# A NEW APPROACH TO GENERALIZED FRACTIONAL in FC [66]. The Caputo fractional derivative has also been

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Bulletin of Mathematical Analysis and Applications

ISSN: 1821-1291, URL: http://www.bmathaa.org

Volume 6 Issue 4 (2014), Pages 1-15

A NEW APPROACH TO GENERALIZED FRACTIONAL

DERIVATIVES

(COMMUNICATED BY CLAUDIO R. HENRÍQUEZ)

UDITA N. KATUGAMPOLA

Abstract. The author (Appl. Math. Comput. 218(3):860-865, 2011) intro- duced a new fractional integral operator given by,(

ρIαa+f ) (x) =

ρ1−α

Γ(α)

∫ x a

τρ−1f(τ)

(xρ − τρ)1−α dτ,

which generalizes the well-known Riemann-Liouville and the Hadamard frac-

tional integrals. In this paper we present a new fractional derivative which gen- eralizes the familiar Riemann-Liouville and the Hadamard fractional deriva-

tives to a single form. We also obtain two representations of the generalized

derivative in question. An example is given to illustrate the results.

1. Introduction

In recent years, the Fractional Calculus (FC) draws increasing attention due to its applications in many fields. The history of the theory goes back to seventeenth century, when in 1695 the derivative of order α = 12 was described by Leibnitz in his letter to L’Hospital [34–36]. Since then, the new theory turned out to be very attractive to mathematicians as well as physicists, biologists, engineers and economists. The first application of fractional calculus was due to Abel in his solution to the Tautocrone problem [1]. It also has applications in biophysics, quantum mechanics, wave theory, polymers, continuum mechanics, Lie theory, field theory, spectroscopy and in group theory, among other applications [22, 24, 25, 41]. In [66], Samko et al. provide an encyclopedic treatment of the subject. Various type of fractional derivatives were studied: Riemann-Liouville, Caputo, Hadamard, Erdélyi-Kober, Grünwald-Letnikov, Marchaud and Riesz are just a few to name [30,43,52,55,66].

In fractional calculus, the fractional derivatives are defined via fractional inte- grals [30, 66]. According to the literature, the Riemann-Liouville fractional deriv- ative (RLFD), hence the Riemann-Liouville fractional integral plays a major role in FC [66]. The Caputo fractional derivative has also been defined via a modified

2010 Mathematics Subject Classification. 26A33, 65R10, 44A15. Key words and phrases. Fractional Calculus, Generalized fractional derivatives, Riemann-

Liouville fractional derivative, Hadamard fractional derivative, Erdélyi-Kober operator, Taylor series expansion.

c©2014 Universiteti i Prishtinës, Prishtinë, Kosovë. Submitted December 2, 2013. Published October 15, 2014.

1

2 U.N. KATUGAMPOLA

Riemann-Liouville fractional integral [30]. Butzer et al. investigate properties of the Hadamard fractional integral and the derivative in [6–8, 29–31, 55, 66]. In [8], they also obtained the Mellin transforms of the Hadamard fractional integral and diffferential operators and in [56], Pooseh et al. obtained expansion formulas of the Hadamard operators in terms of integer order derivatives. Many other interesting properties of those operators and others are summarized in [66] and [30] and the references therein.

In [28], the author introduced a new fractional integral, which generalizes the Riemann-Liouville and the Hadamard integrals into a single form. For further properties such as expansion formulas, variational calculus applications, control theoretical applications, convexity and integral inequalities and Hermite-Hadamard type inequalities of this new operator and similar operators, for example, see [5,20, 22,46–51,56,60]. In the present work, we shall introduce a new fractional derivative, which generalizes the two derivatives in question.

The paper is organized as follows. In the next section, we give definitions and some properties of the fractional integrals and fractional derivatives of various types. More detailed explanation can be found in the book by Samko et al. [66] and the references therein.

2. Definitions

We shall start this section with some historical remarks and definitions to refresh our memories about some of the remarkable milestones in the theory of fractional calculus. As is well known nowadays, the first documented note about a fractional derivative was found in 1695 in the letters of Leibnitz to L’Hospital [34–36, 63]. In 1819, Lacroix obtained the well-known 12 derivative of x [33], using inductive

arguments, to be d 1 2 /dx

1 2 = 2

√ x/π, long before the Riemann-Liouville fractional

derivative surfaced into the realm of fractional calculus. The idea of a derivative that is not of an order of a positive integer was introduced by Liouville in 1832 [37, 40, 54], in a manner that would generalize the relation Dneαx = αneαx to any complex number n. Liouville then used Fourier theory to extend his α-derivative to any function f(z) expanded in a Fourier series [54]. In 1888, Nekrassov [45,53,54], generalizing the Cauchy’s integral formula,

dnf(z)

dzn =

n!

2πi

∮ C

f(ζ)

(ζ − z)n+1 dζ,

where C is a closed contour surrounding the point z and enclosing a region of analyticity of f , came up with a fractional derivative which can be showed to be equal to Riemann-Liouville derivative under certain conditions [52, p.54-55].

The Riemann-Liouville fractional integrals Iαa+f and I α b−f of order α ∈ C,

(Re(α) > 0) are defined by [30,37,58,59,66],

(Iαa+f)(x) = 1

Γ(α)

∫ x a

(x− τ)α−1f(τ)dτ ;x > a, (2.1)

and

(Iαb−f)(x) = 1

Γ(α)

∫ b x

(τ − x)α−1f(τ)dτ ;x < b, (2.2)

respectively. Here Γ(·) is the Gamma function. These integrals are called the left-sided and right-sided fractional integrals, respectively. When α = n ∈ N, the integrals (2.1) and (2.2) coincide with the n-fold integrals [30, chap.2]. The

GENERALIZED FRACTIONAL DERIVATIVES 3

corresponding Riemann-Liouville fractional derivatives Dαa+f and D α b−f of order

α ∈ C, Re(α) ≥ 0 are defined by [66],

(Dαa+f)(x) =

( d

dx

)n ( In−αa+ f

) (x), x > a, (2.3)

and

(Dαb−f)(x) =

( − d dx

)n ( In−αb− f

) (x), x < b, (2.4)

respectively, where n = [Re(α)] + 1. A word about notations is necessary here. We sometimes use the ceiling function, d·e to denote the same quantity [·] + 1, when there is no room for confusion. For simplicity, from this point onwards, we consider only the left-sided integrals and derivatives, except in a few occasions. The interested reader may find more detailed information about right-sided integrals and derivatives in the references, for example in [30,66].

One of the disadvantages of RLFD is that it is not consistant with the physical initial and boundary conditions when it comes to initial or boundary value prob- lems. To overcome this difficulty, M. Caputo coined a variation of RLFD, now known in the litureture as Caputo or Dzherbashyan-Caputo fractional derivative given by [10,15,30],

(cDαa+f)(x) = 1

Γ(n− α)

∫ x a

(x− τ)n−α−1f (n)(τ)dτ, n− 1 < α ≤ n.

Some historical notes about the derivative in question can be found in [42, p. 18- 21]. The interested reader may also find an extended reference lists about Caputo derivative, for example, in the book by Mainardi [42]. An application oriented treatment of the Caputo derivative is given in the book by Diethelm [14]. As pointed out by Hilfer [25], the Caputo derivative was originally introduced by Liouville [38, p.10] though it did not take much attention until Caputo brought the idea back to life in his celebrated paper “Linear models of dissipation whose Q is almost frequency independent, Part II” [9].

An interpolation between the two derivatives mentioned above are defined by the Hilfer fractional derivative of order α and type β given by [24, p.113] [42, p.11],

0D α,β t := I

β(1−α) 0+ ◦D1 ◦ I

(1−β)(1−α) 0+ , 0 < α, β ≤ 1.

The Riemann-Liouville derivative of order α corresponds to the type β = 0, while the Caputo derivative to the type β = 1.

The Weyl-Riesz fractional integration operator of a periodic function f takes the form [44],

Wα = 1

2π

∫ 2π 0

Ψα(τ)f(x− τ) dτ,

where

Ψα(τ) = 2

∞∑ n=1

cosnτ

nα , and 0 ≤ τ ≤ 2π.

Further properties of this operator can be found in [65,66] and the references therein. Jumarie proposed a simple modification to the Riemann-Liouville derivative given by [3, 26,27],

Dαxf(x) = 1

Γ(n− α) dn

dxn

∫ x 0

(x− τ)n−α−1 [ f(τ)− f(0)

] dτ, n− 1 < α ≤ n.

4 U.N. KATUGAMPOLA

with the property that the derivative of a constant function being equal to zero, a property important in Applied Mathematics and Engineering applications.

Another fractional integral that is important in potential theory is the Riesz fractional integral given by [2],

RIαt f(t) = 1

2Γ(α)

∫ b a

|t− τ |α−1f(τ) dτ, α > 0. (2.5)

Two other variations of this integral, one with a factor of 1/ cos(πα/2) and another with a factor of 2 can be found in [66] and [55], respectively. The corresponding Riesz fractional derivative and Riesz-Caputo derivative of (2.5) are defined by [2],

RDαt f(t) = 1

Γ(n− α)

( d dt

)n ∫ b

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