(2τ1y

)+ |C| cos

(2τ2y

)+ E cos

(τ1y)+ F cos

(τ2y)+G cos

((τ2 − τ1)y

)

−b22∣∣y∣∣α cos

(−τ1y +

π

2α)− c22

∣∣y∣∣α cos

(τ2y +

π

2α)− a22

∣∣y∣∣α cos

(π2α)= 0

−|B| sin(2τ1y

)+ |C| sin

(2τ2y

) − E sin(τ1y)+ F sin

(τ2y)+G sin

((τ2 − τ1)y

)

−b22∣∣y∣∣α sin

(−τ1y +

π

2α)− c22

∣∣y∣∣α sin

(τ2y +

π

2α)− a22

∣∣y∣∣α sin

(π2α)= 0,

(4.2)

or

|A| + |B| cos(2τ1y

)+ |C| cos

(2τ2y

)+ E cos

(τ1y)+ F cos

(τ2y)+G cos

((τ2 − τ1)y

)

−b22∣∣y∣∣α cos

(−τ1y − π

2α)− c22

∣∣y∣∣α cos

(τ2y − π

2α)− a22

∣∣y∣∣α cos

(−π

2α)= 0

−|B| sin(2τ1y

)+ |C| sin

(2τ2y


(τ2y)+G sin

((τ2 − τ1)y

)

−b22∣∣y∣∣α sin

(−τ1y − π

2α)− c22

∣∣y∣∣α sin

(τ2y − π

2α)− a22

∣∣y∣∣α sin

(−π

2α)= 0,

(4.3)

where E =∣∣∣a11 a12

b21 b22

∣∣∣ +∣∣∣b11 b12

a21 a22

∣∣∣, F =∣∣∣a11 a12

c21 c22

∣∣∣ +∣∣∣c11 c12

a21 a22

∣∣∣, G =∣∣∣b11 b12

c21 c22

∣∣∣ +∣∣∣c11 c12

b21 b22

∣∣∣.

Proof. First of all we know that

(yi)α =

⎧⎨

⎩

yαeπαi/2, y ≥ 0,∣∣y∣∣αe−παi/2, y < 0,

(4.4)

according to Theorem 3.1, we have

h(yi)=∣∣∣yiI −A − Be−τ1yi − Ceτ2yi

∣∣∣

=

∣∣∣∣∣

(yi)α − a11 − b11e

−τ1yi − c11eτ2yi −a12 − b12e

−τ1yi − c12eτ2yi

−a21 − b21e−τ1yi − c21e

τ2yi −a22 − b22e−τ1yi − c22e

τ2yi

∣∣∣∣∣

= |A| + |B|e−τ1yi + |C|e2τ2yi + |E|e−τ1yi + Feτ2yi +Ge(τ2−τ1)yi

− b22(yi)αe−τ1yi − c22

(yi)αeτ2yi − a22

(yi)α.

(4.5)


then

R[h(yi)]

= |A| + |B| cos(2τ1y

)+ |C| cos

(2τ2y

)+ E cos

(τ1y)+ F cos

(τ2y)

+G cos((τ2 − τ1)y

) − b22∣∣y∣∣α cos

(−τ1y ± π

2α)

− c22∣∣y∣∣α cos

(τ2y ± π

2α)− a22

∣∣y∣∣α cos

(±π

2α)= 0,

I[h(yi)]

= −|B| sin(2τ1y

)+ |C| sin

(2τ2y


(τ2y)

+G sin((τ2 − τ1)y

) − b22∣∣y∣∣α sin

(−τ1y ± π

2α)

− c22∣∣y∣∣α sin

(τ2y ± π

2α)− a22

∣∣y∣∣α sin

(±π

2α)= 0,

(4.6)

where R [z] and I [z] denote the real and imaginary parts of z, respectively. Then usingTheorem 3.1, if there exists y ∈ R, y /= 0, h(yi) = 0, we obtain that

R[h(yi)]

= 0,

I[h(yi)]

= 0,(4.7)

hence system (4.1) exists non-constant periodic solution. This proved the theorem.

5. Examples

In this section we give some concrete examples to illustrate our conclusions.

Example 5.1. We consider the following two-dimensional compound singular fractionaldifferential system with delay:

Dαx1(t) = −x2(t) + x1(t − 1) + x2(t − 1) + x2(t + 1),

0 = x1(t) + x1(t − 1) + x1(t + 1),(5.1)

where H =(

1 0

0 0

), A =

(0 −1

1 0

), B =

(1 1

1 0

), C =

(0 1

1 0

), τ1 = 1, τ2 = 1, α > 1. so we have

|A| = 1, |B| = −1, |C| = −1, E = 0, F = 0, G = −2.Using the discriminant of Theorem 4.1, we have

2 cos(2y)= −1,

0 = 0,(5.2)

the solution is y = kπ + π/3 (k = 0,±1,±2, . . .).


According to Theorem 4.1 , system (5.1) has non-constant periodic solution. Wesuppose that y = π/3, as

((yi)αI −A − Be−yi − Ceyi

)K = 0 (5.3)

exists non-zero solution, it means that

⎛

⎜⎝

[(π

3

)α

cos(πα

2

)−√

32

]

+

[(π

3

)α

sin(πα

2

)+√

32

]

i 0

0 0

⎞

⎟⎠

(k1

k2

)

=

(0

0

)

. (5.4)

So we have k1 = 0, k2 for any real number. Supposed that k2 = 1, then we obtained a non-constant periodic solution of system (5.1):

x(t) = e(π/3)it

(0

1

)

. (5.5)

We can verify that x(t) is a non-constant periodic solution of system (5.1).

Example 5.2. Consider the following two-dimension compounded with singular fractionaldifferential equation delay system:

D1/2x1(t) = −x2(t) + x1(t − 1) + x2(t + 1),

0 = x1(t) + x2(t) + x1(t − 1) + x2(t − 1) + x1(t + 1) + x2(t + 1).(5.6)

where H =(

1 0

0 0

), A =

(0 −1

1 1

), B =

(0 1

1 1

), C =

(0 1

1 1

), τ1 = 1, τ2 = 1, α = 1/2.

So we have |A| = 1, |B| = −1, |C| = −1, E = 0, F = 0, G = −2.Using the discriminant of Theorem 4.1, we obtained

(2 cosy + 1

)∣∣y∣∣1/2 cos

π

4+ 2 cos 2y + 1 = 0,

(2 cosy + 1

)∣∣y∣∣1/2 sin

π

4= 2 sin 2y.

(5.7)

Through the simplication of this equation, we have

√2 sin

(2y + (π/4)

)= −1/2, (5.8)

and the solution is y = kπ + (1/2)arc sin(−1/2√

2) − π/8, k = (0,±1,±2, . . .).According to Theorem 4.1, there exists non-constant periodic solution in the system.


References

[1] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering,Academic Press, London, UK, 1999.


[3] Xuanzhu Lu and Fawang Liu, “The explicit and implicit finite difference approximations for the spacefractional advection diffusion equation,” Computational Mechanics, September 2004, Beijing, China.

[4] A. Arikoglu and I. Ozkol, “Solution of fractional differential equations by using differential transformmethod,” Chaos, Solitons and Fractals, vol. 34, no. 5, pp. 1473–1481, 2007.

[5] M. M. Meerschaert, “Stochastic solution of space-time fractional diffusion equations,” Physical ReviewE, vol. 65, no. 4, Article ID 041103, 4 pages, 2002.

[6] F. Liu, V. V. Anh, and I. Turner, “Time fractional advection-dispersion equation,” Journal of AppliedMathematics & Computing, vol. 13, no. 1-2, pp. 233–245, 2003.

[7] M. Unser and T. Blu, “Fractional splines and wavelets,” SIAM Journal, vol. 42, no. 1, pp. 43–67, 2000.[8] P. Varshney, M. Gupta, and G. S. Visweswaran, “New switched capacitor fractional order integrator,”

Journal of Active and Passive Electronic Devices, pp. 187–197, 2007.[9] T.-M. Hsieh, S.-D. Lin, and H. M. Srivastava, “Some relationships between certain families of ordinary

and fractional differential equations,” Computers and Mathematics with Applications, vol. 46, no. 10-11,pp. 1483–1492, 2003.

[10] K. Nishimoto, Fractional Calculus: Integrations and Differentiations of Arbitrary Order, vol. 1, DescartesPress, Koriyama, Japan, 1984.

[11] M. L. Hbid and R. Qesmi, “Periodic solutions for functional differential equations with periodic delayclose to zero,” Electronic Journal of Differential Equations, vol. 141, pp. 1–12, 2006.

[12] Z.-X. Zhang and W. Jiang, “The periodic solution of compound singular differential equation withdelay,” Journal of Anhui University Natural Science Edition, vol. 30, pp. 4–7, 2006.

[13] V. Kevantuo and C. Lizama, “A characterization of periodic solutions for time-fractional differentialequations in UMD spaces and applications,” Mathematische Nachrichten, pp. 25–28, 2008.

[14] J. L. Kaplan and J. A. Yorke, “Ordinary differential equations which yield periodic solutions ofdifferential delay equations,” Journal of Mathematical Analysis and Applications, vol. 48, pp. 317–324,1993.

[15] W. Deng, C. Li, and J. Lu, “Stability analysis of linear fractional differential system with multiple timedelays,” Nonlinear Dynamics, vol. 48, no. 4, pp. 409–416, 2007.


Research ArticleThe Use of Fractional B-SplinesWavelets in Multiterms Fractional OrdinaryDifferential Equations

X. Huang and X. Lu

School of Mathematical and Computer Sciences, Fuzhou University, Fuzhou 350002, China

Correspondence should be addressed to X. Huang, [email protected]

Received 31 July 2009; Revised 2 November 2009; Accepted 4 November 2009


Copyright q 2010 X. Huang and X. Lu. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

We discuss the existence and uniqueness of the solutions of the nonhomogeneous linear differentialequations of arbitrary positive real order by using the fractional B-Splines wavelets and the Mittag-Leffler function. The differential operators are taken in the Riemann-Liouville sense and the initialvalues are zeros. The scheme of solving the fractional differential equations and the explicitexpression of the solution is given in this paper. At last, we show the asymptotic solution of thedifferential equations of fractional order and corresponding truncated error in theory.

1. Introduction

Recently, there have been several schemes devoted to the solution of fractional differentialequations. These schemes can be broadly classified into two classes, numerical and analytical([1]). As we know, with the help of some special functions, such as Mittag-Leffler functionand Green function, Miller and Ross have obtained the explicit representations of solutions ofsome classes of homogeneous linear fractional differential equations (FDEs) in [2]; throughthe use of the technique of Laplace and Fourier Transform, the analytical solutions have beengiven by Podlubny in [3]. The numerical scheme we have encountered can be divided intotwo groups. In the first group, the solution is approximated over the entire domain usingapproximating functions such as polynomials and orthogonal functions. In the second group,the entire domain is divided into several small domains like in a finite element technique, andthe solution is obtained for variables at the node points ([1]). As Edwards et al. declaimed in[4], several forms of fractional differential equations have been proposed in standard models,and there has been significant interest in developing numerical schemes for their solution.Thus, several papers have been presented in dealing with approximate numerical techniques


for FDEs. Among these, the papers of Diethelm, Edwards, Ford, Freed, and Simpson arenoteworthy (see, e.g., [1, 4–6]). One of these schemes which should be mentioned is theuse of a Predictor-Corrector (more precisely, PECE) method in [6]. In order to obtain higherprecision, they have replaced the PECE method by a P(EC)mE method with m ≥ 2 in [7].In particular, the PECE method is an important numerical scheme which has been appliedin many fields; for example, Yang and Liu have applied the PECE method for simulatingfraction order dynamical control system in [8]. In addition, several numerical schemes havealso been proposed by other authors (see [9, 10]). In this paper, we present a new schemewhich contains the features of both two groups by using the fractional B-splines wavelet.In paper [11], Unser and Blu have proved that the fractional B-splines generates a validmultiresolution analyses of L2 for α > −1/2, which means that the orthogonal fractional B-splines could be obtained by applying the standard technique described in [12]. And theyalso obtained that the fractional B-splines deriving the asymptotic development of the L2

have a fractional order of approximation.In fact, as the theories of wavelets analyses improve day by day, the wavelet has

become a powerful mathematical tool which widely used in signal processing, imagecompression and enhancement, pattern recognition, control systems, and other fields in thepast two decades. But almost no papers or books have applied the theories of waveletsto solve the fractional differential equations. And our fundamental purpose of this paperis applying the fractional B-Splines wavelets to prove the existence and uniqueness of thesolution of the nonhomogeneous linear fractional differential equations (also so-called linearmultiterm fractional differential equations) with its initial conditions. Let us begin to discussthe solution of multiterms fractional ordinary differential equations with the following form:

(bnDαn + bn−1Dαn−1 + · · · + b1D

α1)y(t) = f(t). (1.1)

For convenience, we consider the initial values:

[Dαk−1y(t)

]t=0

= 0,[Dαk−2y(t)

]t=0

= 0, . . . ,[Dαk−rky(t)

]t=0 = 0, (1.2)

where Dα are taken in the Riemann-Liouville sense, αn > αn−1 > · · · > α1, bn /= 0, αn ≥ 1,and 0 ≤ rk − 1 ≤ αk < rk, rk ∈ N, αk ∈ R+, bk ∈ R, k = 1, . . . , n. The function f(t) belongsto the space L2(Ω); without loss of generality, in this paper, we consider the interval as Ω =[0, T], T ∈ R+.

The plan of this paper is as follows. In Section 2, we recall the definitions of fractionalderivative and integral and related properties which will be used in this paper, give therepresentation of Mittag-Leffler function and generalized Mittag-Leffler function, and thenintroduce the fractional B-splines and some related properties of wavelets. In Section 3,by applying the technique of the Laplace Transforms, and considering the proprieties ofthe generalized Mittag-Leffler function, we prove the lemma of the differential equationsof arbitrary positive real order, which make sure the solution belongs to the space L2; byvirtue of the solution that can be expressed as the form of wavelet series and the basisfunction that is the orthogonal fractional B-splines wavelet which yields the Riesz basis forthe space L2, we can prove the uniqueness of the coefficients of the representation of thesolution, which gain the uniqueness of the solution of the fractional differential equationsand validate the representation of solution. Thus, we have finished the proof of the theorem.


In Section 4, the asymptotic solution of the differential equations of fractional order α ∈ Q+

and corresponding truncated error will be discussed.The present paper is essentially based on the works of the Unser and Blu in [11] and

Podlubny in [3], to which more general classes of the fractional differential equations weshall refer in the following research. For more related review of fractional B-spline wavelet,see the papers of the Unser and Blu ([11, 13, 14]), which have discussed some more importantwavelet properties, such as Riesz bounds and two-scale relation.

2. Preliminary and Definitions

2.1. Definitions

We may recall the definition of the Left Riemann-Liouville differential operators of arbitraryorder α > 0, which take the form

0Dαt y(t) :=

1Γ(m − α)

dm

dtm

∫ t0

y(τ)

(t − τ)α−m+1dτ, (2.1)

where m is the integer defined by m − 1 ≤ α < m (see [3]), and Γ(·) is gamma function.Similarly, the left Riemann-Liouville integral operators of order α > 0 is defined as

0D−αt y(t) :=

1Γ(α)

∫ t0(t − τ)α−1y(τ)dτ. (2.2)

And then we should give the following expression for the Laplace transform of the Riemann-Liouville differential operators of the order α > 0, which is

L{

0Dαt y(t); s

}= sαY (s) −

m−1∑k=0

sk[

0Dα−k−1t y(t)

]t=0, (m − 1 ≤ α < m). (2.3)

Let us now introduce the case of the Caputo differential operators of arbitrary orderα > 0, which is defined as

C0 D

αt f(t) =

1Γ(n − α)

∫ t0

f (n)(τ)dτ

(t − τ)α+1−n , (n − 1 < α ≤ n). (2.4)

And the formula of its Laplace transform can be expressed as

L{C0 D

αt f(t); s

}= sαF(s) −

n−1∑k=0

sα−k−1f (k)(0), (n − 1 < α ≤ n). (2.5)

2.2. Mittag-Leffler Functions and Generalized

The Mittag-Leffler functions and its generalized forms have played a special role in solvingthe fractional differential equations. In this section, we just give the definition of the following


series of representation of the Mittag-Leffler function Eα(z) with α > 0, which validates in thewhole complex plane:

Eα(z) :=+∞∑n=0

zn

Γ(αn + 1), α > 0, z ∈ C. (2.6)

And for the generalized Mittag-Leffler function, we use the following definition:

Eα,β(z) :=+∞∑n=0

zn

Γ(αn + β

) , α > 0, β > 0, z ∈ C. (2.7)

Let us now consider the Laplace transforms of the function tαk+β−1Eα,β(±atα) (see [3]), whichis defined by

∫∞

0e−sttαk+β−1E

(k)α,β(at

α)dt =k!sα−β

(sα − a)k+1,(

Re(s) > |a|1/α). (2.8)

2.3. Fractional B-Splines Wavelet

Splines have had a significant impact on the early development of the theory of the wavelettransform (see [13]). And Unser and Blu have first mentioned the fractional B-Splines in [13],who extended Schoenberg’s family of polynomial splines to all fractional degrees α > −1 anddefined the fractional causal B-splines by taking the (α + 1)th fractional difference of the one-sided power function:

βα+(x) :=1

Γ(α + 1)�α+1

+ xα+ =1

Γ(α + 1)

∑k≥0

(−1)k(α + 1

k

)(x − k)α+, (2.9)

where the one-side power function xα+ is defined as follows:

xα+ =

⎧⎨⎩xα, x ≥ 0,

0, otherwise.(2.10)

Then we introduce the fractional B-splines autocorrelation sequence as follows:

β2α+1∗ (k) :=

⟨βα(x), βα(x − k)⟩. (2.11)

From [13], we know the explicit form of the fractional B-splines wavelet:

ψα+

(x2

)=∑k∈Z

(−1)k

2α∑l∈Z

(α + 1

l

)β2α+1∗ (l + k − 1)βα+(x − k), (2.12)


which yields a Riesz basis for L2(Ω). Thus, using the standard orthogonalize techniquedescribed in [12], we have

ϕα+(ω) =

√2ψα+(w)

(∑∞k=0

∣∣ψα+(ω + 2kπ)∣∣2)1/2

, (2.13)

where ψα+(w) is the Fourier Transform of the function of ψα+(x). Thus we obtain the orthogonalfractional B-splines ϕα+(x) which also yields a Riesz basis for L2(Ω). According to the theoriesof wavelet analyses, the function y(t) ∈ L2(Ω) can be expressed as

y(t) =+∞∑k=−∞

c(k)ϕα+(t − k), (2.14)

where the coefficients c(k) are constants.

3. The Existence and Uniqueness of the Solution ofMultiterms Fractional Ordinary Differential Equations

In this section, we will prove the existence and uniqueness properties of the solutions of thenonhomogeneous linear differential equations of arbitrary real order α > 0.

3.1. An Important Lemma

Under the hypothesis of the existence for the solutions of (1.1), we have the following lemma.

Lemma 3.1. Let αn > αn−1 > · · · > α1, and αn ≥ 1, and f(t) ∈ L2(Ω), then the solution of the initialvalue problem (1.1) and (1.2) is also in L2(Ω).

Proof. In order to prove this lemma, we divide the proof into two steps.Firstly, we consider the case of n = 1, then (1.1) can be rewritten as

0Dα1t y(t) = f(t). (3.1)

Taking the transform of (3.1) to both sides, we obtain

y(t)=0D−α1t f(t). (3.2)

By virtue of the functions f(t) ∈ L2(Ω) and D−α : L2(Ω) → L2(Ω) that is a bounded linearoperator (see [15]), we can easily derive the function y(t) ∈ L2(Ω).

And then, let us consider the case of n > 1.To (1.1), using the formula (2.2) and taking the Laplace transform to both sides, we

conclude that

(b1sα1 + b2s

α2 + · · · + bnsαn)Y (s) = F(s), (3.3)


and then

Y (s) =F(s)

b1sα1 + b2sα2 + · · · + bnsαn = F(s)G(s). (3.4)

To prove the functions y(t) ∈ L2(Ω), we change the functions G(s) into the following form:

G(s) =1

bnsαn + bn−1sαn−1

1

1 +(∑n−2

k=0 bksαk/(bnsαn + bn−1sαn−1)

) . (3.5)

Then replacing the factor 1/(bnsαn +bn−1sαn−1) with (b−1

n s−αn−1)/(sαn−αn−1 +bn−1/bn) in (3.5), and

expanding the second factor into a formal of series, we have

G(s) =∞∑m=0

(−1)mb−1n s

−αn−1

(sαn−αn−1 + bn−1/bn)m+1

(n−2∑k=0

(bkbn

)sαk−αn−1

)m

. (3.6)

Using the method described in [3], we obtain the expression of G(s), which is

G(s) =1bn

∞∑m=0

(−1)m∑

k0+k1+···+kn−2=mk0,k1,...,kn−2≥0

(m; k0, k1, . . . , kn−2)n−2∏i=0

(bibn

)ki s−αn−1+∑n−2

i=0 (αi−αn−1)ki

(sαn−αn−1 + bn−1/bn)m+1

,

(3.7)

Substituting G(s) in (3.4), considering the formula (2.8), and taking the inverse Laplacetransform term-by-term, we obtain that the analytical solution of initial values problem (1.1)and (1.2) in the following form:

y(t) = f(t) ∗ g(t) (3.8)

with

g(t) =1bn

∞∑m=0

(−1)m

m!

∑k0+k1+···+kn−2=mk0,k1,...,kn−2≥0

(m; k0, k1, . . . , kn−2)n−2∏i=0

(bibn

)kit(αn−αn−1)m+αn+

∑n−2j=0 (αn−1−αj )kj−1

× E(m)αn−αn−1,αn+

∑n−2j=0 (αn−1−αj )kj

(−bn−1

bntαn−αn−1

),

(3.9)

where (m; k0, k1, . . . , kn−2) = m!/∏n−2

i=0 (ki!) is the multinomial coefficient, the representationof f(t) ∗ g(t) is the convolution of functions f(t) and g(t), and E

(k)λ,μ is the kth derivative of the

Mittag-Leffler function with parameters λ and μ (see [16]).


From the representation (3.9), the function E(m)α,β (z) is bounded in z ∈ C, and the index

satisfies (αn − αn−1)m + αn +∑n−2

j=0 (αn−1 − αj)kj − 1 ≥ 0; so we can easily gain that the functiong(t) is bounded in Ω. And then

∥∥y(t)∥∥L2(Ω) =∥∥f(t) ∗ g(t)∥∥L2(Ω) ≤

∥∥f(t)∥∥L2(Ω)

∥∥g(t)∥∥L2(Ω) ≤ C∥∥f(t)∥∥L2(Ω), (3.10)

it yields that the functions y(t) ∈ L2(Ω), and so we have finished the proof of the lemma.

3.2. The Proof of Existence and Uniqueness

Considering the solution of the fractional differential equation y(t) ∈ L2(Ω) and the fractionalB-spline wavelet ϕα+(x) which generates a Riesz basis for L2(Ω), we can prove the followingtheorem.

Theorem 3.2. Let αn > αn−1 > · · · > α1, and αn ≥ 1, and f(t) ∈ L2(Ω), then the initial valueproblems (1.1) and (1.2) have a unique solution. Further more, the solution has explicit representationof fractional B-splines wavelets series.

Proof. From Lemma 3.1, we have obtained the functions y(t) ∈ L2(Ω), which can be expressedas

y(t) =+∞∑k=−∞

c(k)ϕα+(t − k), (3.11)

where the index satisfies α > 0.Then substituting (2.14) into (1.1), and taking the Fourier transform to both sides, we

obtain

c(ω)[b1(iω)α1 + b2(iω)α2 + · · · + bn(iω)αn

]ψα+(w) = f(ω), (3.12)

where c(ω) =∑+∞

k=−∞ c(k)e−ikw, and (3.12) is equivalent to the following form:

c(ω)ψα+(w) =f(ω)[

b1(iω)α1 + b2(iω)α2 + · · · + bn(iω)αn] . (3.13)

And then, by taking the inverse Fourier transform to (3.13), we have

+∞∑k=−∞

c(k)ϕα+(t − k) = f(t) ∗ g(t). (3.14)

Because the function ϕα+(t) is the orthogonal fractional B-splines, hence, the representation ofthe coefficient c(k) is defined by

c(k) =⟨f(t) ∗ g(t), ϕα+(t − k)

⟩, (3.15)


where 〈•, •〉 is the inner product. Then substituting (3.15) in (2.14), we obtain the solution ofthe initial value problems (1.1) and (1.2).

Let us suppose that the initial value problems (1.1) and (1.2) have another solution,which can be expressed as

y1(t) =+∞∑n=−∞

d(n)ϕα+(t − n). (3.16)

Utilizing the similar scheme to c(k), we acquire the representation of d(n), which is

d(n) =⟨f(t) ∗ g(t), ϕα+(t − n)

⟩. (3.17)

Taking (3.17) into (3.16), we obtain the representation of y1(t). Obviously, we have

y(t) − y1(t) =+∞∑k=−∞

c(k)ϕα+(t − k) −+∞∑n=−∞

d(n)ϕα+(t − n) = 0; (3.18)

it derives that y(t) = y1(t), which means that the solution of the initial value problems (1.1)and (1.2) is unique. Then substituting y1(t) with its coefficient in (1.1), we can easily checkthat the equation is correct. It indicates that the function y1(t) is a solution of the differentialequation (1.1), which yields the existence of solution. Finally, we have completed the proof ofthe solution of existence and uniqueness of the nonhomogeneous linear differential equationsof arbitrary order.

4. The Asymptotic Solution of the Fractional Differential Equationsand Error Estimation

4.1. The Asymptotic Solution of the Fractional Differential Equations

The purpose of this section is to discuss the case of αi ∈ Q+ in (1.1), which appeared inmost practical applications we have encountered, and show the asymptotic solution of thefractional differential equations and error estimation. In fact, the case of αi ∈ Q, i = 1, 2, . . . , nin Theorem 3.2 is equivalent to the following corollary.

Corollary 4.1. Letmv ≥ 1, and f(t) ∈ L2(Ω), then the nonhomogeneous linear fractional differentialequations

(amD

mv + am−1D(m−1)v + · · · + a0D

0)y(t) = f(t) (4.1)

on the initial values

[D(m−k)v−1y(t)

]t=0

= 0,[D(m−k)v−2y(t)

]t=0

= 0, . . . ,[D(m−k)v−rk−1y(t)

]t=0

= 0, (4.2)

where 0 ≤ rk − 1 ≤ (m − k)v < rk, v ∈ Q+, rk ∈N, k = 0, 1, . . . , m, have a unique solution.


In Section 3, we have proved the existence and uniqueness of the nonhomogeneouslinear fractional differential equations of arbitrary positive real order. Obviously, it alsosatisfies the case of positive rational order. To (1.1), let q ∈ Z be the least common multipleof the denominators of αi ∈ Q, i = 1, 2, . . . , n, so αi = qi/q, where qi ∈ Z, i = 1, 2, . . . , n. Andto (4.1), let v = 1/q and m − i0 = qn, m − i1 = qn−1, . . . , m − in−1 = q1, then considering theconditions of am−ij = bn−j , ak = 0, k /=m − ij , where ij < m, ij ∈ Z, j = 1, 2, . . . , n − 1. Hence,the initial value problems (4.1) and (4.2) have been changed into of the problems (1.1) and(1.2) with the order αi ∈ Q+, respectively, which means that Theorem 3.2 in the case of αi ∈ Q+

is equivalent to Corollary 4.1.Similarity to the scheme of Theorem 3.2, we can easily prove Corollary 4.1 and give

the representation of explicit solution of the initial value problems (4.1) and (4.2), which canbe defined by

y(t) =+∞∑k=−∞

c(k)ϕα+(t − k), (4.3)

where c(k) = 〈f(t) ∗ g(t), ϕα+(t − k)〉, and g(t) is the inverse of the Laplace Transform of thefunction G(s), where

G(s) =1

amsmv + am−1s(m−1)v + · · · + a0. (4.4)

To obtain the asymptotic solution of (4.1), we will give the explicit formulation of the functiong(t). Let P(sv) = ams

mv + am−1s(m−1)v + · · · + a0 and x = sv; thus the P(x) is a polynomial of

the degree m. Moreover, we suppose that γ1, γ2,...,γj are distinct zeros of P(x) with the orderof l1, l2,...,lj , respectively; by applying the theories of polynomials, the P(x) can be rewritten inthe following form

P(x) = am(x − γ1

)l1(x − γ2)l2 · · · (x − γj

)lj , (4.5)

where∑j

i=1 li = m, j, li ∈ N, i = 1, 2, . . . , j, Then, substituting the function P(x) in (4.2) andexpanding it to the sum of partial fractions, we have

G(s) =j∑i=1

li−1∑n=0

Ai,li−n(sv − γi

)li−n , (4.6)

where the coefficients Ai,li−n, i = 1, 2, . . . , j; n = 0, 1, . . . , li − 1, are constants.Thus, taking the inverse Laplace Transform to (4.6) and using the formulation (2.8),

we obtain explicit formulation of the function g(t), defined by

g(t) =j∑i=1

li−1∑n=0

t(i+1)v−1

(li − n)!Ai,li−nE(li−n−1)v,v

(γit

v). (4.7)


According to Unser and Blu in [13], the wavelet base generated by fractional B-splineswavelet βα+(x) is denseness of the representation in L2(Ω). Let βα+(x) be the inverse of FourierTransform of ˆβα+(ω), where

ˆβα+(ω) =

√2βα+(w)

(∑∞k=0

∣∣∣βα+(ω + 2kπ)∣∣∣

2)1/2

. (4.8)

With the help of the theories of wavelet analyses, we know that the functions βα+(x) are anorthogonal fractional B-splines wavelet and dense in L2(Ω) with α > 0. Thus the solutions ofinitial value problems (4.1) and (4.2) have the following form:

y(t) =+∞∑k=−∞

c(k)βα+(t − k). (4.9)

Then substituting y(t) in (4.1), we obtain the representation of the coefficients c(k), definedby

c(k) =⟨f(t) ∗ g(t), βα+(t − k)

⟩. (4.10)

Finally, by combining (4.9) and (4.10), the obtained function y(t) is the asymptotic solutionof initial value problems (4.1) and (4.2) that we are looking for.

4.2. Order of the Error Estimation

To estimate the error of asymptotic solution of initial value problems (4.1) and (4.2), weintroduce the following properties of the fractional B-spines βα+(x), which have been provedby Unser and Blu (see [13, Theorem 3.1]).

Proposition 4.2. For all α > 0, we have

βα+(x) =Γ(α + 2) sinπα

πxα+2

∑n≥1

e2niπx

(2niπ)α+1+ o(

1xα+2

), (4.11)

when x tends to +∞.

To (4.8), the function a(ω) =∑∞

k=0 |βα+(ω + 2kπ)|2 is 2π-periodic and symmetric, andso we can restrict its study to ω ∈ [0, π]. In particular, one has

a(ω) ≥∣∣∣sin c

ω

2

∣∣∣2α+2

≥(

2π

)2α+2

, (4.12)

since sin cω/2 is strictly decreasing over [0, π] (see [13]).


Combining (4.12) and (4.8) and taking the inverse of Fourier Transform, we obtain

βα+(t − k) ≤(

2π

)α+1

βα+(t − k). (4.13)

Then according to (4.10), (4.11), and (4.13), the asymptotic solution y(t) can be defined by

y(t) =+∞∑k=−∞

⟨f(t) ∗ g(t), βα+(t − k)

⟩βα+(t − k) ≤

∥∥f(t) ∗ g(t)∥∥L2

+∞∑k=−∞

∣∣βα+(t − k)∣∣2. (4.14)

To calculate the truncated error of the asymptotic solution, let yN(t) be the truncated sum(|k| ≤N) corresponding to asymptotic solution (4.8), where

yN(t) =N∑

k=−N

⟨f(t) ∗ g(t), βα+(t − k)

⟩βα+(t − k). (4.15)

Thus the truncated error y∗(t) will be obtained as follows:

y∗(t) = y(t) − yN(t)

=∑|k|>N

⟨f(t) ∗ g(t), βα+(t − k)

⟩βα+(t − k)

≤ ∥∥f(t) ∗ g(t)∥∥∑|k|>N

∣∣βα+(t − k)∣∣2,

(4.16)

where the function g(t) is bounded in Ω and f(t) ∈ L2(Ω); with the help of (4.12), theinequality of (4.16) can be amplified, which means that

y∗(t) ≤(

2π

)2α+2∥∥f(t) ∗ g(t)∥∥∑|k|>N

∣∣βα+(t − k)∣∣2. (4.17)

Then substituting (4.10) in (4.16), we have

y∗(t) ≤(

2π

)2α+2∥∥f(t) ∗ g(t)∥∥∑|k|>N

∣∣∣∣∣Γ(α + 2) sin πα

π(t − k)α+2

∑n≥1

e2πinx

(2niπ)α+1+ o

(1

(t − k)α+2

)∣∣∣∣∣2

.

(4.18)

To the right side of (4.17), we divided the representation into three parts for discussion.Firstly, because the function g(t) is bounded in Ω and f(t) ∈ L2(Ω), there exists a constant C1

which holds

∥∥f(t) ∗ g(t)∥∥L2 ≤ c∥∥f(t)∥∥L2 ≤ C1. (4.19)


Secondly, we consider the series function

∣∣∣∣∣∑n≥1

e2πinx

(2niπ)α+1

∣∣∣∣∣2

≤∑n≥1

∣∣∣∣∣e2πinx

(2niπ)α+1

∣∣∣∣∣2

≤∑n≥1

1

(2nπ)2α+2, (4.20)

which is convergent in Ω; thus there exists a constant C2 which satisfies

∣∣∣∣∣∑n≥1

e2πinx

(2niπ)α+1

∣∣∣∣∣2

≤ C2. (4.21)

At last, we denote a constant C3 which defined by

C3 =22α+2Γ2(α + 2)sin2πα

π2α+4. (4.22)

Then combining (4.19), (4.21), and (4.22), the inequality of (4.18) can be amplified to thefollowing form:

y∗(t) ≤ C1C22C3

∑|k|>N

∣∣∣∣∣1

(t − k)α+2+ o

(1

(t − k)α+2

)∣∣∣∣∣2

≤ C1C22C3

∑|k|>N

∣∣∣∣∣1 + C0

(t − k)α+2

∣∣∣∣∣2

, (4.23)

when |t − k| tends to +∞.Thus, from the above discussion, it is evident to derive that the truncated error

converges as∑

|k|>N C/(k − t)2α+4, where C = C1C22C3(1 + C0)

2. Note that we should choosea suitable N which should be much more greater than T , where, t ∈ Ω = [0, T]. Finally, wehave obtained the error order of the asymptotic truncated sum in theory.

Remark 4.3. Noting that the process of the proof of the existence and uniqueness of thesolution of the initial value problems (1.1) and (1.2) in the case of Riemann-Liouvillefractional differential operator, and considering the formula of the Laplace transform ofCaputo differential operator, we can be easily replaced the case of Riemann-Liouvillefractional differential operator by Caputo sense with its initial values. It means the followingcorollary is correct.

Corollary 4.4. Let αn > αn−1 > · · · > α1 > 0, and αn ≥ 1, and f(t) ∈ L2(Ω), then the multitermsfractional ordinary differential equations

(bnDαn + bn−1Dαn−1 + · · · + b1D

α1)y(t) = f(t) (4.24)

with its initial values

y(k)(0) = 0, (k = 0, 1, 2, . . . , �αn� − 1), (4.25)

where �αn� = max{m | m ≤ αn,m ∈ Z}, have a unique solution.


5. Conclusion

In this paper, we have proved the existence and uniqueness of the solution of the differentialequations of arbitrary positive real order. And the representation of the solution of (1.1)has been given in the process of proof. We have obtained the asymptotic solution of thedifferential equations of fractional order α ∈ Q+ and corresponding error estimation. Themost notable feature is the order of the asymptotic truncated error, namely, (2α + 4)th,which is effective to calculate the numerical solution of (4.1). In particular, the case ofRiemann-Liouville differential operator is replaced by Caputo sense with its initial valuesin Corollary 4.4. Similarly to the scheme of the proof as showen in the paper, by consideringthe relationship between Riemann-Liouville and Caputo differential operator, we can easilycomplete the proof of the corollary in the case of the Caputo differential operator, whichshows that the method we have discussed can be applied more widely.

References

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[2] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations,A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993.


[4] J. T. Edwards, N. J. Ford, and A. C. Simpson, “The numerical solution of linear multi-term fractionaldifferential equations; systems of equations,” Journal of Computational and Applied Mathematics, vol.148, no. 2, pp. 401–418, 2002.

[5] K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of MathematicalAnalysis and Applications, vol. 265, no. 2, pp. 229–248, 2002.

[6] K. Diethelm, N. J. Ford, and A. D. Freed, “A predictor-corrector approach for the numerical solutionof fractional differential equations,” Nonlinear Dynamics, vol. 29, no. 1–4, pp. 3–22, 2002.

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[10] V. Daftardar-Gejji and A. Babakhani, “Analysis of a system of fractional differential equations,”Journal of Mathematical Analysis and Applications, vol. 293, no. 2, pp. 511–522, 2004.

[11] M. Unser and T. Blu, “Construction of fractional spline wavelet bases,” in Wavelets Applications inSignal and Image Processing VII, vol. 3813 of Proceedings of SPIE, pp. 422–431, Denver, Colo, USA, July1999.

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[13] M. Unser and T. Blu, “Fractional splines and wavelets,” SIAM Review, vol. 42, no. 1, pp. 43–67, 2000.[14] M. Unser, A. Aldroubi, and M. Eden, “A family of polynomial spline wavelet transforms,” Signal

Processing, vol. 30, pp. 141–162, 1993.[15] V. J. Ervin and J. P. Roop, “Variational formulation for the stationary fractional advection dispersion

equation,” Numerical Methods for Partial Differential Equations, vol. 22, no. 3, pp. 558–576, 2005.[16] F. Mainardi and R. Gorenflo, “On Mittag-Leffler-type functions in fractional evolution processes,”

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Research ArticleSolvability of Nonlinear Langevin EquationInvolving Two Fractional Orders with DirichletBoundary Conditions

Bashir Ahmad1 and Juan J. Nieto2

1 Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203,Jeddah 21589, Saudi Arabia

2 Departamento de Analisis Matematico, Facultad de Matematicas, Universidad de Santiago de Compostela,15782 Santiago de Compostela, Spain

Correspondence should be addressed to Bashir Ahmad, bashir [email protected]

Received 8 August 2009; Accepted 14 November 2009

Academic Editor: Nikolai Leonenko

Copyright q 2010 B. Ahmad and J. J. Nieto. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

We study a Dirichlet boundary value problem for Langevin equation involving two fractionalorders. Langevin equation has been widely used to describe the evolution of physical phenomenain fluctuating environments. However, ordinary Langevin equation does not provide the correctdescription of the dynamics for systems in complex media. In order to overcome this problem anddescribe dynamical processes in a fractal medium, numerous generalizations of Langevin equationhave been proposed. One such generalization replaces the ordinary derivative by a fractionalderivative in the Langevin equation. This gives rise to the fractional Langevin equation with asingle index. Recently, a new type of Langevin equation with two different fractional orders hasbeen introduced which provides a more flexible model for fractal processes as compared with theusual one characterized by a single index. The contraction mapping principle and Krasnoselskii’sfixed point theorem are applied to prove the existence of solutions of the problem in a Banachspace.

1. Introduction

Fractional differential equations have recently gained much importance and attention.The study of fractional differential equations ranges from the theoretical aspects ofexistence and uniqueness of solutions to the analytic and numerical methods for findingsolutions. Fractional differential equations appear naturally in a number of fields suchas physics, polymer rheology, regular variation in thermodynamics, biophysics, bloodflow phenomena, aerodynamics, electrodynamics of complex medium, viscoelasticity,


Bodes analysis of feedback amplifiers, capacitor theory, electrical circuits, electronanalyticalchemistry, biology, control theory, fitting of experimental data, etc. An excellent accountin the study of fractional differential equations can be found in [1–3]. For more detailsand examples, see [4–13] and the references therein. Some new and recent aspectson fractional calculus can be seen in [14–16]. In [15], it was shown that fractionalNambu systems can be proposed as a generalization of fractional Hamiltonian sys-tems.

Langevin equation is widely used to describe the evolution of physical phenomenain fluctuating environments [17]. However, for the systems in complex media, ordinaryLangevin equation does not provide the correct description of the dynamics. One of thepossible generalizations of Langevin equation is to replace the ordinary derivative by afractional derivative in it. This gives rise to fractional Langevin equation, see for instance[18, 19] and the references therein. In [18], the authors studied a new type of Langevinequation with two different fractional orders. The solution to this new version of fractionalLangevin equation gives a fractional Gaussian process parameterized by two indices, whichprovides a more flexible model for fractal processes as compared with the usual onecharacterized by a single index. In [19], the fractional oscillator process with two indiceswas discussed.

In this paper, we study a Dirichlet boundary value problem of Langevin equation withtwo different fractional orders. This work is motivated by recent work of Lim et al. [18, 19].Precisely, we consider the problem

cDβ(cDα + λ) x(t) = f(t, x(t)), 0 < t < 1, 0 < α, β ≤ 1,

x(0) = γ1, x(1) = γ2,(1.1)

where cD is the Caputo fractional derivative, f : [0, 1] × X → X, λ is a real number andγ1, γ2 ∈ X. Here, (X, ‖ · ‖) is a Banach space and C = C([0, 1], X) denotes the Banach space ofall continuous functions from [0, 1] → X endowed with a topology of uniform convergencewith norm defined by ‖x‖ = sup{|x(t)|, t ∈ [0, 1]}.

In Section 1, we prove a new result for linear differential equations involvingtwo fractional orders. Section 2 deals with the theory of nonlinear differential equationswith two fractional orders. We first use the contraction mapping principle to prove theexistence and uniqueness of the solution of problem (1.1) in a Banach space. We thenemploy Krasnoselskii’s fixed point theorem to establish another new existence result forproblem (1.1). We also give an example for the illustration of the theory established in thispaper.

A function x ∈ C with its Caputo derivative of fractional order existing on (0, 1) is asolution of (1.1) if it satisfies (1.1).

Relative to (1.1), we now introduce the following linear problem:

cDβ(cDα + λ)x(t) = σ(t), 0 < t < 1, 0 < α, β ≤ 1,

x(0) = γ1, x(1) = γ2,(1.2)

where σ ∈ C[0, 1].


Lemma 1.1. The unique solution of the boundary value problem (1.2) is given by

x(t) =∫ t

0

(t − u)α−1

Γ(α)

(∫u

0

(u − s)β−1

Γ(β) σ(s)ds − λx(u)

)du

− tα[∫1

0

(1 − u)α−1

Γ(α)

(∫u

0

(u − s)β−1


)du

]+(γ2 − γ1

)tα + γ1.

(1.3)

Proof. As argued in [2, Section 5.4], the general solution of

cDβ(cDα + λ)x(t) = σ(t) (1.4)

can be written as

x(t) =∫ t

0

(t − u)α−1

Γ(α)

(∫u

0

(u − s)β−1


)du − c0

Γ(α + 1)tα − c1. (1.5)

Using the boundary conditions for (1.2), we find that

c1 = −γ1,c0

Γ(α + 1)=∫1

0

(1 − u)α−1

Γ(α)

(∫u

0

(u − s)β−1


)du − γ2 + γ1.

(1.6)

Substituting (1.6) in (1.5), we obtain the solution given by (1.3). This completes the proof.

Now, we state a known result due to Krasnoselskii (see [20]) which is needed to provethe existence of at least one solution of (1.1).


Theorem 1.2. Let M be a closed convex and nonempty subset of a Banach space X. Let A,B be theoperators such that (i) Ax +By ∈ M whenever x, y ∈ M; (ii) A is compact and continuous; (iii) Bis a contraction mapping. Then there exists z ∈ M such that z = Az + Bz.

2. Existence of Solutions

Theorem 2.1. Let f : [0, 1] ×X → X be a jointly continuous function satisfying the condition

∣∣f(t, x) − f(t, y

)∣∣ ≤ L∣∣x − y

∣∣, ∀t ∈ [0, 1], x, y ∈ X, (2.1)

Then the boundary value problem (1.1) has a unique solution provided Λ < 1, where

Λ =2L

Γ(α + β + 1

) +2|λ|

Γ(α + 1). (2.2)

Proof. Define � : C → C by

(�x)(t) =∫ t

0

(t − u)α−1

Γ(α)

(∫u

0

(u − s)β−1

Γ(β) f(s, x(s))ds − λx(u)

)du

− tα[∫1

0

(1 − u)α−1

Γ(α)

(∫u

0

(u − s)β−1


)du

]

+(γ2 − γ1

)tα + γ1, t ∈ [0, 1].

(2.3)

Let us set supt∈[0,1]|f(t, 0)| = M and choose

r ≥ 11 − δ

(2M

Γ(α + β + 1

) +(∣∣γ2

∣∣ + 2∣∣γ1

∣∣)), (2.4)


where δ is such that Λ ≤ δ < 1. Now we show that �Br ⊂ Br, where Br = {x ∈ C : ‖x‖ ≤ r}.For x ∈ Br, we have

‖(�x)(t)‖

= supt∈[0,1]

∣∣∣∣∣∫ t

0

(t − u)α−1

Γ(α)

(∫u

0

(u − s)β−1


)du

− tα[∫1

0

(1 − u)α−1

Γ(α)

(∫u

0

(u − s)β−1


)du

]+(γ2 − γ1

)tα + γ1

∣∣∣∣∣

≤ supt∈[0,1]

(∫ t

0

(t − u)α−1

Γ(α)

(∫u

0

(u − s)β−1

Γ(β) (∣∣f(s, x(s)) − f(s, 0)

∣∣ + ∣∣f(s, 0)∣∣)ds + |λx(u)|)du

+ tα[∫1

0

(1 − u)α−1

Γ(α)

(∫u

0

(u − s)β−1

Γ(β) (∣∣f(s, x(s)) − f(s, 0)

∣∣ + ∣∣f(s, 0)∣∣)ds

+ |λx(u)|)du

]+(∣∣γ2

∣∣ + ∣∣γ1∣∣)tα + ∣∣γ1

∣∣)

≤ supt∈[0,1]

(∫ t

0

(t − u)α−1

Γ(α)

(∫u

0

(u − s)β−1

Γ(β) (

L|x(s)| + ∣∣f(s, 0)∣∣)ds + |λx(u)|)du

+ tα[∫1

0

(1 − u)α−1

Γ(α)

(∫u

0

(u − s)β−1

Γ(β) (

L|x(s)| + ∣∣f(s, 0)∣∣)ds + |λx(u)|)du

]

+(∣∣γ2

∣∣ + ∣∣γ1∣∣)tα + ∣∣γ1

∣∣)

≤ supt∈[0,1]

(∫ t

0

(t − u)α−1

Γ(α)

(∫u

0

(u − s)β−1

Γ(β) ds

(L|x(u)| + ∣∣f(u, 0)∣∣) + |λx(u)|

)du

+ tα[∫1

0

(1 − u)α−1

Γ(α)

(∫u

0

(u − s)β−1

Γ(β) ds

(L|x(u)| + ∣∣f(u, 0)∣∣) + |λx(u)|

)du

]

+(∣∣γ2

∣∣ + ∣∣γ1∣∣)tα + ∣∣γ1

∣∣)

≤ supt∈[0,1]

∫ t

0

(t − u)α−1

Γ(α)

∫u

0

(u − s)β−1

Γ(β) dsdu(L‖x‖ + |M|) + sup

t∈[0,1]

∫ t

0

(t − u)α−1

Γ(α)du|λ|‖x‖

+∫1

0

(1 − u)α−1

Γ(α)

∫u

0

(u − s)β−1

Γ(β) dsdu(L‖x‖ +M) +

∫1

0

(1 − u)α−1

Γ(α)du|λ|‖x‖ + ∣∣γ2

∣∣ + 2∣∣γ1

∣∣


≤ 2(Lr +M)∫1

0

(1 − u)α−1

Γ(α)

∫u

0

(u − s)β−1

Γ(β) dsdu

+ 2|λ|r∫1

0

(1 − u)α−1

Γ(α)du +

(∣∣γ2∣∣ + 2

∣∣γ1∣∣)

=2(Lr +M)Γ(α)Γ

(β + 1

)∫1

0(1 − u)α−1uβdu +

2|λ|rΓ(α + 1)

+(∣∣γ2

∣∣ + 2∣∣γ1

∣∣).

(2.5)

Using (2.2), (2.4), and the relation for Beta function B(·, ·):

B(β + 1, α

)=∫1

0(1 − u)α−1uβdu =

Γ(α)Γ(β + 1

)Γ(α + β + 1

) , (2.6)

we find that

‖(�x)(t)‖ ≤ (Λ + 1 − δ)r ≤ r. (2.7)

Now, for x, y ∈ C and for each t ∈ [0, 1], we obtain

∥∥(�x)(t) − (�y

)(t)

∥∥

= supt∈[0,1]

∣∣(�x)(t) − (�y

)(t)

∣∣

≤ supt∈[0,1]

(∫ t

0

(t − u)α−1

Γ(α)

(∫u

0

(u − s)β−1

Γ(β) ∣∣f(s, x(s)) − f

(s, y(s)

)∣∣ds)du

+ |λ|∫ t

0

(t − u)α−1

Γ(α)(∣∣x(u) − y(s)

∣∣)du

+ tα[∫1

0

(1 − u)α−1

Γ(α)

(∫u

0

(u − s)β−1

Γ(β) ∣∣f(s, x(s)) − f

(s, y(s)

)∣∣ds)du

+ |λ|∫1

0

(1 − u)α−1

Γ(α)

∣∣x(u) − y(u)∣∣du

])


≤ supt∈[0,1]

(L

∫ t

0

(t − u)α−1

Γ(α)

∫u

0

(u − s)β−1

Γ(β) dsdu

∣∣x(t) − y(t)∣∣

+ |λ|∫ t

0

(t − u)α−1

Γ(α)du

∣∣x(t) − y(t)∣∣)

+ L

∫1

0

(1 − u)α−1

Γ(α)

∫u

0

(u − s)β−1

Γ(β) dsdu

∥∥x − y∥∥

+ |λ|∫1

0

(1 − u)α−1

Γ(α)du

∥∥x − y∥∥

≤ ∥∥x − y∥∥[

2L∫1

0

(1 − u)α−1

Γ(α)

∫u

0

(u − s)β−1

Γ(β) dsdu + 2|λ|

∫1

0

(1 − u)α−1

Γ(α)du

]

= Λ∥∥x − y

∥∥,(2.8)

where

Λ =2L

Γ(α + β + 1

) +2|λ|

Γ(α + 1), (2.9)

which depends only on the parameters involved in the problem. As Λ < 1, then � isa contraction. Thus, the conclusion of the theorem follows by the contraction mappingprinciple. This completes the proof.

Theorem 2.2. Assume that f : [0, 1] × X → X is a jointly continuous function and maps boundedsubsets of [0, 1] ×X into relatively compact subsets of X. Furthermore, assume that

(H1) |f(t, x) − f(t, y)| ≤ L|x − y|, for all t ∈ [0, 1], x, y ∈ X;

(H2) |f(t, x)| ≤ μ(t), for all (t, x) ∈ [0, 1] ×X, and μ ∈ L1([0, 1], R+).

If

(L

Γ(α + β + 1

) +|λ|

Γ(α + 1)

)< 1, (2.10)

then the boundary value problem (1.1) has at least one solution on [0, 1].

Proof. Let us fix

r ≥(

2∥∥μ∥∥L1/Γ

(α + β + 1

)+∣∣γ2

∣∣ + 2∣∣γ1

∣∣1 − 2|λ|/Γ(α + 1)

)(2.11)


and consider Br = {x ∈ C : ‖x‖ ≤ r}. We define the operators Φ and Ψ on Br as

(Φx)(t) =∫ t

0

(t − u)α−1

Γ(α)

(∫u

0

(u − s)β−1


)du,

(Ψx)(t) = −tα[∫1

0

(1 − u)α−1

Γ(α)

(∫u

0

(u − s)β−1


)du

]

+(γ2 − γ1

)tα + γ1.

(2.12)

For x, y ∈ Br, we find that

∥∥Φx + Ψy∥∥ ≤

(2∥∥μ∥∥L1

Γ(α + β + 1

) +2|λ|r

Γ(α + 1)+∣∣γ2

∣∣ + 2∣∣γ1

∣∣)

≤ r. (2.13)

Thus, Φx + Ψy ∈ Br. From the assumption

(L

Γ(α + β + 1

) +|λ|

Γ(α + 1)

)< 1, (2.14)

it follows that Ψ is a contraction mapping. The continuity of f implies that the operator Φ iscontinuous. Also, Φ is uniformly bounded on Br as

‖Φx‖ ≤∥∥μ∥∥L1

Γ(α + β + 1

) +|λ|r

Γ(α + 1). (2.15)

Now we prove the compactness of the operator Φ. Setting Ω = [0, 1] × Br, we definesup(t,x)∈Ω|f(t, x)| = f, and consequently we have

‖(Φx)(t1) − (Φx)(t2)‖ =

∥∥∥∥∥∫ t1

0

(t1 − u)α−1

Γ(α)

(∫u

0

(u − s)β−1


)du

−∫ t2

0

(t2 − u)α−1

Γ(α)

(∫u

0

(u − s)β−1


)du

∥∥∥∥∥

≤ f

Γ(α + β + 1

)∣∣∣tα+β1 − tα+β2

∣∣∣ + |λ|rΓ(α + 1)

∣∣tα1 − tα2∣∣,

(2.16)

which is independent of x. Thus, Φ is equicontinuous. Using the fact that f maps boundedsubsets into relatively compact subsets, we have that Φ(A)(t) is relatively compact in X forevery t, where A is a bounded subset of C. So Φ is relatively compact on Br. Hence, by theArzela Ascoli theorem, Φ is compact on Br. Thus all the assumptions of Theorem 1.2 aresatisfied and the conclusion of Theorem 1.2 implies that the boundary value problem (1.1)has at least one solution on [0, 1]. This completes the proof.


Example 2.3. Consider the boundary value problem

cD1/4(

cD1/2 +14

)x(t) =

1

(t + 3)2

|x|1 + |x| , 0 < t < 1,

x(0) = γ1, x(1) = γ2.

(2.17)

Here, f(t, x) = (1/(t + 3)2)(|x|/(1 + |x|)), α = 1/2, β = 1/4 and λ = 1/4. Clearly |f(t, x) −f(t, y)| ≤ (1/9)|x − y| with L = 1/9. Further,

Λ =8

27Γ(3/4)+

1√π

< 1. (2.18)

Thus, by Theorem 2.1, the boundary value problem (2.17) has a unique solution on [0, 1].

3. Conclusions

The existence of solutions for a Dirichlet boundary value problem involving Langevinequation with two different fractional orders has been discussed. We apply the concepts offractional calculus together with fixed point theorems to establish the existence results. Firstof all, we find the unique solution for a linear Dirichlet boundary value problem involvingLangevin equation with two different fractional orders, which in fact provides the platformto prove the existence of solutions for the associated nonlinear fractional Langevin equationwith two different orders. Our approach is simple and is applicable to a variety of real worldproblems.

Acknowledgments

The research of J. J. Nieto has been supported by Ministerio de Educacion y Cienciaand FEDER, project MTM2007-61724, and by Xunta de Galicia and FEDER, projectPGIDIT06PXIB207023PR.

References

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[2] V. Lakshmikantham, S. Leela, and J. Vasundhara Devi, Theory of Fractional Dynamic Systems,Cambridge Academic, Cambridge, UK, 2009.

[3] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif., USA, 1999.[4] B. Ahmad and J. J. Nieto, “Existence results for nonlinear boundary value problems of fractional

integrodifferential equations with integral boundary conditions,” Boundary Value Problems, vol. 2009,Article ID 708576, 11 pages, 2009.

[5] B. Ahmad and J. J. Nieto, “Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations,” Abstract and Applied Analysis, vol. 2009, Article ID494720, 9 pages, 2009.


[6] B. Ahmad and J. J. Nieto, “Existence results for a coupled system of nonlinear fractional differentialequations with three-point boundary conditions,” Computers & Mathematics with Applications, vol. 58,no. 9, pp. 1838–1843, 2009.

[7] B. Ahmad and J. J. Nieto, “Existence of solutions for anti-periodic boundary value problems involvingfractional differential equations via Leray-Schauder degree theory,” to appear in Topological Methodsin Nonlinear Analysis.

[8] B. Ahmad, “Existence of solutions for irregular boundary value problems involving nonlinearfractional differential equations,” Applied Mathematics Letters, 2009.

[9] B. Ahmad and J. J. Nieto, “Existence of solutions for impulsive anti-periodic boundary value problemsof fractional order,” to appear in Taiwanese Journal of Mathematics.

[10] Y.-K. Chang and J. J. Nieto, “Some new existence results for fractional differential inclusions withboundary conditions,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 605–609, 2009.

[11] V. Daftardar-Gejji and S. Bhalekar, “Boundary value problems for multi-term fractional differentialequations,” Journal of Mathematical Analysis and Applications, vol. 345, no. 2, pp. 754–765, 2008.

[12] R. Hilfer, Ed., Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000.[13] S. Z. Rida, H. M. El-Sherbiny, and A. A. M. Arafa, “On the solution of the fractional nonlinear

Schrodinger equation,” Physics Letters A, vol. 372, no. 5, pp. 553–558, 2008.[14] A. Arara, M. Benchohra, N. Hamidi, and J. J. Nieto, “Fractional order differential equations on an

unbounded domain,” Nonlinear Analysis, vol. 72, pp. 580–586, 2010.[15] D. Baleanu, A. K. Golmankhaneh, and A. K. Golmankhaneh, “Fractional Nambu mechanics,”

International Journal of Theoretical Physics, vol. 48, no. 4, pp. 1044–1052, 2009.[16] M. R. Ubriaco, “Entropies based on fractional calculus,” Physics Letters A, vol. 373, no. 30, pp. 2516–

2519, 2009.[17] W. T. Coffey, Yu. P. Kalmykov, and J. T. Waldron, The Langevin Equation: With Applications to Stochastic

Problems in Physics, Chemistry and Electrical Engineering, vol. 14 of World Scientific Series in ContemporaryChemical Physics, World Scientific, River Edge, NJ, USA, 2nd edition, 2004.

[18] S. C. Lim, M. Li, and L. P. Teo, “Langevin equation with two fractional orders,” Physics Letters A, vol.372, no. 42, pp. 6309–6320, 2008.

[19] S. C. Lim and L. P. Teo, “The fractional oscillator process with two indices,” Journal of Physics A, vol.42, no. 6, Article ID 065208, 34 pages, 2009.

[20] D. R. Smart, Fixed Point Theorems, Cambridge University Press, Cambridge, UK, 1980.


Research ArticleOn the Selection and Meaning of Variable OrderOperators for Dynamic Modeling

Lynnette E. S. Ramirez and Carlos F. M. Coimbra

School of Engineering, University of California, P.O. Box 2039, Merced, CA 95344, USA

Correspondence should be addressed to Carlos F. M. Coimbra, [email protected]

Received 4 August 2009; Accepted 8 October 2009

Academic Editor: Nikolai Leonenko

Copyright q 2010 L. E. S. Ramirez and C. F. M. Coimbra. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

We review the application of differential operators of noninteger order to the modeling of dynamicsystems. We compare all the definitions of Variable Order (VO) operators recently proposed inliterature and select the VO operator that has the desirable property of continuous transitionbetween integer and non-integer order derivatives. We use the selected VO operator to connect themeaning of functional order to the dynamic properties of a viscoelastic oscillator. We conclude thatthe order of differentiation of a single VO operator that represents the dynamics of a viscoelasticoscillator in stationary motion is a normalized phase shift. The normalization constant is found bytaking the difference between the order of the inertial term (2) and the order of the spring term(0) and dividing this difference by the angular phase shift between acceleration and position inradians (π), so that the normalization constant is simply 2/π .

1. Introduction

The integer order differential operators of classical calculus (such as the first or second orderderivatives) are familiar to anyone who has an active interest in understanding dynamicsystems. These differential operators are used to formulate models that accurately describethe majority of physical phenomena and are ubiquitous in the mathematical description ofdynamic behavior. However effective these integer order differential operators are in general,there are more complex systems that are better characterized by dynamic behavior that lies inbetween the normal integer order description. A case in point is the so-called “viscoelastic”behavior, which has characteristics of both elastic (order zero) and viscous (order one)elements. It is thus natural to assume that differential operators of noninteger order, suchas a 0.25, 0.50, or 0.75 would provide a convenient mathematical description to analyze theseintermediate behaviors. The study of these noninteger differential operators falls under thegeneral subject of what became known as Fractional Calculus, though the orders studied arenot strictly limited to rational numbers.


A further generalization of the concept of noninteger order derivatives that isapplicable to more complex systems is that of a derivative of varying order. One can findsystems where the order of dynamics associated with each element is a function of time orfrequency or position or any derivative of the position vector [1–3]. The objective of this workis to identify the most appropriate definition of a variable-order (VO) operator for modelingdynamic systems and to assign to the order of the derivative a physical meaning that willfacilitate the understanding of its use in problems of vibration and control. First, we compareall VO operator definitions recently proposed in literature in order to select a definition thatis better suited for modeling purposes. We then use the familiar example of viscoelasticharmonic oscillators to connect the order of the derivative to the dynamic properties of theoscillators.

Unlike ordinary derivatives, noninteger order derivatives are integrodifferentialoperators with either a power-law (in the case of fractional derivatives) or a variable-exponent (for VO derivatives) kernel. Thus, noninteger derivatives are nonlocal by definitionand are ideally suited for modeling systems characterized by nonlocal (or memory-laden)behavior. Multiple definitions for a fractional derivative have been proposed, and thereis no straightforward geometric or physical interpretation for the meaning of a fractionalderivative, although a few interpretations have been proposed [4]. The subject of FractionalCalculus has developed rapidly, especially since in the last four decades, with quite a fewrecent books dedicated exclusively to the subject (see, e.g., [4–8]). Some applications thatinvolve fractional derivatives include particle motion at small but finite Reynolds numbers[9–12], viscoelastic constitutive equations [13–16], transport dynamics with anomalousdiffusion [17], and fractional order control systems [4, 18].

The concept of a VO operator is a much more recent development and is less widely-known. The order of the derivative/integral is allowed to vary over the domain of interest,thus they are suitable for modeling systems with evolving dynamics. Such systems includedeformation of viscoelastic materials [19–22] and the mechanics of variable viscoelasticoscillators [1, 3]. Similar to the state of affairs in Fractional Calculus, multiple definitionsof a VO derivative have been suggested [1, 2, 19, 23, 24], each preferred for different reasons.Since the kernel of the VO operators have a variable-exponent, analytical solutions to VOdifferential equations (VODEs) are more difficult to obtain, and have not been the focus ofmuch attention. However, numerical solutions for the VODEs that have been formulatedwith the various VO operators have been developed [1–3]. In addition, rather than seekinga constant or multiple constants for the order of the derivative in the model that wouldaccurately represent the data, in the VO case a function needs to be determined throughmathematical and/or physical arguments. Previous applications of VO derivatives haveeither explicitly chosen the function [1–3] or found an approximation numerically [19, 20, 22].This complication can be lessened if the functional form of the order is determined throughphysical arguments as in [21], where a model based on statistical mechanics was developedto describe the compression of a viscoelastic material.

The aim of this work is to compare the Variable Order operator definitions that havebeen proposed and to select the operator with the characteristics that are critical for thesuccess of a dynamic model. We compare the various VO operators based on a very simplecriteria: the VO operator must return the correct fractional derivative that corresponds tothe argument of the functional order. In other words, if the argument q(t) is equal to,say, 0, 0.50 or 1, then the function itself (zero-order derivative), the half-derivative, or thefirst derivative must be returned as the output of the operator. We will see that only twodefinitions previously proposed satisfy this elementary requirement. Of the two operator


definitions that satisfy this property, one is more efficient from the numerical standpoint,and is therefore adopted in the remainder of this work. The appropriate operator then is usedto study the somewhat familiar problem of a harmonically forced oscillator with viscoelasticdamping. The goal of this second part of this work is to illustrate how a familiar problem indynamics can be used to understand the meaning of a VO operator, and to understand howthe dynamics in this familiar problem is affected by the physical parameters of the systemusing a VO analysis.

The next section presents an overview of the various VO operator definitions and abrief comparison of the VO operators applied to a harmonic and other bounded function.Subsequent to selecting the operator, we propose a VO model for the harmonically forcedoscillator with viscoelastic damping of order p (0 < p < 1) and conduct a stationary analysisthat yields a very concrete meaning to the order of the operator.

2. Variable Order Operators

The VO operator definitions that have been proposed are either direct extensions ofthe fractional calculus definitions or generalizations that arise from Laplace or Fouriertransformations. In the direct extension approach, the constant exponent in the fractionaloperator is replaced with a function. For example, a VO integral is defined in [23] as

D−q(t)c+ f(t) =1

Γ[q(t)]∫ t

c

(t − σ)q(t)−1f(σ)dσ. (2.1)

When q(t) = α = constant, then the αth-order fractional integral is recovered. Otherdefinitions can be formulated by changing the form of the argument of the exponent to beq = q(σ) or q = q(t − σ) and considering the Gamma function under the integral sign [24, 25]:

1D−q(t)0 f(t) =∫ t

0

(t − σ)q(t)−1

Γ[q(t)] f(σ)dσ, (2.2)

2D−q(t)0 f(t) =∫ t

0

(t − σ)q(σ)−1

Γ[q(σ)

] f(σ)dσ, (2.3)

3D−q(t)0 f(t) =∫ t

0

(t − σ)q(t−σ)−1

Γ[q(t − σ)

] f(σ)dσ. (2.4)

In the cases above, the lower terminal is set equal to 0, and it is assumed that f(0) = 0 fort < 0. Since (2.3) and (2.4) involve the variable of integration within the exponent, then thisimplies memory in the order, with the past states having a stronger effect on the order fordefinition (2.4) [24]. Also, the full convolution form of (2.4) enables use of the convolutionproperties to study the operator.

Similarly, a VO derivative definition can be obtained by directly substituting q = q(t)in the Riemann-Liouville fractional derivative definition [23] valid for 0 < q < 1:

Dq(t)c+ f(t) =

1Γ[1 − q(t)

]ddt

∫ t

c

f(σ)

(t − σ)q(t)dσ. (2.5)


Further definitions are obtained by taking m first-order derivatives of the integrals defined in(2.2)–(2.4)

Dq(t)0 f(t) =

dm

dtm

∫ t

0

(t − σ)m−q(t,σ)−1

Γ[q(t, σ)

] f(σ)dσ, (2.6)

where m − 1 < q(t) < m. Caputo-type VO operators are defined by taking the derivative ofthe function under the integrals

Dq(t)0 f(t) =

∫ t

0

(t − σ)m−q(t,σ)−1

Γ[q(t, σ)

] f (m)(σ)dσ, (2.7)

where f (m) denotes the mth integer order derivative of f(t). VO operators based onother fractional derivative definition forms have also been proposed. Samko and Ross [23]introduce a VO operator based on the Marchaud fractional derivative:

Dq(t)c f(t) =

f(t)

Γ[1 − q(t)

](t − c)q(t)

+q(t)

Γ[1 − q(t)

]∫ t

c

f(t) − f(σ)(t − σ)1+q(t)

dσ, (2.8)

where 0 < Re q(t) < 1.Using a different approach, Coimbra [1] begins with the Laplace transform of the

Caputo operator to obtain a VO operator definition. Treating q(t) as a running parameterand inverting back into the time domain yields the following definition valid for q(t) < 1:

Dq(t)f(t) =1

Γ[1 − q(t)

]∫ t

0+(t − σ)−q(t)f (1)(σ)dσ +

(f(0+) − f(0−)

)t−q(t)

Γ[1 − q(t)

] . (2.9)

Extension of definition (2.9) to values of q(t) larger than unity is possible as long as the higher-order derivatives are defined and integrable. Although this definition began with the Laplacetransform of the Caputo operator, it is not strictly a Caputo generalization because of theaddition of the initial condition term. As a result, Dq(0)f(0)/= 0 for any function f(t) as is thethe case for a Caputo-based operator. Soon et al. [3] show that a properly weighted sum offractional order derivatives terms approximates this single term VO operator when a largenumber of terms are used, which implies convergence of both the operator and the numericalmethod.

Although definitions (2.8) and (2.9) were defined independently through differentmethods, they are similar. After integrating (2.8) by parts and simplifying, we arrive at

1Γ[1 − q(t)

]∫ t

c

f (1)(σ)

(t − σ)q(t)dσ +

f(c)

Γ[1 − q(t)

](t − c)q(t)

. (2.10)

The difference between (2.9) and (2.10) lies in the term that is evaluated at the lower terminal.For situations in which the system is assumed to be in dynamical equilibrium for t < 0 suchthat D2f(t < 0) = 0 for any f(t), definitions (2.9) and (2.10) return the same result. However,


if f(0) is a true constant, such that f(0−) = f(0+) = a, then (2.9) would return 0 for thederivative, whereas (2.10) will not.

In summary, there are a total of nine VO operator definitions to be compared:

1Dq(t)0 f(t) =

1Γ[1 − q(t)

]ddt

∫ t

0

f(σ)

(t − σ)−q(t)dσ, (2.11)

2Dq(t)0 f(t) =

f(t)

Γ[1 − q(t)

](t)q(t)

+q(t)

Γ[1 − q(t)

]∫ t

0

f(t) − f(σ)(t − σ)1+q(t)

dσ, (2.12)

3−5Dq(t)0 f(t) =

ddt

(∫ t

0

(t − σ)q(t,σ)−1

Γ[q(t, σ)

] f(σ)dσ

)

, (2.13)

6−8Dq(t)0 f(t) =

∫ t

0

(t − σ)q(t,σ)−1

Γ[q(t, σ)

] f (1)(σ)dσ, (2.14)

9D−q(t)0 f(t) =1

Γ[1 − q(t)

]∫ t

0+(t − σ)−q(t)f (1)(σ)dσ +

(f(0+) − f(0−)

)t−q(t)

Γ[1 − q(t)

] , (2.15)

where q(t, σ) in definitions (2.13) and (2.14) signify the three arguments: q(t, σ) = q(t),q(t, σ) = q(σ), and q(t, σ) = q(t−σ). Each of the above definitions is defined for real derivativeorders between 0 and 1. The value at all the lower terminals is set to 0, since we are interestedin applying the operators to physical processes not necessarily at steady state (we examinea stationary problem in the next section). As is the case with fractional derivatives, there isno single VO derivative (or integral) definition that is widely considered to be the “correct”definition. Samko and Ross prefer definition (2.12) because the operator retains the symmetryon power functions that is found in the case of constant orders, that is:

Dq(t)c+ + (t − c)α =

Γ[α + 1]Γ[α + 1 − q(t)

](t − c)α−q(t) (2.16)

for 0 < Re q(t) < 1 and α > −1 [6]. Lorenzo and Hartley prefer the full convolution VOintegral definition (2.4) because it satisfies the index rule for certain functions [25] and istime-invariant [24].

The approach chosen here is to determine which operator when acting upon afunction returns the fractional derivative of the function at the corresponding time. This isan important characteristic from the aspect of physical modeling because it signifies that theoperator yields a continuous transition of all orders of differentiation between integer orders.Thus, a smooth transition from zero-order dynamics to first-order dynamics is possible. Fora qualitative comparison of the the VO operators, we look at the q(t) = t derivative oftwo bounded functions: sin(2πt), and erfc(t). The q(t) = t derivative of both functions iscomputed numerically using a product trapezoidal rule to evaluate the convolution integrals[3, 26]. Plots of the t derivative of sin(2πt) and erfc(t) are shown in Figures 1-2. The 0, 0.25,0.50, 0.75, and 1st-order derivatives of the sinusoidal function are shown for comparisonsince both the Riemann-Liouville and Caputo fractional derivatives return the same result.We show only the 0- and 1st-order derivatives of the erfc function since the different


−4

−2

0

2

4

6Dtsi

n(2π

t)

0 0.2 0.4 0.6 0.8 1

t

(a)

−2

0

2

4

6

Dtsi

n(2π

t)

0 0.2 0.4 0.6 0.8 1

t

(b)

−4

−2

0

2

4

6

Dtsi

n(2π

t)

0 0.2 0.4 0.6 0.8 1

t

(c)

Figure 1: Plots of the t derivatives of sin(2πt). The points are the α = 0, 0.25, 0.50, 0.75, and 1st-orderfractional derivatives at t = α. (a) Definition (2.13) with q(t, σ) = q(t) (thin line), q(t, σ) = q(σ) (mediumline), and q(t, σ) = q(t − σ) (thick line). Note that none of the definitions match the 1st-order derivative att = 1. (b) Caputo-type VO operators (2.14) with q(t, σ) = q(t) (thin line), q(t, σ) = q(σ) (medium line), andq(t, σ) = q(t − σ) (thick line). Note that the variant of (2.14) with argument t matches the corresponding αfractional derivatives at t = α. (c) Definition (2.11) (thick line) and Coimbra’s operator (2.15) (thin line).The t-derivative defined by (2.15) is equivalent to the corresponding fractional derivatives at all the points.

fractional derivative definitions yield different results. Also, note that Samko’s Marchaud-based definition and Coimbra’s definition coincide for these conditions, and therefore onlydefinition (2.15) is plotted.

As expected, the Caputo-based VO operators have values of 0 at t = 0, similar tothe fractional derivative case. They do not return the zeroth-order derivative at t = 0 forany functions that deviate from this (see Figure 2). Definition (2.14) is equivalent to (2.15)for the sine function, since f(t) is continuous at t = 0 and q(t) is only a function of t. TheRiemann-Liouville and Caputo type operators with the argument (t − σ), also are shown to


−1.5

−1

−0.5

0

0.5

1

1.5Dt e

rfc(t)

0 0.2 0.4 0.6 0.8 1

t

(a)

−1

−0.5

0

0.5

1

Dt e

rfc(t)

0 0.2 0.4 0.6 0.8 1

t

(b)

−0.5

0

0.5

1

1.5

2

Dt e

rfc(t)

0 0.2 0.4 0.6 0.8 1

t

(c)

Figure 2: Plots of the t derivatives of erfc(t). The points are the 0- and 1st-order derivatives at t = 0 and t = 1.Also shown for comparison is the t derivative from Coimbra’s operator (2.15) (dashed line). (a) Riemann-Liouville type operator (2.13) with q(t, σ) = q(t) (thin line), q(σ) (medium line), q(t − σ) (thick line). (b)Caputo-type definition (2.14) with q(t, σ) = q(t) (thin line), q(σ) (medium line), q(t − σ) (thick line). Inthis case the operator (2.14) with exponent q(t) does not match Coimbra’s operator, but is equivalent tothe first derivative at t = 1. (c) Definition (2.11) (thin line).

be equivalent because f(0) = 0. This is analogous to the similarity of the Riemann-Liouvilleand Caputo fractional derivatives of functions when f (0)(0) = 0 for 0 ≤ q < 1 [4]. Definitions(2.12) and (2.15) are the only operators that have the desirable property of returning thecorresponding qth order fractional derivative of x(t) when q(t) = p for both the sine and erfcfuncitons. However, the convergence of (2.12) to that of (2.15) is slower due to the strongersingularity that must be evaluated in the convolution integral. Also, in the case of a trueconstant function f(t) where f(0−) = f(0+), (2.12) would not return 0 as the derivative, so weconclude that definition (2.15) is preferable for modeling dynamic systems.


Note that the operator defined in (2.15) is dynamically consistent with the causalbehavior of the initial conditions. In other words, when x(t) is a true constant from −∞ tothe initial time (t = 0+), the operator in (2.15) returns zero for all values of q(t). However,if f(t) is not continuous between t = 0− and t = 0+, the operator returns the appropriateHeaviside contribution to the integral value of Dq(t)f(t). In accordance with this causaldefinition, we take the value of the physical variable f(t) to be identically null from −∞to 0− as a representation of dynamic equilibrium. A nonzero initial condition is treated as aHeaviside function at t = 0, and therefore included in the second term of the definition of theoperator (2.15).

Through the direct comparison of the various proposed VO operator definitions, weselected the operator that has fundamental characteristics that are desirable for physicalmodeling. Definition (2.15) proposed in [1] represents a continuous transition between theinteger order derivatives, and returns a zero value for the derivative of a function that isconstant from −∞ < t < ∞; so we select it as the most appropriate definition. Now withthe chosen operator, we proceed to connect the behavior of a VO operator with a physicalquantity that is characteristic of a selected memory-laden system.

3. Stationary Analysis for Viscoelastic Oscillators

One of the drawbacks of VO modeling is that to date there is no clear physical understandingof what a VO derivative represents. The objective of this section is to illustrate how a variableorder differential operator may be used to understand a familiar problem: the stationaryanalysis of a constant order viscoelastic oscillator. We will use a single VO operator of orderq to replace multiple terms of constant order differential operators, including the viscoelasticterm of constant order p. We seek an analytical expression for q to examine the effects of theparameters of the system on the dynamics of the oscillator. The exact expression for q(ω)is obtained from the stationary analysis of the problem. In order words, we look for twofunctions q that would allows us to replace the multiterm differential equation describingthe steady motion of the oscillator with a single-term variable order equation. Note thatthe objective of this section is not to rehash the analysis of the constant order viscoelasticoscillator, rather we seek to find meaning for a VO operator by comparing its order with thephysical parameters in a well-known problem. The reader who is interested in the details ofthe dynamics of constant order viscoelastic oscillators should consult [4, 27, 28].

The equation of motion for the constant order viscoelastic oscillator is

mD2x(t) + cDpx(t) + kD0x(t) = F(t), (3.1)

where F(t) = F0 cos(Ω t). When p = 1, the system is an oscillator with viscous damping, andfor 0 < p < 1, the system is said to be characterized by viscoelastic damping. The equation isrecast in dimensionless form using the following scaled parameters:

x =x

Lc, t = tωn, Ω =

Ωωn

, (3.2)


where Lc is a characteristic length (or amplitude of the motion) and ωn =√k/m is the

undamped natural frequency of the system. The dimensionless equation of motion is

D2 x + ξ Dp x + x = Fo cos(Ω t), (3.3)

where F0 = F0/m ω2n Lc is the dimensionless amplitude of the forcing, and ξ is the damping

ratio and is

ξ =ckp/2−1

mp/2. (3.4)

Since we are concerned only with stationary behavior, η and q are not functions oftime, so we look for a relationship such as

η(p, Ω, ξ

)Dq(p,Ω,ξ )x =

(D2 + ξDp +D0

)x = Re

(F0 exp

(iΩt)). (3.5)

The stationary VO derivative of eiΩ t is

Dq(p,Ω,ξ)AeiΩt =(

i Ω)q(p,Ω,ξ)

AeiΩt, (3.6)

where definition (2.15) with a lower terminal of −∞ is used since we are dealing with astationary problem where the initial conditions are irrelevant. Rewriting (3.6) with (3.3) asthe stationary solution exp(iΩt) yields

η(

iΩ)q (p,Ω,ξ)

= 1 − Ω2 +(

iΩ)pξ. (3.7)

We now equate the real and imaginary parts of the above equation to arrive at

q(p, Ω, ξ

)= tan−1

⎛

⎜⎝

ξ Ωp sin((π/2)p

)

(1 − Ω2

)+ Ωpξ cos

((π/2)p

)

⎞

⎟⎠

2π,

η(p, Ω, ξ

)=

1

Ωq

√(1 − Ω2

)2+(Ωpξ)2

+ 2(

1 − Ω2)Ωpξ cos(π/2)p.

(3.8)


The stationary solution for (3.5) can be written as x(t) = A cos(Ω t − φ) where the amplitudeA and phase shift φ are

A =F0

√(1 − Ω2

)2+(Ωpξ)2

+ 2(

1 − Ω2)Ωpξ cos(π/2)p

,

φ = tan−1

⎛

⎜⎝

ξΩp sin((π/2)p

)

(1 − Ω2

)+ Ωpξ cos

((π/2)p

)

⎞

⎟⎠.

(3.9)

The procedure used to obtain the functional form for q and η is easily extendable to systemsthat consist of multiple viscoelastic terms. For n terms, the expression for q is

q = tan−1

⎛

⎜⎝

∑nk=1 ξkΩ

pk sin(π/2)pk(

1 − Ω2)+∑n

k=1 ξkΩpk cos(π/2)pk

⎞

⎟⎠

2π. (3.10)

Similarly, the expression for η is

η =1

Ωq

√√√√(

1 − Ω2 +n∑

k=1

ξkΩpk cosπ

2pk

)2

+

(n∑

k=1

ξkΩpk sinπ

2pk

)2

. (3.11)

Equations (3.8)-(3.9) clearly show that q represents a scaled phase shift and η isthe scaled ratio of the amplitude of the forcing to the response. Thus, for the stationarysolution of the oscillator the order of the derivative is connected to a physical quantitythat is characteristic of the system. The multiterm equation in frequency that represents thestationary motion of a viscoelastic operator can be replaced by a single term parametricoperator in frequency, where both the order of the derivative and the scaling function ηhave physical meaning. A phase shift of π/2 implies that the response is proportional tothe velocity and hence to the 1st-order derivative, while a phase shift of π implies that it isproportional to the acceleration or to the 2nd-order derivative. The order of the VO operatorthat captures the whole dynamics of the systems is thus naturally connected with the phaseshift between the response and the forcing.

Plots of q and η versus Ω for damping orders of p = 0.25, 0.50, 0.75, and 1 and variousvalues of the damping ratio are shown in Figures 3, 4, 5, and 6. Also shown are the maps of qand η.

The expression for q reveals regions in which the three terms of the original equation ofmotion are dominant. For example, regions in which the order of the derivative is near 2 (suchas systems with lower damping ratio and higher forcing frequency) are primarily dominatedby the inertial term. Lower values of damping ratio and orders of viscoelastic damping reachthe asymptotic value more quickly, suggesting that the change in the dynamics (and alsothe phase shift) is more sensitive to changes in frequency in those cases. The dependence of


0

0.5

1

1.5

2

q

0 1 2 3 4 5∼

Ω

(a)

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0

0.5

1

1.5

2

2.5

3

ξ n

0 0.5 1 1.5 2 2.5 3∼

Ω

(b)

0

0.5

1

1.5

2

2.5

η

0 1 2 3 4 5∼

Ω

(c)

0.5

1

.5

.5

1.5

2

2.5

3 3.5

0

0.5

1

1.5

2

2.5

3ξ n

0 0.5 1 1.5 2 2.5 3∼

Ω

(d)

Figure 3: Plots of q and η for p = 0.25. (a) q(Ω, ξ) versus Ω for ξ = 0.1, 0.5, 1, 1.5, and 2 (going from light todark). Note that q aysmptotes to 2 as Ω increases. (b) Map of q(Ω, ξ) where lower values are representedby darker colors. When Ω = 1, q = 0.25 for any value of ξ. (c) η(Ω, ξ) versus Ω for ξ = 0.1, 0.5, 1, 1.5, and2 (going from light to dark). (d) Map of η(Ω, ξ). The lowest values of η occur near Ω = 1 and for smalldamping ratio.

the order of the derivative on the damping ratio also changes when Ω = 1, correspondingto the case where the driving frequency is the same as the natural frequency of the system.When Ω < 1, then the systems with higher damping ratios have a higher value for q. OnceΩ > 1 then the behavior is switched with the systems with higher damping ratio having lowervalues for the order. For all cases, when Ω = 1, then q = p for any damping ratio.

A scaled behavior identical to q is shown in the plots of η. The normalized amplituderatio reaches an asymptotic value of η = 1 as Ω increases. Thus, at higher frequencies the


0

0.5

1

1.5

2

q

0 1 2 3 4 5∼

Ω

(a)

0.2

0.4

0.8

1

1.20.6

1.4

1.61.8

0

0.5

1

1.5

2

2.5

3

ξ n

0 0.5 1 1.5 2 2.5 3∼

Ω

(b)

0

0.5

1

1.5

2

2.5

η

0 1 2 3 4 5∼

Ω

(c)

0.5

12

2.5

33.5

1.5

0

0.5

1

1.5

2

2.5

3ξ n

0 0.5 1 1.5 2 2.5 3∼

Ω

(d)

Figure 4: Plots of q and η for p = 0.50. (a) q(Ω, ξ) versus Ω for ξ = 0.1, 0.5, 1, 1.5, and 2 (going from lightto dark). The asymptotic behavior is similar to the case when p = 0.25. (b) Map of q(Ω, ξ) where lowervalues are represented by darker colors. When Ω = 1, q = 0.50 for any value of ξ. (c) η(Ω, ξ) versus Ω forξ = 0.1, 0.5, 1, 1.5, and 2 (going from light to dark). (d) Map of η(Ω, ξ).

amplitude ratio is proportional to Ω−p, and the order of the damping and the damping ratiodo not have any effect on the amplitude of the stationary motion. Similar to the case for q, ηapproaches the asymptotic value more quickly for small viscoelastic damping orders p andlower damping ratios. Also for the cases with damping order p < 1, the peak amplituderesponse shifts to higher frequencies for increasing damping ratio. For increasing p anddamping ratio, the peak response begins to flatten out. The maps of η show that as the orderof the viscoelastic damping increases, the regions where the amplitude of the response isdamped increase.


0

0.5

1

1.5

2

q

0 1 2 3 4 5∼

Ω

(a)

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0

0.5

1

1.5

2

2.5

3

ξ n

0 0.5 1 1.5 2 2.5 3∼

Ω

(b)

0

0.5

1

1.5

2

η

0 1 2 3 4 5∼

Ω

(c)

0.5

1

1.5 2

2.5

3

0

0.5

1

1.5

2

2.5

3ξ n

0 0.5 1 1.5 2 2.5 3∼

Ω

(d)

Figure 5: Plots of q and η for p = 0.75. (a) q(Ω, ξ) versus Ω for ξ = 0.1, 0.5, 1, 1.5, and 2 (going from lightto dark). (b) Map of q(Ω, ξ) where lower values are represented by darker colors. When Ω = 1, q = 0.75for any value of ξ. (c) η(Ω, ξ) versus Ω for ξ = 0.1, 0.5, 1, 1.5, and 2 (going from light to dark). (d) Map ofη(Ω, ξ).

4. Conclusions

This work advances our understanding of the use of variable order (VO) differentialoperators in dynamics in two substantial ways. First, we compare several definitionsproposed in literature, and select the most suitable definition based on a few criteria: (1) theVO operator must be able to return all intermediate values between 0 and 1 that correspondto the argument of the order of differentiation, (2) the VO operator must be effectivelyevaluated numerically, and (3) all derivatives of a true constant (a function that is constantfrom −∞ to +∞) must be zero. The operator defined in (2.15) satisfies these criteria formodeling dynamic systems. We then proceed to illustrate the meaning of a variable order


0

0.5

1

1.5

2

q

0 1 2 3 4 5∼

Ω

(a)

0.2

0.4 1.4

1.60.6

0.8

1

1.2

1.8

0

0.5

1

1.5

2

2.5

3

ξ n

0 0.5 1 1.5 2 2.5 3∼

Ω

(b)

0

0.5

1

1.5

2

η

0 1 2 3 4 5∼

Ω

(c)

0.5

1

22.5

1.5

0

0.5

1

1.5

2

2.5

3ξ n

0 0.5 1 1.5 2 2.5 3∼

Ω

(d)

Figure 6: Plots of q and η for p = 1, signifying the oscillator with viscous damping. (a) q(Ω, ξ) versus Ω forξ = 0.1, 0.5, 1, 1.5, and 2 (going from light to dark). (b) Map of q(Ω, ξ) where lower values are representedby darker colors. When Ω = 1, q = 0.75 for any value of ξ. (c) η(Ω, ξ) versus Ω for ξ = 0.1, 0.5, 1, 1.5, and 2(going from light to dark). (d) Map of η(Ω, ξ).

of differentiation by analyzing a familiar dynamical problem (the stationary analysis of aviscoelastic oscillator). We determine that the order of differentiation for a single operatordescribing all dynamic elements in the stationary equation of motion (mass, damping andspring) is equal to the normalized phase shift. The normalization is easily understood as thequantity that transforms the maximum phase shift between acceleration and position (π)into the maximum difference between the order of acceleration (2) and the order of position(0). The normalization constant is thus 2/π , and the variable order differentiation is seenas being just the normalized phase shift in this problem, which gives us a straightforwardinterpretation of the meaning of variable orders of differentiation in dynamic systems.


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[2] D. Ingman and J. Suzdalnitsky, “Control of damping oscillations by fractional differential operatorwith time-dependent order,” Computer Methods in Applied Mechanics and Engineering, vol. 193, no. 52,pp. 5585–5595, 2004.

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