Research paper
Variable order fractional systems

https://doi.org/10.1016/j.cnsns.2018.12.003Get rights and content

Highlights

  • It presents a revision of previously proposed variable order fractional derivatives.

  • It introduces a new approach based on the inverse Laplace transform.

  • Variable order fractional linear systems are defined.

  • A variable order Mittag-Leffler function is introduced.

  • Variable order two-sided fractional derivatives.

Abstract

Fractional Calculus had a remarkable evolution during recent decades, and paved the way towards the definition of variable order derivatives. In the literature we find several different alternative definitions of such operators. This paper presents an overview of the fundamentals of this topic and addresses the questions of finding out which of them are reasonable according to simple criteria used for constant order fractional derivatives. This approach leads to the definitions of variable order fractional derivative based on the Grünwald–Letnikov and the Liouville formulations defined on R, as well as to the definition of a Mittag-Leffler function for variable orders, and to the application of these definitions to dynamical systems.

Introduction

Fractional Calculus exhibited a remarkable evolution during the last 30 years and became popular in many scientific areas [9], [10], [11]. The progress in applications vis-a-vis theoretical developments motivated recently the re-evaluation of past mathematical formulations [22]. The concepts of fractional derivative (FD) and fractional integral (FI) proposed so far assume various forms and we can question if they are completely equivalent, or to what extent they are compatible with each other. The Riemann–Liouville (RL) derivative [28] was mostly used during this period in the context of mathematical studies, but, in the last decade, the Caputo (C) derivative [35] became popular in applied sciences due to the way it handles the initial conditions [18]. On the other hand, the Grünwald–Letnikov (GL) derivative was considered mostly for approximations in the scope of numerical calculations.

The first proposal of a variable order (VO) fractional derivative was developed by Ross and Samko [26], [27], [30]. Their formulation was based on the VO RL integral and the Marchaud (M) derivative. Lorenzo and Hartley, apparently in an independent way, proposed several versions of the VO RL derivative/integral [7]. However, they observed that the derivative is not the left inverse of the integral. In fact, the problem was also noted by Ross and Samko in their initial formulation, and that limitation justified the use of the M derivative. Nonetheless, the M derivative is not the left inverse of the VO RL integral [1], [29]. Coimbra proposed a Caputo based VO derivative [3]. Later, Valério and Sá da Costa presented a review of the state of the art including expressions based on the GL derivative [34], [35], [36] that were generalized in [31], [32]. Several applications using VO derivatives were considered in [3], [13], [14], [24], [33]. In a recent paper [2], Bohannan presented a critical view of the VO based on physical considerations.

A brief analysis of all these derivatives based on the criteria suggested in Ortigueira and Machado [21] reveals some concerns: none of the proposed derivatives obey most of those criteria. In particular, the index law does not hold for any combination of orders. This limitation motivates the formulation of a suitable definition of VO derivative. Loosely speaking a VO FD is an operator that results from the constant order derivatives by making the order variable.

In the follow-up we will start from the GL derivative/integral, by proposing a formulation that does not need the property of additivity of the orders. This approach is based on the Laplace transform (LT) that allows the definition VO linear systems and the computation their output.

The paper is organized as follows. Sections 2 and 3 introduce the fundamental mathematical aspects and the new approach for handling VO derivatives, respectively. Section 4 discusses the application of the VO in linear systems. Sections 5 and 6 generalize the concepts of Mittag-Leffler function and the two-sided derivative in the light of the proposed VO, respectively. Finally, Section 7 outlines the conclusions.

We assume that:

  • We work on R .

  • We use the two-sided Laplace transform (LT) [25]F(s)=Rf(t)estdthaving a non-empty region of convergence (ROC); the complex variable s is the LT of t. The inversion of the LT is given by the Bromwich integralf(t)=12πiσiσ+iF(s)estds,where σ is in the ROC and i=1. This integral converges uniformly in the ROC [25].

  • The Fourier transform (FT) is obtained from the LT through the substitution s=iω with ωR [25]F(ω)=f(t)eiωtdt,and its inverse isf(t)=12πF(ω)eiωtdω

  • Functions and distributions have Laplace and/or Fourier transforms.

  • The fractional derivative order is any bounded real function.

  • The values of the multi-valued expressions sα and (s)α are obtained with branch-cut lines at the negative real half axis for sα and at the positive real half axis for (s)α. In both cases, the first Riemann surface is chosen.

  • The Heaviside, or unit step, function isɛ(t)={1t>01/2t=0,0t<0tR.Its derivative is the Dirac impulse:Dɛ(t)=δ(t),tR,

  • The unit slope ramp isr(t)=tɛ(t)

  • The signum function is defined bysgn(t)=2ɛ(t)1={1t>00t=0,1t<0tR.

  • Given a real number x, the ceiling function (that rounds x up) is denoted at ⌈x⌉, the floor function (that rounds x down) as ⌊x⌋, and the round function (that round x to the nearest integer) as [x].

Section snippets

Main formulations

In this section, the standard RL, C, and GL derivatives are extended to VO [34], [35]. In what follows, the differentiation order will no longer be restricted to a constant value αR, and the more general case α(t)R will be assumed. The variable tR will be read as “time”, but it can have any other physical meaning. We consider the derivative order as function of the independent variable, that is, we have α(t)=g(t), since it is relevant for the theoretical formulation to be developed. However,

A new approach to VO derivatives

The considerations in the previous section motivate a new definition of VO FD, proceeding in agreement with [21].

Definition 8

Let f(t) be a function defined on R, with Laplace transform (LT), and α a real number. An operator T[f(t), α] verifying the properties of the above stated criterion is said a fractional derivative of order α.

This definition is the base for introducing the notion of VO FD:

Definition 9

In the conditions of Definition 8, let us assume that α(t)=g(t), where g(t),tR, is any bounded real function. We

Variable order linear systems

Rapaić and Pisano proposed in [24] a method for describing fractional commensurate VO systems. An adaptive parameter estimation scheme for estimating the order was suggested, but only a VO integral, identical to (27), was used.

We introduce a general framework for dealing with VO linear systems that extends the results described in Moghaddam and Machado [14].

Definition 12

Let x(t) and y(t) be the input and output functions of a VO fractional linear system defined byk=0NakDαk(t)y(t)=k=0MbkDβk(t)x(t)with tR.

The VO Mittag-Leffler function

With the transfer function defined in (37) we can invert (38) directly, by means of the following the steps [18]:

  • 1.

    Transform H(s, α(t)) into H(u), by substitution of sα(t) for u.

  • 2.

    Perform the expansion of H(u) in partial fractions.

  • 3.

    Substitute u back for sα(t), to obtain the partial fraction decomposition [25]:H(s)=k=0NpAk(sα(t)pk)nk

where pkC are called pseudo-poles, Np is the number of parcels, and nkN0 are the corresponding multiplicities.

Remark 5.1

We observe that, having solved the problem where

Two-sided derivatives

Two constant-order centered derivatives based on the GL formulation were introduced in [15], [16], [20]. They can be considered as the composition of a left and a right derivative and expressed in a constant order unified general two-sided derivative given byDθαf(t)=limh0+hαn=(1)nΓ(α+1)Γ(α+θ2n+1)Γ(αθ2+n+1)f(tnh),where α is the derivative order and θ is a parameter (sometimes called skewness) that we will denote as dissymmetry because it determines the symmetry of the derivative. The

Conclusions

This paper discussed the concept of variable order derivative. A review of several formulations motivated the analysis of their properties. Based on these expressions, the definition that better suits the desirable properties was developed, and applied to several illustrative examples. Furthermore, the formulation was extended to linear system models, to transfer functions, and to the Mittag–Lefler function. The proposed definitions lead to a deeper discussion of this emerging topic and to

Acknowledgments

This work was partially funded by National Funds of the Foundation for Science and Technology under project PEst-UID/EEA/00066/2013, and through IDMEC, under LAETA, project UID/EMS/50022/2013, as well as through grant SFRH/BSAB/142920/2018 to author D. Valério.

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