Local Fractional Derivatives and Fractal Functions of Several Variables

4 Pages·0.1 MB·English

by  Kiran M. Kolwankar

(physics/9801010)
Local fractional derivatives and
fractal functions of several variables
Kiran M. Kolwankar∗ and Anil D. Gangal†
Department of Physics, University of Pune, Pune 411 007, India.
Thenotionofalocalfractionalderivative(LFD)wasintroducedrecentlyforfunctionsofasingle
variable. LFD was shown to be useful in studying fractional differentiability properties of fractal
and multifractal functions. It was demonstrated that the local H¨older exponent/ dimension was
8
directly related to themaximum order for which LFD existed. Wehaveextended thisdefinition to
9
directional-LFDforfunctionsofmanyvariablesanddemonstrateditsutilitywiththehelpofsimple
9
examples.
1

n I. INTRODUCTION Fractionalcalculus [13–15]is a study whichdeals with
a
generalization of differentiation and integration to frac-
J
0 Fractalandmultifractalfunctionsandthecorrespond- tionalorders. Therearenumberofways(notneccesarily
1 ing curves or surfaces are found in numerous places in equivalent)ofdefining fractionalderivativesandintegra-
nonlinear and nonequillibrium phenomenon. For exam- tions. We use the Riemann-Liouville definition. We be-

] ple, isoscalarsurfacesfor advectedscalarsin certaintur- gin by recalling the Riemann-Liouville definition of frac-
h
bulence problems [1,2], typical Feynman [3,4] and Brow- tionalintegralofarealfunction,whichisgivenby[13,15]
p
- nianpaths[5,6],attractorsofsomedynamicalsystems[7] dqf(x) 1 x f(y)
h are,amongmanyothers,examplesofoccurenceofcontin- = dy for q <0, (2)
at uous but highly irregular (nondifferentiable) curves and [d(x−a)]q Γ(−q)Za (x−y)q+1
m surfaces. Velocity field of a turbulent fluid [8]at low vis- where the lower limit a is some real number and of the
[ cosityisawell-knownexampleofamultifractalfunction. fractional derivative
Ordinarycalculus isinadequate tocharacterizeandhan- dqf(x) 1 dn x f(y)
1 dle such curves and surfaces. Some recent papers [9–12] = dy (3)
v [d(x−a)]q Γ(n−q)dxnZ (x−y)q−n+1
indicate a connectionbetweenfractionalcalculus [13–15] a
0
and fractal structure [5,6] or fractal processes [16–18]. for n − 1 < q < n. Fractional derivative of a simple
1
0 However the precise nature of the connection between function f(x)=xp p>−1 is given by [13,15]
1 the dimension of the graph of a fractal curve and frac-
dqxp Γ(p+1)
0 tional differentiability properties was recognizedonly re- = xp−q for p>−1. (4)
8 cently. A new notion oflocal fractional derivative (LFD) dxq Γ(p−q+1)
9
wasintroduced[19,20]tostudyfractionaldifferentiability Furtherthefractionalderivativehastheinterestingprop-
/
s properties of irregular functions. An intresting observa- erty (see ref [13]), viz,
c
tion of this work was that the local Ho¨lder exponent/
si dimension was related to the maximum order for which dqf(βx) =βqdqf(βx) (5)
y LFD existed. In this paper we briefly review the con- dxq d(βx)q
h
cept of LFD as applied to a function of one variable and
p whichmakesitsuitableforthestudyofscaling. Notethe
generalize it for functions of severalvariables.
: nonlocal character of the fractional integral and deriva-
v Anirregularfunction ofone variableisbest character-
i tive in the equations (2) and (3) respectively. Also it is
X ized locally by a Ho¨lder exponent. We will use the fol-
clear from equation (4) that unlike in the case of integer
lowing general definition of the Ho¨lder exponent which
r derivatives the fractional derivative of a constant is not
a has been used by various authors [21,22] recently. The
zero in general. These two features make the extraction
exponent h(y) of a function f at y is given by h such
of scaling information somewhat difficult. The problems
that there exists a polynomial P (x) of ordern, where n
n wereovercomebytheintroductionofLFDin[19]. Inthe
is the largest integer smaller than h, and
followingsection we briefly review the notion ofLFD for
|f(x)−P (x−y)|=O(|x−y|h), (1) the real valued functions of real variable. In the section
n
III we generalize this definition to real valued functions
for x in the neighbourhood of y. This definition serves of many variables and demonstrate it with the help of
to classify the behavior of the function at y. some simple examples.
∗email: [email protected]
†email: [email protected]
II. LOCAL FRACTIONAL DIFFERENTIABILITY according to definitions 2 and 3 is γ. It is clear that the
critical order of the C∞ function is ∞.
In [19] it was shown that the Weierstrass nowhere dif-
Unfortunately, as noted in section I, fractional deriva-
ferentiable function, given by
tives are not local in nature. On the other hand it is de-
sirableandoccasionallycrucialtohavelocalcharacterin
∞
wide range of applications ranging from the structure of W (t)= λ(s−2)ksinλkt, (8)
λ
differentiable manifolds to various physical models. Sec- X
k=1
ondly the fractional derivative of a constant is not zero,
consequently the magnitude of the fractional derivative whereλ>1,1<s<2andtreal,iscontinuouslylocally
changeswiththeadditionofaconstant. Theappropriate fractional differentiable for orders below 2−s and not
newnotionoffractionaldifferentiabilitymustbypassthe for orders between 2−s and one. This implies that the
hindrance due to these two properties. These difficulties criticalorderofthis function is 2−s at allpoints. Inter-
were remedied by introducing the notion LFD in [19] as esting consequence of this result is the relation between
follows: box dimension s [6] of the graph of the function and the
critical order. This result has for the first time given a
Definition 1 If, for a function f :[0,1]→IR, the limit
directrelationbetweenthe differentiability propertyand
the dimension. In fact this observationwas consolidated
dq(f(x)−f(y))
IDqf(y)= lim (6) into a general result showing equivalence between criti-
x→y d(x−y)q cal order and the local Ho¨lder exponent of any continu-
ousfunction. TheLFDwasfurthershowntobeusefulin
exists and is finite, then we say that the local fractional
thestudyofpointwisebehaviourofmultifractalfunctions
derivative (LFD) of order q (0<q <1), at x=y, exists.
and for unmasking the singularities masked by stronger
In the above definition the lower limit y is treated as singularities.
a constant. The subtraction of f(y) corrects for the Whenever it is required to distinguish between limits
from right and left sides we can write the definition for
fact that the fractional derivative of a constant is not
zero. Whereas the limit as x → y is taken to remove LFD in the following form.
the nonlocal content. Advantage of defining local frac-
tional derivative in this manner lies in its local nature IDqf(y)= lim dqFN(x,y) (9)
and hence allowing the study of pointwise behaviour of ± x→y±[d±e(x−y)]q
functions. This will be seen more clearly after the devel-
opment of Taylor series below. The importance of the notion of LFD lies in the fact
that it appears naturally in the fractional Taylor’sseries
Definition 2 We define critical order α, at y, as with a remainder term for a real function f, given by,
α(y)=Sup{q|allLFDsof orderlessthanq existaty}. N f(n)(y) IDqf(y)
f(x)= ∆n+ + ∆q +R (y,∆) (10)
q
Γ(n+1) Γ(q+1)
These definitions were subsequently generalized [20] for nX=0
q >1 as follows.
where x−y =∆> 0 and R (y,∆) is a remainder given
q
Definition 3 If, for a function f :[0,1]→IR, the limit by
IDqf(y)=xli→mydq(f(x)−P[dNn(x=0−Γfy((nn))]+(qy1))(x−y)n) (7) Rq(y,∆)= Γ(q1+1)Z0∆ dF(y,dtt;q,N)(∆−t)qdt (11)
and
exists and is finite, where N is the largest integer for
which Nth derivative of f(x) at y exists and is finite, dq(f(x)− N f(n)(y)∆n)
then we say that the local fractional derivative (LFD) of n=0 Γ(n+1)
F(y,∆;q,N)= (12)
order q (N <q ≤N +1), at x=y, exists. P[d∆]q
Weconsiderthisasthegeneralizationofthelocalderiva- We note that the local fractional derivative as defined
tive for order greater than one. Note that when q is a above(notjustfractionalderivative),alongwiththefirst
positive integer ordinary derivatives are recovered. The N derivatives, provides an approximation of f(x) in the
definition of the critical order remains the same since, vicinity of y. We further remark that the terms on the
for q < 1, (6) and (7) agree. This definition extends RHS of eqn(10) are nontrivial and finite only in the case
the applicability of LFD to C1−functions which are still whenqequalsα,thecriticalorder. Osler[23]constructed
irregular due to the nonexistence of some higher order afractionalTaylorseriesusingusual(Riemann-Liouville)
derivative (i.e. belong to class Cγ, γ > 1). For example fractional derivatives which was applicable only to ana-
thecriticalorderoff(x)=a+bx+c|x|γ,γ >1,atorigin, lyticfunction. FurtherOsler’sformulationinvolvesterms
withnegativeordersalsoandhenceisnotsuitableforap- If we choose y = 0, i.e., we examine fractional differen-
proximating schemes. When ∆< 0, a similar expansion tiability at a point on x axis and if we choose v = 0,
y
can be written for IDqf(y) by replacing ∆ by −∆. i.e., we are looking at LFD in the direction of increasing
−
When 0<q <1 we get as a special case x then we get
IDqf(y)
f(x)=f(y)+ (x−y)q+Remainder (13) ∞
Γ(q+1) Φ(x,t)= λ(s−2)k[sinλk(x+v t)−sinλk(x)]
x
providedtheRHSexists. Ifwesetqequaltooneinequa- Xk=1
tion (13) one gets the equation of the tangent. All the
curves passing through a point y and having same the which is known [19] to have critical order 2−s.
tangent,formanequivalence class(whichis modelledby
Ifwekeepy =0butkeepv andv non-zero,i.e.,ifwe
x y
alinearbehavior). Analogouslyallthefunctions(curves)
examine fractional differentiability at a point on x-axis
withthesamecriticalorderαandthesameIDα willform
but in any direction then we get
an equivalence class modeled by the power law xα. This
is how one may generalize the geometric interpretation
∞
of derivatives in terms of tangents. This observation is Φ(x,t)= λ(s−2)k[sinλk(x+v t+v t)−sinλk(x)]
x y
useful in the approximationof anirregularfunction by a
X
k=1
piecewise smooth (scaling) function. One may recognize
the introduction of suchequivalence classes as a starting
point for fractional differential geometry. Thisagainhascriticalorder2−s,exceptwhenvx =−vy
(in which case it is ∞ since the function Φ identically
vanishes).
III. GENERALIZATION TO HIGHER Infactitcanbeshownthatthegivenfunctionhas2−s
DIMENSIONAL FUNCTIONS
ascriticalorderatany pointandin anydirectionexcept
the direction with v =−v .
x y
ThedefinitionoftheLocalfractionalderivativecanbe Example 2: Another example we consider is
generalized for higher dimensional functions in the fol-
lowing manner. ∞
Consider a function f :IRn →IR. We define W (x)=W (x,y)= λ(s−2)ksinλk(xy), (17)
λ λ
X
Φ(y,t)=f(y+vt)−f(y), v∈IRn, t∈IR. (14) k=1
Thenthedirectional-LFDoff atyoforderq,0<q <1,
where λ>1 and 1<s<2.
in the direction v is given (provided it exists) by
dqΦ(y,t) W (x+vt)=W (x+v t,y+v t)
IDvqf(y)= dtq |t=0 (15) λ ∞λ x y
= λ(s−2)ksin(λk(x+v t)(y+v t)).
wherethe RHSinvolvesthe usualfractionalderivativeof x y
X
equation(3). Thedirectional-LFDsalongtheunitvector k=1
e will be called ith partial-LFD.
i
Now let us consider two examples.
Example 1: Let Φ(x,t)=W (x+vt)−W (x)
λ λ
∞ ∞
Wλ(x)=Wλ(x,y)= λ(s−2)ksinλk(x+y), (16) = λ(s−2)k[sin(λk(xy+yvxt+xvyt+vxvyt2))
kX=1 Xk=1
with λ > 1 and 1 < s < 2. Let v = (v ,v ) be a unit −sin(λk(xy))]
x y
2-vector. Then
Nowifwechoosey =0andv =0thenthecriticalorder
W (x+vt)=W (x+v t,y+v t) y
λ λ x y
is ∞.
∞
= λ(s−2)ksinλk(x+vxt+y+vyt). If we choosey =0 but vx and vy non-zerothen we get
X
k=1
∞
Φ(x,t)=Wλ(x+vt)−Wλ(x) Φ(x,t)= λ(s−2)k[sin(λk(xvyt+vxvyt2))]
X
∞ k=1
= λ(s−2)k[sinλk(x+v t+y+v t)
x y
X
k=1 Thereforeusingresultsof[19]thecriticalorderalongany
−sinλk(x+y)] otherdirectionv, atapointonx-axisisseentobe 2−s.
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