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J. Math. Anal. Appl. 270 (2002) 66–79
www.academicpress.com
Oncalculus of local fractional derivatives
A. Babakhani and Varsha Daftardar-Gejji∗
Department of Mathematics, University of Pune, Ganeshkhind, Pune 411007, India
Received 23 January 2001
Submitted by H.M. Srivastava
Abstract
Local fractional derivative (LFD) operators have been introduced in the recent liter-
ature (Chaos 6 (1996) 505–513). Being local in nature these derivatives have proven
useful in studying fractional differentiability properties of highly irregular and nowhere
differentiable functions. In the present paper we prove Leibniz rule, chain rule for LFD
operators. Generalization of directional LFD and multivariable fractional Taylor series to
higher orders have been presented. 2002 Elsevier Science (USA). All rights reserved.
Keywords: Riemann–Liouville fractional derivatives/integrals; Local fractional derivatives; Local
fractional Taylor series
1. Introduction
Fractional calculus [1,2] developed since 17th century thorough the pioneer-
ingworksofLeibniz,Euler,Lagrange,Abel,Liouvilleandmanyothersdealswith
generalizationofdifferentiationandintegrationto fractionalorder.In recentyears
the term “fractional calculus” refers to integration and differentiation to an arbi-
trary order. Complex analytic version of fractional differentiation/integration has
beendiscussedbySrivastavaandOwa[3].Interestinglythesederivatives/integrals
are not mere mathematical curiosities but have applications in visco-elasticity,
feedback amplifiers, electrical circuits, electro-analytical chemistry, fractional
* Corresponding author.
E-mail addresses: ababakhani@hotmail.com (A. Babakhani), vsgejji@math.unipune.ernet.in
(V. Daftardar-Gejji).
0022-247X/02/$ – see front matter 2002 Elsevier Science (USA). All rights reserved.
PII:S0022-247X(02)00048-3
A. Babakhani, V. Daftardar-Gejji / J. Math. Anal. Appl. 270 (2002) 66–79 67
multipoles, neuron modelling and related areas in physics, chemistry, and bio-
logical sciences [2]. It is well known that the fractional derivatives/integrals have
beendefinedinavarietyofwaysas[1,2]givenbyRiemann,Liouville,Grunw˝ ald,
Weylandothers.Thesedefinitions,however,arenon-localinnature,whichmakes
them unsuitable for investigating properties related to local scaling or fractional
differentiability [4]. Kolwankar and Gangal [4,5] have proposed local fractional
derivative (LFD) operator through renormalization of Riemann–Liouville defini-
tion. LFD follows as a natural generalization of the usual derivatives to fractional
orderconservingthelocalnatureofthederivativesincontrasttotraditionaldefin-
itions of fractional derivatives and used further to explore local scaling properties
of highly irregular and nowhere differentiable Weierstrass functions [4]. LFD op-
erators engender a new kind of differential equations, referred as local fractional
differential equations (LFDE) different from the conventionalfractional differen-
tial equations.Thefractionalanalog[6]oftheFokker–Planckequation[7]involv-
ing LFDs has been used in modelling phenomena involving fractal time. LFDs
therefore provide a much needed tool for calculus of fractal space–time.
As a pursuit of these we herein investigate the formal properties of LFD
operators. In the present work we prove Leibniz rule for a product of functions
and subsequently derive chain rule for evaluating LFD of composite function.
Generalizations of directional LFD and fractional multivariable Taylor series to
higher orders have also been presented.
Thepaperhasbeenorganisedasfollows.InSection2wegivebasicdefinitions
in Riemann–Liouville fractional calculus and LFD operator. Leibniz rule and
chain rule for LFD have been derived in Section 3 and Section 4. Extensions
of directional LFDs and local fractional Taylor series to higher orders have been
presented in Sections 5 and 6.
2. Basic definitions and preliminaries
2.1. Riemann–Liouvillefractional calculus
Definitionsof Riemann–Liouvillefractionalderivative/integraland their prop-
erties are given below.
Riemann–Liouvillefractionalderivativeof areal function f is givenfor x>a
as [1,2]
x
dαf(x) = 1 dn f(t) dt, n1αa. (3)
α α+1
d(xa) (α)a (xt)
β
Notethat [1,2] if f(x)=(x a) , β>1,x>a,then
dαf(x) (β+1) βα
α = (x a) . (4)
d(xa) (βα+1)
From(3)and(4)itfollowsthat
α α
d 1 =(xa) ,x>a, (5)
α
d(xa) (1α)
where α is any real number. Composition of the Riemann–Liouville fractional
derivative with integer-order derivatives for f ∈ Cn, α>0,n∈ N [2] gives
α+n α (n) n1 (j) jαn
d f(x) = d f (x) +f (a)(xa) ,x>a. (6)
α+n α
d(xa) d(xa) j=0 (1+jαn)
α α
If the fractional derivative d f(x)/d(x a) of a function f(x)is integrable,
then [2]
α dαf(x) n αj αj
d α d f(x) (x a)
d(xa) =f(x) , (7)
α αj
d(xa) j=1 d(xa) x=a (αj+1)
wheren1αa.
Leibniz rule for fractional differentiation is given below [2].
If f(x) is continuous in [a,b] and ϕ(x) ∈ Cn+1[a,b], then the fractional
derivative of the product ϕ(x)f(x)is given by
α n αk
d (ϕ(x)f(x)) = α ϕ(k)(x) d f(x) Rα(x),
α αk n
d(xa) k=0 k d(xa)
0<αn1, (8)
where
x x
α 1 α1 (n+1) n
R (x)= (x u) f(u)du ϕ (r)(ur) dr (9)
n n! (α)
a u
α 1
and k isthegeneralizedbinomialcoefficient(= (α+1)(k! (αk+1)) ).
A. Babakhani, V. Daftardar-Gejji / J. Math. Anal. Appl. 270 (2002) 66–79 69
The fractional derivative of the composite analytic function ϕ(x)= f(h(x))
[2] turns out to be
α α ∞ kα k
d ϕ(x) (x a) α k!(xa) (m)
α = ϕ(x)+ f h(x)
d(xa) (1α) k=1 k (kα+1)m=1
k
(r) ar
× 1 h (x) , (10)
r=1 ar! r!
where the sum
extends over all combinations of non-negative integral values
of a ,a ,...,a such that k ra =nand k a =m.
1 2 k r=1 r r=1 r
2.2. Local fractional derivative
Kolwankar and Gangal [4] have defined LFD as follows. If for a function
f :[0,1]→R,thelimit
dα(f(y)f(x))
Dαf(x)= lim , 0<α<1, (11)
± y→x± α
d(±(yx))
exists and is finite, then f is said to have right (left) LFD of order α at y = x.If
for a function f :[0,1]→R, the limit
α
n (k) k
Dαf(x)= lim d f(y) k=0(f (x)/ (k+1))(yx) (12)
± y→x± α
d(±(yx))
exists and is finite, where n is the largest integer for which nth-order derivative
of f(y)at x exist and is finite, then Dα f(x)are called as the right (left) LFD of
order α(n<α
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