RESEARCH PAPER ON FRACTIONAL LYAPUNOV EXPONENT...

RESEARCH PAPER

ON FRACTIONAL LYAPUNOV EXPONENT

FOR SOLUTIONS OF LINEAR

FRACTIONAL DIFFERENTIAL EQUATIONS

Nguyen Dinh Cong 1, Doan Thai Son 1,2, Hoang The Tuan 1

Abstract

Our aim in this paper is to investigate the asymptotic behavior of solu-tions of linear fractional differential equations. First, we show that the clas-sical Lyapunov exponent of an arbitrary nontrivial solution of a boundedlinear fractional differential equation is always nonnegative. Next, usingthe Mittag-Leffler function, we introduce an adequate notion of fractionalLyapunov exponent for an arbitrary function. We show that for a linearfractional differential equation, the fractional Lyapunov spectrum whichconsists of all possible fractional Lyapunov exponents of its solutions pro-vides a good description of asymptotic behavior of this equation. Con-sequently, the stability of a linear fractional differential equation can becharacterized by its fractional Lyapunov spectrum. Finally, to illustratethe theoretical results we compute explicitly the fractional Lyapunov ex-ponent of an arbitrary solution of a planar time-invariant linear fractionaldifferential equation.

MSC 2010 : Primary 34A08; Secondary 34D08, 34D20

Key Words and Phrases: fractional calculus, linear fractional differen-tial equations, Mittag-Leffler type functions, Lyapunov exponent, stability

c© 2014 Diogenes Co., Sofiapp. 285–306 , DOI: 10.2478/s13540-014-0169-1

286 N.D. Cong, T.S. Doan, H.T. Tuan

1. Introduction

In the recent years, the fractional differential equations have attractedincreasing interests due to the fact that many mathematical problems in sci-ence and engineering are represented by mathematical models of fractionalorder, see e.g. [8, 18, 21]. Along with this interest, researchers’ enthusiasmis addressed to understanding the qualitative behavior of their solutions,i.e. the behavior of the solutions when the time tends to infinity, such asstability theory [7, 12], linear theory [3, 16], invariant manifold theory [4],linearization theory [15], etc.

It is well known that the Lyapunov exponent provides a powerful toolto investigate the dynamical behaviors of linear ordinary differential equa-tions, see e.g. [1, 17]. More precisely, our state space can be decomposedas a direct sum of subspaces in which each subspace is given as the setof all solutions corresponding to a characteristic Lyapunov exponent. Thedimensions of these subspaces are called multiplicities of (the correspond-ing) Lyapunov exponents. The Lyapunov exponents together with theirmultiplicities form the Lyapunov spectrum of this system. The qualitativebehavior of this linear system, such as stability and hyperbolicity, can becharacterized adequately by its Lyapunov spectrum.

For linear fractional differential equations, the Lyapunov exponent isfirst discussed in [14] in which the authors use the asymptotic behavior (incomparison with the exponential function) of the nonsingular eigenvaluesof the fundamental matrix to define the Lyapunov spectrum. This notionof the Lyapunov spectrum is used to investigate the chaotic behavior in aclass of fractional differential systems, see e.g. [5, 13, 20].

In this paper, we first show in Subsection 3.1 that the classical Lya-punov exponents of solutions of linear fractional differential equations arealways nonnegative. Therefore, the stability of a linear fractional differ-ential can not be characterized adequately by this Lyapunov spectrum.Next, in Subsection 3.2 we provide a new notion of fractional Lyapunovexponent by comparing the solutions with the Mittag-Leffler exponentialfunction. This notion has an advantage that the stability of a linear frac-tional differential equation can be characterized by this spectrum. The lastSection 4 is devoted to time-invariant linear planar systems; we computeexplicitly the fractional Lyapunov spectrums, thus have an illustration ofthe theoretical results in Section 3.

To conclude this introductory section, we introduce notations which areused throughout this paper. Let R≥0,R>0 and R<0 denote the set of allnonnegative, positive and negative real numbers, respectively. For z ∈ C,let <z and =z denote the real part and imaginary part of z, respectively.

RESEARCH PAPER

NUMERICAL SOLUTION OF FRACTIONAL

STURM–LIOUVILLE EQUATION IN INTEGRAL FORM

Tomasz Blaszczyk 1, Mariusz Ciesielski 2

Abstract

In this paper a fractional differential equation of the Euler–Lagrange/Sturm–Liouville type is considered. The fractional equation with deriva-tives of order α ∈ (0, 1] in the finite time interval is transformed to theintegral form. Next the numerical scheme is presented. In the final partof this paper examples of numerical solutions of this equation are shown.The convergence of the proposed method on the basis of numerical resultsis also discussed.

MSC 2010 : Primary 26A33; Secondary 34A08, 65L10Key Words and Phrases: fractional Euler–Lagrange equation, fractional

Sturm–Liouville equation, fractional integral equation, numerical solution

1. Introduction

The fractional differential equations, both ordinary and partial ones,are very useful tools for modelling many phenomena in physics, mechanics,control theory, biochemistry, bioengineering and economics [10, 12, 21, 22,24, 38]. Therefore, the theory of fractional differential equations is an areathat has developed extensively over the last decades. In the monographs[13, 14, 16, 30, 31, 11] one can find a review of methods of solving fractionaldifferential equations.


308 T. Blaszczyk, M. Ciesielski

In the recent years, a subtopic of the theory of fractional differentialequations gains importance: it concerns the variational principles for func-tionals involving fractional derivatives. These principles lead to equationsknown in the literature as the fractional Euler–Lagrange equations. Theequations of this type were derived when fractional integration by partsrule [13] has been applied.

This approach was initiated by Riewe in [36], where he used non-integerorder derivatives to describe nonconservative systems in mechanics. Next,Klimek [15] and Agrawal [1] noticed that such equations can be investigatedin the sequential approach. A fractional Hamiltonian formalism for thecombined fractional calculus of variations was introduced in [27]. In thework [29] Green theorem for generalized partial fractional derivatives wasproved. Other applications of fractional variational principles are presentedin [2, 19, 25, 26, 28].

Recently, the fractional Sturm–Liouville problems were formulated byKlimek and Agrawal in [18] and Rivero et al. in [37]. The authors in thesepapers considered several types of the fractional Sturm–Liouville equationsand investigated the eigenvalues and eigenfunctions properties of the frac-tional Sturm–Liouville operators.

Unfortunately, the fractional Euler–Lagrange/Sturm–Liouville equationscontain the composition of the left- and right-sided derivatives. It is an ad-ditional drawback for computation of an exact solution (even with simpleLagrangian, see [5, 16, 17]). Consequently, numerous studies have been de-voted to numerical schemes for the fractional equations (see [4, 6, 8, 23, 39]).For numerical methods in the fractional calculus of variations we refer thereader to [33, 34, 35].

In our previous works [7, 8, 9] we proposed numerical scheme on thebasis of a finite difference method of solution for a special case of the prob-lem, namely the fractional oscillator equation. In this paper we proposea numerical solution of the fractional Sturm–Liouville equation. We in-vestigate a new integral form of this equation and a numerical method ofsolution of considered equation in conjunction with analysis of a rate ofconvergence. Another integral form of the fractional Euler-Lagrange equa-tions (containing the Caputo derivatives) has been recently considered in[20].

2. Statement of the problem and definitions

We consider the fractional differential equation with derivatives of orderα ∈ (0, 1] in the finite time interval t ∈ [0, b], for parameter λ ∈ R andvariable potential determined by function q (t)

CDαb− Dα

0+ f (t) + (λ + q (t)) f (t) = 0, (2.1)

RESEARCH PAPER

OPTIMAL RANDOM SEARCH, FRACTIONAL DYNAMICS

AND FRACTIONAL CALCULUS

Caibin Zeng 1, YangQuan Chen 2

Abstract

What is the most efficient search strategy for the random located targetsites subject to the physical and biological constraints? Previous resultssuggested the Levy flight is the best option to characterize this optimalproblem, however, which ignores the understanding and learning abilitiesof the searcher agents. In this paper we propose the Continuous Time Ran-dom Walk (CTRW) optimal search framework and find the optimum forboth of search length’s and waiting time’s distributions. Based on frac-tional calculus technique, we further derive its master equation to show themechanism of such complex fractional dynamics. Numerous simulationsare provided to illustrate the non-destructive and destructive cases.

MSC 2010 : Primary 26A33; Secondary 82b41, 34A08, 49KxxKey Words and Phrases: random search, fractional dynamics, continu-

ous time random work, fractional calculus, Levy flight

1. Introduction

Over the recent years the accumulating experimental evidences showthat the moving organisms are ubiquitous. For instance, the foraging be-havior of the wandering albatross (Diomedea exulans) on the ocean surfacewas found to obey a power-law distribution [19]; the foraging patterns offree-ranging spider monkeys (Ateles geoffroyi) in the forests of the YucatanPeninsula was also found to be a power law tailed distribution of steps


322 C. Zeng & Y.Q. Chen

consistent with a Levy walk [12, 2]. More experimental findings can befound in [20, Part II]. A number of foundational but important questionsarise naturally: How to model these organisms’ movement trajectories?What factors determine the shape and the statistical properties of suchtrajectories? How to optimize the efficiency to search of randomly locatedtargets? These questions have been studied from many different points ofview. For example, Levy flight search was claimed to be an optimal strat-egy in sparsely target site with an inverse square power-law distributionof flight lengths [18]. Then composite Brownian walk searches were foundto be more efficient than any Levy flight when searching is non-destructiveand when the Levy walks are not responsive to conditions found in thesearch [1]. In particular, the movement patterns have scale-free and super-diffusive characteristics. So the fractional Brownian motions and fractionalLevy motions are possible to account for the movement patterns [13]. How-ever, the above strategies ignore one important factor, waiting time betweenthe successive movement steps, since the search agents need some time tounderstand the visited target sites. The relation between waiting time andflight length for efficient search was discussed [5].

This fact inspires us to propose a potentially transformative frameworkfor optimal random search based on continuous time random walk (CTRW).The CTRW strategy is composed of the flight lengths of a movement stepwith a random direction, as well as the waiting time elapsing between twosuccessive movement steps, both of which are independent random vari-ables, identically, distributed according to their probability densities. Inaddition, CTRW is also the stochastic solution of non-integer order dif-fusion equation based on the fractional calculus [21], which is a part ofCalculus dealing with derivatives and integration of arbitrary order, e.g.[11]. Different from the analytical results on linear integer-order differen-tial equations, which are represented by the combination of exponentialfunctions, the analytical results on the linear fractional order differentialequations are represented by the Mittag-Leffler function, which exhibits apower-law asymptotic behavior, [9]. Therefore, fractional calculus is beingwidely used to analyze the random signals with power-law distributions orpower-law decay of correlations, [15]. By choosing the flight lengths sub-ject to heavy-tailed distribution and finite characteristic waiting time, theCTRW encompasses the Levy flight as a special case. Therefore, in thispaper we propose the CTRW optimal search framework, which may providenew insights into the optimal random search in unpredictable environments.

The paper is organized of as follows: In Section 2 we review the Levyflight optimal random search strategy. Then we formulate the CTRW op-timal random search framework and find the optimum for both of search

RESEARCH PAPER

NONLINEAR MULTI-ORDER FRACTIONAL

DIFFERENTIAL EQUATIONS WITH PERIODIC/

ANTI-PERIODIC BOUNDARY CONDITIONS

Sangita Choudhary 1, Varsha Daftardar-Gejji 2

Abstract

In the present manuscript we analyze non-linear multi-order fractionaldifferential equation

L(D)u(t) = f(t, u(t)), t ∈ [0, T ], T > 0,

where

L(D) = λncDαn + λn−1

cDαn−1 + · · ·+ λ1cDα1 + λ0

cDα0 ,

λi ∈ R (i = 0, 1, · · · , n), λn 6= 0, 0 ≤ α0 < α1 < · · · < αn−1 < αn < 1,

and cDα denotes the Caputo fractional derivative of order α. We find theGreens functions for this equation corresponding to periodic/ anti-periodicboundary conditions in terms of the two-parametric functions of Mittag-Leffler type. Further we prove existence and uniqueness theorems for thesefractional boundary value problems.

MSC 2010 : Primary 26A33; Secondary 33E12, 34A08, 34K37, 35R11

Key Words and Phrases: fractional calculus, multi order Mittag-Lefflerfunctions, fractional differential equations, periodic/ anti-periodic bound-ary conditions


334 S. Choudhary, V. Daftardar-Gejji

1. Introduction

Fractional Calculus deals with differentiation and integration of arbi-trary orders. Fractional order derivatives provide a new approach for mod-elling many complex phenomena in Physics, Chemistry, Biology and Engi-neering sciences [8],[13], especially when dealing with memory effects. Com-pared to integer order models, the fractional order models provide betterdescription of underlying processes. Fractional order differential equations,therefore, have attracted attention of researchers and analysis of fractionaldifferential equations has widely been studied during last decade [6]-[8], [12]-[14]. The existence of periodic solutions is one of the important aspects ofdifferential systems. Several results in this context have been reported inthe literature. In this respect, Kaslik and Sivasundaram [10] have recentlyproved the non-existence of exact periodic solutions in a wide class of frac-tional order dynamical systems. Extensive work has been carried on linear/non linear boundary value problems for fractional differential equations, e.g.[7], [9], [16]. Belmekki et. al [4] have introduced and studied a new con-cept of periodic boundary conditions in the context of Riemann-Liouvillederivatives. Existence results for non-linear fractional differential equationswith integral boundary conditions [5] and anti-periodic fractional bound-ary conditions [1], [15] have been investigated. For partially ordered metricspaces, Baleanu et. al [2] recently have proved existence and uniqueness ofsolutions in case of non-linear fractional differential equations.

In this article we analyze solutions for the multi-order fractional differ-ential equations. This equation is a generalization of the classical relaxationequation, and governs some fractional relaxation processes. In this regard,Bazhlekova [3] has studied linear (non homogeneous) initial value problemand derived fundamental solution [corresponding to f(t) = 0, u(0) = 1] andimpulse response solution [f(t) = δ(t), u(0) = 0]. Further asymptotic prop-erties of these solutions are studied. In the present paper we consider anonlinear case along with periodic and anti-periodic boundary conditions.We find the Greens functions in terms of multi-order Mittag-Leffler func-tions. Further we prove existence and uniqueness theorems for solutionsunder various conditions.

The organization of the present work is as follows. In Section 2, pre-liminaries and notations are presented. In Section 3 we derive the Greenfunction for multi-order fractional differential equations followed by exis-tence and uniqueness theorems for the fractional boundary value problems.

2. Preliminaries

In this section, we introduce notations, definitions and preliminarieswhich are used in the paper [8, 13].

RESEARCH PAPER

A FULLY HADAMARD TYPE INTEGRAL BOUNDARY

VALUE PROBLEM OF A COUPLED SYSTEM

OF FRACTIONAL DIFFERENTIAL EQUATIONS

Bashir Ahmad 1, Sotiris K. Ntouyas 2,∗

Abstract

This paper is concerned with the existence and uniqueness of solutionsfor a coupled system of Hadamard type fractional differential equations andintegral boundary conditions. We emphasize that much work on fractionalboundary value problems involves either Riemann-Liouville or Caputo typefractional differential equations. In the present work, we have considered anew problem which deals with a system of Hadamard differential equationsand Hadamard type integral boundary conditions. The existence of solu-tions is derived from Leray-Schauder’s alternative, whereas the uniquenessof solution is established by Banach’s contraction principle. An illustrativeexample is also included.

MSC 2010 : Primary 34A08; Secondary 34A12, 34B15

Key Words and Phrases: Hadamard fractional derivative, fractionaldifferential systems, integral boundary conditions, fixed point theorems

1. Introduction

In this paper, we study a coupled system of Hadamard type fractionaldifferential equations and integral boundary conditions given by


A FULLY HADAMARD TYPE INTEGRAL BOUNDARY . . . 349

cDαu(t) = f(t, u(t), v(t)), 1 < t < e, 1 < α ≤ 2,

cDβv(t) = g(t, u(t), v(t)), 1 < t < e, 1 < β ≤ 2,

u(1) = 0, u(e) = Iγu(σ1) =1

Γ(γ)

∫ σ1

1

(log

σ1

s

)γ−1 u(s)s

ds,

v(1) = 0, v(e) = Iγv(σ2) =1

Γ(γ)

∫ σ2

1

(log

σ2

s

)γ−1 v(s)s

ds,

(1.1)

where γ > 0, 1 < σ1 < e, 1 < σ2 < e, D(·) is the Hadamard fractionalderivative of fractional order, Iγ is the Hadamard fractional integral oforder γ and f, g : [1, e]× R× R→ R are continuous functions.

Here we remark that the problem (1.1) is new in the context of Hadamardfractional differential equations and Hadamard nonlocal integral boundaryconditions. To the best of the authors’ knowledge, it is the first paper ad-dressing the given problem. We show the existence of solutions for the prob-lem (1.1) by applying some standard fixed point principles. Our method ofproof is the standard one yet its application in the framework of the presentproblem is new. Section 2 contains some basic concepts and an auxiliarylemma-an important result for establishing our main results. In Section 3,we present the main results.

Fractional calculus is the field of mathematical analysis, which dealswith the investigation and applications of integrals and derivatives of anarbitrary order. Fractional differential equations arise in the mathemati-cal modeling of systems and processes occurring in many engineering andscientific disciplines such as physics, chemistry, aerodynamics, electrody-namics of complex medium, polymer rheology, economics, control theory,signal and image processing, biophysics, blood flow phenomena, etc. [15],[16], [19], [20]. Fractional order differential equations are also regarded as abetter tool for the description of hereditary properties of various materialsand processes than the corresponding integer order differential equations.With this advantage, fractional-order models have become more realisticand practical than the corresponding classical integer-order models. Forsome recent development on the topic, see [1], [2], [4], [5], [6], [7], [8], [9],[10], [11], [12], [14], [17], and the references therein. The study of coupledsystems of fractional order differential equations is also very significant assuch systems appear in a variety of problems of applied nature, especially inbiosciences. For details and examples, the reader is referred to the papers[3], [18], [21], [22], [23] and the references cited therein.

DISCUSSION SURVEY

TOWARDS A GEOMETRIC INTERPRETATIONOF GENERALIZED FRACTIONAL INTEGRALS –

ERDELYI-KOBER TYPE INTEGRALS ON RN ,AS AN EXAMPLE

Richard Herrmann

Abstract

A family of generalized Erdelyi-Kober type fractional integrals is inter-preted geometrically as a distortion of the rotationally invariant integralkernel of the Riesz fractional integral in terms of generalized Cassini oval-oids on RN . Based on this geometric point of view, several extensions arediscussed.

Key Words and Phrases: fractional calculus, Riesz fractional integrals,Erdelyi-Kober fractional integrals, generalized fractional calculus, Cassiniovaloids

1. Introduction

For a long time, the open question if there exists any geometrical orphysical interpretation of the operators of the (classical) fractional calcu-lus (FC) stayed as a challenge and was discussed at the first conferencesdedicated to that topic, as at the University of New Haven - USA (1974),University of Strathclyde, Ross Priory - Scotland (1984), Nihon University,Tokyo - Japan (1989), etc. Then the experts’ hypotheses were pessimistic,and the fractional calculus was still considered as an elegant but exotictheory with the idea of extending Calculus.


362 R. Herrmann

However, at the end of the past century, more and more examples havebeen collected for useful applications of this theory to solving practicalproblems in various areas of natural sciences, engineering, control theory,economics, biomedicine, etc. The mathematicians, physicists, chemists, en-gineers, etc. started to talk each other using the language of differentiationand integration of arbitrary non-integer (i.e. fractional) order and modelsbased on FC as the better and more adequate description of the real worldand life phenomena. The discussions turned the trend and suggested pos-itive answers to the mentioned open problem. The fractal geometry andthe self-similar processes became also a popular topic related to FC, as inNigmatullin and Baleanu [10], etc. See for example, the notes taken duringa round-table discussion held at the international conference ”TMSF ’96”,Varna - Bulgaria, Kiryakova [5]. Several papers followed shortly after thatto confirm the optimistic trends, as the popular paper by Podlubny [13],and continue to appear also recently.

In the following we want to present a geometric approach that couldhelp for a deeper understanding of the concepts and strategies used inthe generalized fractional calculus, based on the Erdelyi-Kober operators(Erdelyi [1], Kober [6], Sneddon [17], etc.) and on compositions of suchoperators written either by means of repeated (multiple) integrals or bythe use of special functions (Meijer’s G- and Fox’s H-functions) as kernels,Kiryakova [4].

We will collect arguments in support of the idea that a generalization ofthe fractional calculus may be considered from a geometrical point of viewas a distortion of the isotropic kernel commonly used in standard fractionalcalculus, mediated by one or more additional fractional parameters.

For example, the fractional integral Iα acting on a function f(x) on RN

is generalized to a multi-parameter fractional integral, where the additionalparameters can be considered as a measure of distortion:

Iαf(~x) → Iα,γβ f(~x) → I

(αk),(γk)(βk),m f(~x), k = 1, 2, ..., m. (1)

It is well known that the fractional integrals are of convolution type andexhibit weakly singular kernels of power-law type, see e.g. Mainardi [8].

As a first step, in this paper we will investigate the specific geometricproperties of the kernels or the weight-functions of a set of generalizedmulti-dimensional (multiple) fractional integrals of Erdelyi-Kober type.

For this case, we will demonstrate that a geometric approach allows adirect classification and interpretation of the generalized multi-parameterfractional integrals in a straightforward manner, in terms of the Cassiniand Maxwell ovaloids.

RESEARCH PAPER

A NEW EQUIVALENCE OF STEFAN’S PROBLEMS

FOR THE TIME FRACTIONAL DIFFUSION EQUATION

Sabrina Roscani 1, Eduardo Santillan Marcus 2

Abstract

A fractional Stefan’s problem with a boundary convective condition issolved, where the fractional derivative of order α ∈ (0, 1) is taken in theCaputo sense. Then an equivalence with other two fractional Stefan’s prob-lems (the first one with a constant condition on x = 0 and the second witha flux condition) is proved and the convergence to the classical solutionsis analyzed when α ↗ 1 recovering the heat equation with its respectiveStefan’s condition.

MSC 2010 : Primary 26A33; Secondary 33E12, 35R11, 35R35, 80A22Key Words and Phrases: Caputo’s fractional derivative, fractional dif-

fusion equation, Stefan’s problem

1. Introduction

In 1695 L’Hopital inquired of Leibnitz, the father of the concept of theclassical differentiation, what meaning could be ascribed to the derivativeof order 1

2 . Leibnitz replied prophetically: “[...] this is an apparent paradoxfrom which, one day, useful consequences will be drawn.”

From 1819, mathematicians as Lacroix, Abel, Liouville, Riemann andlater Grunwald and Letnikov attempted to establish a definition of frac-tional derivative.

We use here the definition introduced by Caputo in 1967, and we willcall it fractional derivative in Caputo’s sense, which is given by


372 S. Roscani, E. Santillan Marcus

Ca Dαf(t) = Dαf(t) =

1Γ(n− α)

∫ t

a(t− τ)n−α−1f (n)(τ)dτ,

where α > 0 is the order of derivation, n = dαe and f is a differentiablefunction up to order n in [a, b]. To simplify notation, we use from here thenotation Dα for the fractional derivative in Caputo’s sense.

The one-dimensional heat equation has become the paradigm for theall-embracing study of parabolic partial differential equations, linear andnonlinear. Cannon [2] did a methodical development of a variety of as-pects of this paradigm. Of particular interest are the discussions on theone-phase Stefan problem, one of the simplest examples of a free-boundary-value problem for the heat equation (see Datzeff [3]). In mathematics andits applications, particularly related to phase transitions in matter, a Stefanproblem is a particular kind of boundary value problem for a partial differ-ential equation, adapted to the case in which a phase boundary can movewith the time. The classical Stefan problem aims to describe the tempera-ture distribution in a homogeneous medium undergoing a phase change, forexample ice passing to water: this is accomplished by solving the heat equa-tion imposing the initial temperature distribution on the whole medium,and a particular boundary condition, the Stefan condition, on the evolvingboundary between its two phases. Note that in the one-dimensional casethis evolving boundary is an unknown curve: hence, the Stefan problemsare examples of free boundary problems. A large bibliography on free andmoving boundary problems for the heat-diffusion equation was given inTarzia [18]. Other references for the general Stefan problem are [14], [17].

In this paper, we deal with three one-phase Stefan problems with timefractional diffusion equation, obtained from the standard diffusion equationby replacing the first order time-derivative by a fractional derivative of orderα > 0 in the Caputo sense:

Dαu(x, t) = λ2 ∂2u

∂x2(x, t), −∞ < x < ∞, t > 0, 0 < α < 1,

and the Stefan conditionds(t)dt

= kux(s(t), t), t > 0, by the fractionalStefan condition

Dαs(t) = kux(s(t), t), t > 0.

The fractional diffusion equation has been treated by a number of au-thors (see [5], [8], [9],[13], [15]) and, among the several applications thathave been studied, Mainardi [6] studied the application to the theory oflinear viscoelasticity.

RESEARCH PAPER

NUMERICAL SOLUTIONS OF THE INITIAL VALUE

PROBLEM FOR FRACTIONAL DIFFERENTIAL

EQUATIONS BY MODIFICATION OF THE ADOMIAN

DECOMPOSITION METHOD

Neda Khodabakhshi 1, S. Mansour Vaezpour 1

and Dumitru Baleanu 2,3,4

Abstract

In this paper, we extend a reliable modification of the Adomian de-composition method presented in [34] for solving initial value problem forfractional differential equations.

In order to confirm the applicability and the advantages of our ap-proach, we consider some illustrative examples.

MSC 2010 : 34K28, 34K37Key Words and Phrases: fractional derivative, modified decomposition

method, Adomian polynomials

1. Introduction

Recently, fractional differential calculus has attracted a lot of attentionby many researchers of different fields, such as physics, chemistry, biology,economics, control theory and biophysics, etc. [27, 29, 33]. Since most frac-tional differential equations do not have exact analytic solutions, approx-imate and numerical techniques, are used extensively, such as homotopyanalysis method [10, 16, 31], homotopy perturbation method [20, 21], varia-tional iteration method [15, 17, 18, 19], Chebyshev spectral method [12, 13],


NUMERICAL SOLUTIONS OF THE INITIAL VALUE . . . 383

new iterative method [9, 22, 23, 24], orthogonal polynomial method [33, 36],Oldham-Spanier L1 method [32], Grunwald-Letnikov method [33, 14], frac-tional Adams method [11], and several other methods [33, 27, 28, 46].

The Adomian decomposition method (ADM) [1, 2, 3, 4, 5, 6, 7, 42, 43,44, 45, 38, 39, 40] is a powerful tool for solving both linear and nonlinearfunctional equations.

Consider the equation

Lu + Ru + Nu = g, (1.1)

where L is an invertible operator, which is taken as the highest order deriv-ative, R is the remainder of the linear operator, N represents the nonlinearterms and g is the specified analytic input function. Applying the inverseoperator L−1 on both sides of equation (1.1) yields

u = φ + L−1(g)− L−1(Ru)− L−1(Nu), (1.2)

where φ is determined by using the given initial values. This method de-composes the solution u(x) into a rapidly convergent series of solution com-ponents, and then decomposes the analytic nonlinearity Nu into the seriesof the Adomian polynomials [1, 2, 3]

u(x) =∞∑

n=0

un, (1.3)

Nu(x) =∞∑

n=0

An, (1.4)

where An = An(u0, u1, ..., un) are the well-known Adomian polynomials,whose definitional formula

An =1n!

dn

dλnN

( ∞∑

k=0

ukλk)|λ=0, n ≥ 0, (1.5)

was first published by Adomian and Rach in 1983 [1]. Then the standardAdomian recursion scheme:

u0(x) = φ + L−1g,

un+1 = −L−1(Run + An),is given.

Shawagfeh [41], Daftardar-Gejji and Jafari [8, 26, 25] have employedADM for solving nonlinear fractional differential equations and a systemof fractional differential equations, respectively. Momani [30] presented analgorithm for the numerical solution of linear and nonlinear multi-orderfractional differential equations. The algorithm is based on Adomian’s de-composition approach.

RESEARCH PAPER

FRACTIONAL SOBOLEV TYPE SPACES ASSOCIATED

WITH A SINGULAR DIFFERENTIAL OPERATOR

AND APPLICATIONS

Mourad Jelassi 1 and Hatem Mejjaoli 2

Abstract

In this paper we introduce and we study fractional Sobolev type spacesassociated with a singular second order differential operator on (0, ∞) andpropose several results. As applications we give certain properties includingestimates for the solution of the generalized wave equation and generalizedfractional operator.

MSC 2010 : 43A62, 44A15, 44A35Key Words and Phrases: second order differential operator4A, Sobolev

spaces, generalized fractional operator, Fourier transform associated with4A, wave equation

1. Introduction

The Sobolev spaces have served as a very useful tool in the theory ofpartial differential equations, mostly those related to problems from con-tinuum mechanics or physics. Their use and the study of their propertiesare based on the theory of distributions and Fourier analysis. The Sobolevspace W s,p(R+) is defined by the use of the classical Fourier transformas the set of all tempered distributions u such that its classical Fouriertransform u satisfies

(1 + |ξ|2) s2 u ∈ Lp(R+).

Generalization of the Sobolev space have been studied by replacing theclassical Fourier transform by a generalized one.


402 M. Jelassi, H. Mejjaoli

In this paper we consider the differential operator on (0,∞),

4A =d2

dx2+

A′(x)A(x)

d

dx+ ρ2, ρ > 0,

where A is the Chebli-Trimeche function (cf. [4], Section 3.5) defined on[0,∞) and satisfying the following conditions:

i) There exists a positive even infinitely differentiable function B onR, with B(x) ≥ 1,x ∈ R+, such that A(x) = x2α+1B(x), α > −1

2 .ii) A is increasing on R+ and lim

x→∞A(x) = ∞.

iii)A′

Ais decreasing on (0,∞), and lim

x→∞A′(x)A(x)

= 2ρ.

iv) There exists a constant σ > 0, such that for all x ∈ [x0,∞), x0 > 0,we have

A′(x)

A(x)=

{2ρ + e−σxF (x), if ρ > 02α + 1

x+ e−σxF (x) , if ρ = 0,

where F is C∞ on (0,∞), bounded together with its derivatives.

For A(x) = x2α+1, α > −12 and ρ = 0, we regain the Bessel operator

lαf =d2f

dx2+

(2α + 1

x

)df

dx.

For A(x) = sinh2α+1(x) cosh2β+1(x), α ≥ β ≥ −12 , α 6= −1

2 and ρ =α + β + 1, we regain the Jacobi operator

lα,βf =d2f

dx2+

[(2α + 1) cothx + (2β + 1) tanhx

] df

dx+ ρ2.

The purpose of this paper is to introduce and study new spaces associ-ated with the singular operator 4A: the Sobolev space Ws,p

A (R+) and thegeneralized fractional Sobolev space Hs,p

A (R+) that generalize the corre-sponding classical spaces. The Bessel case was treated by Assal-Nessibi [1],[2], while Ben Salem-Dachraoui [3] studied the generalized Sobolev spacesin the Jacobi setting theory.

The paper is organized as follows. In Section 2, we recall the mainresults about the harmonic analysis associated with the operator ∆A. InSection 3, fractional Sobolev spaces and fractional operators on the Chebli-Trimeche hypergroup are studied. Some properties including completenessand fractional Sobolev embedding theorems are established. Next, in Sec-tion 4, other Sobolev type spaces on the Chebli-Trimeche hypergroup aredefined and investigated. In Section 5, we give some applications. First,

RESEARCH PAPER

FRACTIONAL RELAXATION WITH

TIME-VARYING COEFFICIENT

Roberto Garra 1, Andrea Giusti 2,Francesco Mainardi 3, Gianni Pagnini 4

Abstract

From the point of view of the general theory of the hyper-Bessel op-erators, we consider a particular operator that is suitable to generalizethe standard process of relaxation by taking into account both memoryeffects of power law type and time variability of the characteristic coeffi-cient. According to our analysis, the solutions are still expressed in terms offunctions of the Mittag-Leffler type as in case of fractional relaxation withconstant coefficient but exhibit a further stretching in the time argumentdue to the presence of Erdelyi-Kober fractional integrals in our operator.We present solutions, both singular and regular in the time origin, that arelocally integrable and completely monotone functions in order to be con-sistent with the physical phenomena described by non-negative relaxationspectral distributions.

MSC 2010 : Primary 26A33; Secondary 33E12, 34A08, 76A10Key Words and Phrases: fractional derivatives, fractional relaxation,

Mittag–Leffler functions, fractional power of operators, hyper-Bessel differ-ential operators

1. Introduction

The applications of fractional calculus in the mathematical theory ofthe relaxation processes have a long history and have gained great interestin different fields of the applied science. These models, usually applied for


FRACTIONAL RELAXATION WITH . . . 425

anomalous (that is non-exponential) relaxation in linear viscoelastic anddielectric media, consider memory effects by replacing in the governingevolution equation the ordinary derivatives with fractional derivatives butkeeping constant the coefficients, see e.g. [3, 16] and references therein. Inthis paper we discuss some mathematical results about fractional relaxationmodels with power law time-varying coefficients. We apply the McBride-Lamb theory of the fractional powers of Bessel-type operators. By doingso, we can give an explicit representation of the fractional order operatorin terms of Erdelyi-Kober and Hadamard integrals. For this we refer thereader to the books [11, 17] and references recalled in the following.

In order to be consistent with most relaxation processes described bynon-negative relaxation spectral distributions, we concentrate our attentionto the cases for which the solutions are locally integrable and completelymonotone functions. Our solutions turn out to be both singular and reg-ular in the time origin. The regularization is carried out with a Caputo-like counterpart of our operators. The solutions are still represented viafunctions of the Mittag-Leffler type with one or two parameters as in thestandard fractional relaxation with constant coefficient but with a furthertime stretching due to the Erdelyi-Kober integrals involved in our operator.

The plan of the paper is the following. In Section 2 we first providethe definition of the time operator (tθd/dt)α that for 0 < α ≤ 1 and θ ∈ Ris assumed by us to generalize the usual fractional derivative (both in theRiemann-Liouville and Caputo sense) in the more common fractional relax-ation equation with constant coefficient often investigated in the literature,see for example the survey by Gorenflo and Mainardi [8]. Our definitionis justified in the framework of the theory of fractional powers of hyper-Bessel operators recalled in Appendix (Section 5) along with the relevantreferences, for reader’s convenience. Indeed our main purpose is to findthe solutions of the relaxation-type equation obtained by our operator thatturns out to be related to the Erdelyi-Kober and Hadamard integrals. Be-cause of the effect of these integrals on the power laws we find the powerseries representations of the solutions (both singular and regular in the timeorigin) that we easily recognize as functions of the Mittag-Leffler type.

In Section 3 we point out the relevance of the completely monotonicityto ensure that solutions will be suitable to represent physical relaxationprocesses with a non-negative spectral distribution. In view of this require-ment we devote our attention to investigate the zones in the parameterplane {α, θ} where the solutions are both locally integrable and completelymonotone. We then exhibit some illustrative plots of the solutions versustime in a few case-studies in order to remark the role of the parameter θ

SURVEY PAPER

TIME RESPONSE ANALYSIS OF FRACTIONAL-ORDER

CONTROL SYSTEMS: A SURVEY ON RECENT RESULTS

Mohammad Saleh Tavazoei

Abstract

The aim of this paper is to provide a survey on the recently obtainedresults which are useful in time response analysis of fractional-order con-trol systems. In this survey, at first some results on error signal analysisin fractional-order control systems are presented. Then, some previouslyobtained results which are helpful for system output analysis in fractional-order control systems are summarized. In addition, some results on theanalysis of the control signal and the system response to the load distur-bances in fractional-order control systems are reviewed.

MSC 2010 : Primary 93C05; Secondary 34A08, 26A33, 37N35Key Words and Phrases: fractional-order systems, fractional-order con-

trollers, time response analysis

1. Introduction

Nowadays, no one can deny the effective role which has been playedby fractional-order dynamics in design and practice of control systems. Al-though the first ideas about benefiting from fractional-order dynamics incontrol system design go back to more than half a century ago [8], [66], [36],[37], design and practice of fractional-order control systems have gainedincreasing attention from the control system community in the past two

c© 2014 Diogenes Co., Sofiapp. 440–461 , DOI: 10.2478/s13540-014-0179-z

TIME RESPONSE ANALYSIS OF FRACTIONAL-ORDER . . . 441

decades. In these years by considering the fractional calculus as a power-ful tool for generalizing the structure of traditional controllers [58], somepopular and simple structures have been proposed for fractional-order con-trollers (For example, fractional-order PID controllers [46] [22] and theirsubclasses, i.e. fractional-order PI and PD controllers [33], [26], differentgenerations of CRONE controllers [41], [27], [28], fractional-order phase-lead/lag compensators [39], [65], [49], etc). Also, different methods havebeen presented for design and tuning of these controllers (For example, [34],[42], [5], [44], and [68]). Moreover, more advanced control techniques suchas smith predictor control [18], [50], sliding mode control [15], [69], opti-mal control [7], [12], adaptive control [31], [1], and model predictive control[13], [4] have been generalized by using the fractional calculus concepts.Till now, fractional-order controllers have been applied in many practicalapplications such as control of hard disk drive servo systems [35], controlof cement milling processes [16], suppression of chaos in chaotic electricalcircuits [64], control of power electronic converters [9], control of compositehydraulic cylinders [70], control of irrigation canals [19], and so on.

Fractional-order dynamics not only have enhanced the traditional con-trollers, but also has provided an enriched environment for modeling andidentification of real-world systems [40], [17], [20]. Using fractional-orderdynamics in modeling and identification of practical systems brings aboutachieving more precise models for describing the behavior of these systems[10], [21]. Clearly, the performance of a control system can be improvedif its controller is designed based on a more precise model of the process.This is another influence of the fractional-order dynamics in improving theperformance of traditional control systems.

Due to the increasing interest in design of fractional-order control sys-tems, these systems have been analyzed in literature from different aspectsof view. The aim of this paper is to present a survey on the results recentlyobtained on time response analysis of fractional-order control systems.

This paper is organized as follows. Some preliminaries are given inSection 2. In Section 3, some results on error signal analysis in fractional-order systems are reviewed. Section 4 is devoted to presenting some re-sults which are useful in analysis of the system output in a fractional-ordercontrol system. Also, additional results on analysis of control signal andsystem response to load disturbances are summarized in Section 5. Finally,the paper is concluded in Section 6.

RESEARCH PAPER

ADAPTIVE GAIN-ORDER FRACTIONAL CONTROL FOR

NETWORK-BASED APPLICATIONS

Ines Tejado 1, S. Hassan HosseinNia 2, Blas M. Vinagre 3

Abstract

This paper deals with the application of adaptive fractional order con-trol to networked control systems (NCSs) to compensate the effects of time-varying network-induced delays. In essence, it adapts both the gains andthe orders of a local PIαDµ controller in accordance with the current net-work condition in order to avoid a decreased control performance. A fre-quency domain framework is provided to analyze the system stability on thebasis of the switching systems theory. The velocity control of a servomo-tor through the Internet is given to show the effectiveness of the proposedadaptive controller, including a comparison with non- and gain scheduledcontrollers.

MSC 2010 : 93C40, 93D99Key Words and Phrases: networked control system, fractional order

control, gain scheduling, variable order, stability analysis, servomotor, ve-locity

1. Introduction

Within the challenge of networked control systems (NCSs), there is aneed for control algorithms that can deal with communication imperfectionsand constraints, which can degrade the performance of the control loopsignificantly and even lead to instability (see e.g. [1, 2, 3]). To this respect,


ADAPTIVE GAIN-ORDER FRACTIONAL CONTROL . . . 463

and considering network-induced delays as the main drawback, much ofthe available literature is recently dominated by the compensation of time-varying delays in real-time, especially using gain scheduling (GS) [4, 5, 6, 7].In many instances, this method of control is clearly reasonable and can bejustified by appropriate experimental results.

Major advancements over the last decades in fractional order control(FOC) have led to recognize that fractional order PID controllers, namelyPIαDµ, possess the same ease of use of standard PID controllers but providemore flexibility in the design and, consequently, improvements in the per-formance of the control system –e.g. refer to [8, 9] for fundamentals and asurvey on FOC. In the past years, the application of FOC to network-basedcontrol is becoming more popular and its success and advantages have beendemonstrated in many works [10, 11, 12, 13, 14, 15].

Likewise, more recently variable order (VO) fractional calculus has at-tracted significant interests in the field. Many valuable studies run to useVO fractional operators in different applications, such as diffusion processes[16, 17], signal processing [18, 19, 20] and control [21, 22]. However, up tonow a few works were developed to apply variable operators in PIαDµ con-trollers, i.e., to generalize them to VO controllers of the form PIα(t)Dµ(t). Abrief study of VO integrators and differentiators, their possible applicationand extension to PIα(t)Dµ(t) controllers are reported in [23, 24]. Despite thegeneral interest, no experimental applications of PIα(t)Dµ(t) can be foundin the technical literature.

Motivated by the widespread success of GS in NCSs and the promisingpotential of PIα(t)Dµ(t) controllers as a form of adaptive control, this paperfocuses on the design of fractional order PIαDµ controllers with variablegains and orders to minimize the effects of time-varying delays in real-timecontrol over networks. This approach, which will be called as gain-orderscheduling (GOS), involves the automatic adjustment in real-time of boththe controller gains and the orders α and µ (0 < α, µ < 2) with respect tothe current network condition. Essentially, it will allow more robust systemperformance to be attained. The contributions of this paper are three:

(1) The development of a tuning method for the automatic adjustmentof controller parameters for efficient network-based control. Thebasis and initial results of this strategy can be found in our previousworks [25, 26, 27].

(2) The development of a frequency domain framework to analyse thestability of the controlled system, based on switching systems theoryand assuming slow variations in the scheduling variables.

(3) The experimental validation of the proposed strategy with a servo-motor.

RESEARCH PAPER

MAXIMUM PRINCIPLE FOR THE FRACTIONAL

DIFFUSION EQUATIONS WITH

THE RIEMANN-LIOUVILLE FRACTIONAL DERIVATIVE

AND ITS APPLICATIONS

Mohammed Al-Refai 1, Yuri Luchko 2

Abstract

In this paper, the initial-boundary-value problems for the one-dimensi-onal linear and non-linear fractional diffusion equations with the Riemann-Liouville time-fractional derivative are analyzed. First, a weak and astrong maximum principles for solutions of the linear problems are de-rived. These principles are employed to show uniqueness of solutions of theinitial-boundary-value problems for the non-linear fractional diffusion equa-tions under some standard assumptions posed on the non-linear part of theequations. In the linear case and under some additional conditions, thesesolutions can be represented in form of the Fourier series with respect to theeigenfunctions of the corresponding Sturm-Liouville eigenvalue problems.

MSC 2010 : Primary 26A33; Secondary 33E12, 35S10, 45K05

Key Words and Phrases: Riemann-Liouville fractional derivative, ex-tremum principle for the Riemann-Liouville fractional derivative, maximumprinciple, linear and non-linear time-fractional diffusion equations, unique-ness and existence of solutions


484 M. Al-Refai, Yu. Luchko

1. Introduction

The maximum principles are one of the few known techniques for ob-taining information about solutions of differential equations without anyexplicit knowledge of the solutions themselves. Until recently, the maxi-mum principles were formulated and proved only for the conventional ordi-nary and partial differential equations (see e.g. the books [17] or [18] thatcontain a detailed survey of the results related to the maximum principlesand their applications for the ordinary and partial differential equations).

Over the last few years, the maximum principle method started to beemployed for analysis of the fractional differential equations, too. In [4] and[6], some arguments related to a kind of a maximum principle have beenused for analysis of the fractional diffusion equation without an explicit for-mulation of this principle. In [8], a maximum principle for the generalizedfractional diffusion equation was formulated and proved for the first timeand in [10] the applications of this maximum principle to the problem ofuniqueness and existence of solutions to the initial-boundary-value prob-lems for the multi-dimensional time-fractional diffusion equation have beenprovided. Extended maximum principles for the multi-term time-fractionaldiffusion equation and for the time-fractional diffusion equation of the dis-tributed order were introduced and applied in [11] and [9], respectively. In[20], a maximum principle for the multi-term time-space fractional differ-ential equations with the modified Riesz space-fractional derivative in theCaputo sense was introduced and employed to show uniqueness and con-tinuous dependence of the solutions to the initial-boundary-problems forthe one-dimensional time-space fractional differential equations. In [3], amaximum principle along with the method of lower and upper solutionshas been used to establish some existence and uniqueness results for a classof eigenvalue problems of the fractional order α, 1 < α < 2. Finally, werefer to [12] for a survey of the maximum principles for different kinds ofthe time-fractional diffusion equations and their applications.

In all papers related to the maximum principle for the fractional dif-ferential equations mentioned above, the fractional derivatives were takenin the Caputo sense. The main reason for this choice of the fractional de-rivative is connected with the fact that the Caputo fractional derivativepossesses a kind of an extremum principle that says that the Caputo frac-tional derivative of the order α, 0 < α ≤ 1 at the maximum point of acontinuous function on some closed interval is non negative (see [8] or [10]).In [2], an even stronger inequality for the Caputo fractional derivative ofthe order α, 0 < α < 1 at the maximum point has been shown for thespace of the continuous differentiable functions. In contrast to the Ca-puto fractional derivative, the Riemann-Liouville fractional derivative at

RESEARCH PAPER

EXISTENCE AND UNIQUENESS OF SOLUTIONS FOR

A FRACTIONAL BOUNDARY VALUE PROBLEM

ON A GRAPH

John R. Graef 1, Lingju Kong 2, Min Wang 3

Abstract

In this paper, the authors consider a nonlinear fractional boundaryvalue problem defined on a star graph. By using a transformation, an equiv-alent system of fractional boundary value problems with mixed boundaryconditions is obtained. Then the existence and uniqueness of solutions areinvestigated by fixed point theory.

MSC 2010 : Primary 34B15; Secondary 34B45Key Words and Phrases: fractional calculus, boundary value problems,

Green’s function, differential equations on graphs

1. Introduction

Differential equations on graphs have extensive applications in manyareas such as physics, engineering, and ecology. A graph G consists of afinite or countably infinite set of nodes V = {γi : i = 0, 1, . . . } and aset of edges E connecting these nodes, i.e., G = V ∪ E. On each edge, alocal coordinate system is assigned with the origin at a node. A differentialequation on a graph is a differential equation defined on each edge of Gbased on the local coordinate system; a problem consisting of such anequation and certain conditions defined at the boundary nodes is calleda boundary value problem (BVP) on a graph. For more details about


500 J. R. Graef, L. Kong, M. Wang

differential equations and BVPs on graphs, as well as their applications,the reader is referred to [3, 5, 6, 9, 10, 21, 22, 23] and the references therein.

Fractional differential equations have extensive applications in variousfields of science and engineering. Many phenomena in viscoelasticity, elec-trochemistry, control theory, porous media, electromagnetism, and otherfields, can be modeled by fractional differential equations. We refer thereader to [18, 24] and references therein for some applications. FractionalBVPs defined on intervals have been studied by many authors. Many re-sults on the existence, uniqueness, multiplicity, and nonexistence of solu-tions for fractional differential equations subject to various boundary con-ditions (BCs) have been obtained; see for example [1, 2, 4, 7, 8, 11, 12, 13,14, 15, 16, 17, 19, 20, 25, 26, 27].

Figure 1: A star graph with n edges

To the best of our knowledge, no work has been done for fractionalBVPs on graphs. This is in part due to the fact that many useful tech-niques for integer order differential equations fail to work for the corre-sponding fractional order problems. One example of this is the construc-tion of Green’s functions (see the discussion in Section 2 below). In thispaper, we consider a fractional BVP on a star graph, which is a graph con-sisting of n edges with a common node γ0 (see Figure 1). For the purposeof simplification, we focus on a star graph consisting of three nodes andtwo edges, i.e., G = V ∪ E with V = {γ0, γ1, γ2} and E = {−−→γ1γ0,

−−→γ2γ0},

RESEARCH PAPER

STABILIZABILITY OF FRACTIONAL

DYNAMICAL SYSTEMS

Krishnan Balachandran 1, Venkatesan Govindaraj 1,Luis Rodrıguez-Germa 2, Juan J. Trujillo 2

Abstract

In this paper, we establish that the controllability and observabilityproperties of fractional dynamical systems in a finite dimensional space aredual. Using this duality result and the Mittag-Leffler matrix function, wepropose the stabilizability of fractional MIMO (Multiple-input Multiple-output) systems. Some numerical examples are provided to show the effec-tiveness of the obtained results.

MSC 2010: 93B05, 93B07, 70K20, 34A08Key Words and Phrases: controllability, observability, stability, frac-

tional differential equations, Mittag-Leffler matrix function, MIMO systems

1. Introduction

Many real systems are better characterized by using a non-integer orderdynamic model based on fractional calculus. Fractional order system is asystem described by an integro-differential equation involving fractional or-der derivatives of its input(s) and/or output(s). The research in this area isinherently multi-disciplinary and its application is found across diverse dis-ciplines in automatic control [23, 26, 28, 33], that is, system identification,observation and control of fractional systems, etc. In control theory, thequalitative behaviors of dynamical systems are controllability, observability


512 K. Balachandran, V.Govindaraj, L.Rodrıguez-Germa, J.J. Trujillo

and stability. Now the study of such qualitative behaviors in fractional dy-namical systems are important issues for many applied problems becausethe use of fractional order derivatives and integral in control theory leadsto better results than integer order ones.

Controllability is one of the fundamental concepts in control theory, itmeans that it is possible to steer a dynamical system from an arbitrary ini-tial state to arbitrary final state using the set of admissible controls. Theproblem of controllability of linear systems represented by fractional differ-ential equations in finite dimensional spaces has been extensively studied bymany authors [1, 12, 21]. Balachandran et al. [6, 7, 8, 9, 10] established suf-ficient conditions for the controllability of nonlinear fractional dynamicalsystems with or without delays in finite dimensional spaces using fixed-point techniques. More recently, controllability of higher order nonlinearfractional dynamical systems are also studied by Balachandran et al. [4, 5].

The dual notion of controllability is observability. It is defined as thepossibility to deduce the initial state of the system from observing its input-output behavior. This means that from the system’s outputs it is possibleto determine the behavior of the entire system. Several authors (see, forinstance [11, 30]) have established the results for observability of linearfractional dynamical systems using Grammian matrix and rank conditions.

The concept of stability is extremely important because almost everyworkable control system is designed to be stable. Stability is an importantqualitative behavior of a dynamical system. It means that system remainsin a constant state unless affected by an external action and returns to aconstant state when the external action is removed. The stability of solu-tion is important in physical applications, because deviations in mathemat-ical model inevitably result from errors in measurement. A stable solutionwill be usable despite such deviations. The study of the stability of frac-tional system can be carried out by studying the solutions of the differentialequations that characterize them. In [17], it is observed that the analysis ofstability in fractional order systems is more complicated than in integer or-der systems even though it may have additional attractive feature over theinteger order system. Recently the stability of linear fractional dynamicalsystems has attracted many researchers [22, 24, 25, 29, 31, 32]. Li et al. [16]studied stability of nonlinear fractional dynamical systems by introducingthe concept of the Mittag-Leffer stability and fractional Lyapunov directmethod.

In the field of fractional order control systems, one of the challengingproblems related to stability theory is stabilizability. It means that if asystem is not stable by itself, then the question arises whether it can be

RESEARCH PAPER

FRACTIONAL SKELLAM PROCESSES

WITH APPLICATIONS TO FINANCE

Alexander Kerss 1, Nikolai N. Leonenko 2, Alla Sikorskii 3

Abstract

The recent literature on high frequency financial data includes modelsthat use the difference of two Poisson processes, and incorporate a Skellamdistribution for forward prices. The exponential distribution of inter-arrivaltimes in these models is not always supported by data. Fractional gener-alization of Poisson process, or fractional Poisson process, overcomes thislimitation and has Mittag-Leffler distribution of inter-arrival times. Thispaper defines fractional Skellam processes via the time changes in Poissonand Skellam processes by an inverse of a standard stable subordinator. Anapplication to high frequency financial data set is provided to illustrate theadvantages of models based on fractional Skellam processes.

MSC 2010 : Primary 60E05, 60G22; Secondary 60G51, 26A33Key Words and Phrases: fractional Poisson process, fractional Skellam

process, Mittag-Leffler distribution, high frequency financial data

1. Introduction

The advent of high frequency financial data has spurred new model-ing techniques to describe characteristics of trade by trade data. Recentliterature on the subject includes [1, 2, 3, 10] where models based on thedifference of two point processes are proposed. Difference of Poisson pro-cesses is considered in [3, 10], and Hawkes processes are discussed in [1, 2].This paper extends the models where the forward price of a risky asset


FRACTIONAL SKELLAM PROCESSES . . . 533

is modeled via the difference of two independent Poisson processes, alsoknown as Skellam processes. A drawback of the existing models, whichmay be at odds with empirical facts, is exponential inter-arrival time, ortime between trades. Mainardi et al. [21, 22] studied the fractional Poissonprocess, where the exponential waiting time distribution is replaced by aMittag-Leffler distribution, see also [4, 17, 28, 32]. Meerschaert et al. [23]showed that the same fractional Poisson process can also be obtained via aninverse stable time change. Using a time change in models for financial datahas been popularized in the last decade based on the idea that it is not thecalendar time that drives the changes in price, but rather the informationflow or activity time is what matters for modeling of the prices [15, 14, 18].In this paper we define fractional Skellam processes via time changes inordinary Skellam processes. The resulting fractional Skellam models incor-porate Mittag-Leffler distribution of inter-arrival times and may providebetter fit to high frequency financial data.

2. Preliminaries

This section collects definitions and some results on the Skellam process,subordinators and the fractional Poisson process. These results will be usedin the next section for the construction of the fractional Skellam processes.

2.1. Skellam processes

Definition 2.1. A Skellam process is defined as

S(t) = N (1)(t)−N (2)(t), t ≥ 0,

where N (1)(t), t ≥ 0 and N (2)(t), t ≥ 0 are two independent homogeneousPoisson processes with intensities λ(1) > 0 and λ(2) > 0, respectively.

The Skellam distribution has been introduced in [30] and [16], and theSkellam processes are considered in [3]. The probability mass function ofS(t) is of the form

sk(t) = P(S(t) = k) = e−t(λ(1)+λ(2))

(λ(1)

λ(2)

)k/2

I|k|(2t√

λ(1)λ(2)

),

k ∈ Z = {0,±1,±2, . . .},(2.1)

where Ik is the modified Bessel function of the first kind [31, p. 114]

Ik(z) =∞∑

n=0

(z/2)2n+k

n!(n + k)!.

The Skellam process is a Levy process withE[exp{−θS(t)}] = exp{−tψS(1)(θ)},

HISTORICAL SURVEY

SOME PIONEERS OF THE APPLICATIONS

OF FRACTIONAL CALCULUS

Duarte Valerio 1, Jose Tenreiro Machado 2, Virginia Kiryakova 3

Abstract

In the last decades fractional calculus (FC) became an area of intensiveresearch and development. This paper goes back and recalls important pio-neers that started to apply FC to scientific and engineering problems duringthe nineteenth and twentieth centuries. Those we present are, in alphabet-ical order: Niels Abel, Kenneth and Robert Cole, Andrew Gemant, AndreyN. Gerasimov, Oliver Heaviside, Paul Levy, Rashid Sh. Nigmatullin, YuriN. Rabotnov, George Scott Blair.

MSC 2010 : Primary 26A33; Secondary 01A55, 01A60, 34A08Key Words and Phrases: fractional calculus, applications, pioneers,

Abel, Cole, Gemant, Gerasimov, Heaviside, Levy, Nigmatullin, Rabotnov,Scott Blair

1. Introduction

In 1695 Gottfried Leibniz asked Guillaume l’Hopital if the (integer)order of derivatives and integrals could be extended. Was it possible ifthe order was some irrational, fractional or complex number? “Dreamcommands life” and this idea motivated many mathematicians, physicistsand engineers to develop the concept of fractional calculus (FC). Dur-ing four centuries many famous mathematicians contributed to the theo-retical development of FC. We can list (in alphabetical order) some im-portant researchers since 1695 (see details at [1, 2, 3], and posters athttp://www.math.bas.bg/∼fcaa):


SOME PIONEERS OF THE APPLICATIONS . . . 553

• Abel, Niels Henrik (5 August 1802 - 6 April 1829), Norwegian math-ematician

• Al-Bassam, M. A. (20th century), mathematician of Iraqi origin• Cole, Kenneth (1900 - 1984) and Robert (1914 - 1990), American

physicists• Cossar, James (d. 24 July 1998), British mathematician• Davis, Harold Thayer (5 October 1892 - 14 November 1974), Amer-

ican mathematician• Djrbashjan, Mkhitar Mkrtichevich (11 September 1918 - 6 May

1994), Armenian (also Soviet Union) mathematician; family nametranscripted also as Dzhrbashian, Jerbashian; short CV can befound in Fract. Calc. Appl. Anal. 1, No 4 (1998), 407–414

• Erdelyi, Arthur (2 October 1908 - 12 December 1977), Hungarian-born British mathematician

• Euler, Leonhard (15 April 1707 - 18 September 1783), Swiss math-ematician and physicist

• Feller, William (Vilim) (7 July 1906 - 14 January 1970), Croatian-American mathematician

• Fourier, Jean Baptiste Joseph (21 March 1768 - 16 May 1830),French mathematician and physicist

• Gelfand, Israel (Israıl) Moiseevich (2 September 1913 - 5 October2009), Soviet mathematician (of Jewish origin, born in Russian Em-pire in a southern Ukrainian town of Okny, moved later to the USA)

• Gemant, Andrew (1895 - 1983), American physicist• Gerasimov, Andrey Nikolaevich (20th century), Russian (then So-

viet) physicist working in the field of mechanics• Grunwald, Anton K. (1838 - 1920), German mathematician• Hadamard, Jacques Salomon (8 December 1865 - 17 October 1963),

French mathematician• Hardy, Godfrey Harold (7 February 1877 - 1 December 1947), Eng-

lish mathematician• Heaviside, Oliver (18 May 1850 - 3 February 1925), English electri-

cal engineer, mathematician, and physicist• Holmgren, Hjalmar J. (1822 - 1885), Swedish mathematician• de l’Hopital, Guillaume Francois Antoine (1661 - 2 February 1704),

French mathematician• Kilbas, Anatoly Aleksandrovich (20 July 1948 - 28 June 2010), Be-

larusian mathematician; biographical and memorial notes can befound in Fract. Calc. Appl. Anal. 11, No 4 (2008), an issue ded-icated to his 60th anniversary, and in 13, No 2 (2010), 221–223;a founding editor of this journal with a lot of publications therein

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