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8/18/2019 PHYSICAL AND GEOMETRICAL INTERPRETATION OF GRUNWALD-LETNIKOV DIFFERINTEGRALS-MEASUREMENT OF PA…

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DISCUSSION SURVEY

PHYSICAL AND GEOMETRICAL INTERPRETATION

OF GRÜNWALD-LETNIKOV DIFFERINTEGRALS:

MEASUREMENT OF PATH AND ACCELERATION

Radoslaw Cioć

Abstract

A function f (t) of the independent variable t changing with every in-crement dt can be formulated as a functional sequence. If g

f (t)

is a

derivative or an integral of f (t) and the value of dt is interpreted subjectto an error ∆T , then g

f (t)

is Grünwald-Letnikov differintegral of that

sequence with an order closely related to dt and ∆T . This paper illustratesthis relationship and proposes a geometrical and physical interpretation of

a fractional order Grünwald-Letnikov differintegrals using the example of path and acceleration measurements of a point in motion.

MSC 2010 : Primary 26A33; Secondary 28E05, 33E30, 34A25

Key Words and Phrases : fractional calculus, Grünwald-Letnikov differ-integrals, fractional order interpretation, measure theory

1. Introduction

Some geometrical and physical interpretations of fractional order deriva-tives and integrals are described by I. Podlubny [11], who bases on S. Samkoet al. [14], R.S. Rutman [12] and others. R. Herrmann [4], J.F. Gómez-Aguilar et al. [3], A.G. Butkovskii et al. [1], J. Sabatier et al. [13], N.Heymans et al. [6], R. Hilfer [5] have published also on the subject re-cently. None provides an unambiguous interpretation of fractional calculusthat would refer to the physical interpretation of derivative and integralas a path and acceleration measurement of a point in motion. This is thefocus of this paper.

c 2016 Diogenes Co., Sofiapp. 161–172 , DOI: 10.1515/fca-2016-0009

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162 R. Cioć

2. A general definition of Grünwald-Letnikov differintegralof a functional sequence

Let f (t) be a function of real variable t. The first derivative of f (t) isdefined as:

f

(t) = f (1)(t) = df (t)

dt = lim

dt→0

f (t + dt) − f (t)

dt = tan α, (2.1)

where:dt is the increment of the independent variable t,df (t) is the increment of a function dependent on t.

The second derivative is:

f

(t) = f (2)(t) = limdt→0

f

(t + dt) − f

(t)dt

= limdt→0

f (t) − 2f (t + dt) + f (t + 2dt)

(dt)2 . (2.2)

The derivative of any order n ∈ N is formulated as:

f (n)(t) = limdt→0

nm=0(−1)

mnm

f (t − mdt)

(dt)n , (2.3)

where:nm

= n!

m!(n−m)! for n m.

The subsequent values of (t − mdt) are indexed:

tm = t − mdt, (2.4)where:dt = t0 − t1 = t1 − t2 = · · · = tl−1 − tl,m = 0, 1, 2, . . . , l,l = t0−tl

dt (the floor function).

A function f (t − mdt) can be expressed as a functional sequence, con-sidering (2.4):

f (t − mdt) = {f (t)}0...l = {f 0(tl), f 1(tl−1) . . . f l(t0)}. (2.5)

By substituting (2.5) to (2.3), the nth order derivative (n ∈ N) is pro-duced, referred to as Grünwald-Letnikov differintegral [2, 7, 9, 10] of func-tional sequence (GLs for short):

{f (n)(t)} = limdt→0

1(dt)n

lm=0

(−1)m

nm

{f l−m(tm)} ≡ Dntl t0{f (t)}0...l, (2.6)

where:{f l−m(tm)} is the m element of the sequence {f (t)}0...l,l ≥ n,

Dntl t0{f (t)}0...l is Davis’ notation of the derivative.


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PHYSICAL AND GEOMETRICAL INTERPRETATION . . . 163

3. Physical and geometrical interpretationof Grünwald-Letnikov differintegral of a positive order

functional sequence (GLs+)

For n > 0, the binomial coefficient can be presented in terms of theGamma function:

n

m

=

n!

m!(n − m)! =

Γ(n + 1)

m!Γ(n − m + 1) . (3.1)

By generalizing (2.6) with the aid of (3.1) to orders (n > 0) ∈ R,Grünwald-Letnikov differintegral of a positive order of functional sequence(GLs+ for short) is obtained, see [2, 7, 9, 10]:

Dηtl t0{f (t)}0...l = limdt→0

1

(dt)η

lm=0

(−1)m Γ(η + 1)

m!Γ(η − m + 1){f l−m(tm)} . (3.2)

The first derivative of functional sequence (2.5) determined for l = 1corresponds to η = 1 order GLs+ and is equal to (2.1):

D1tl t0{f (t)}0...1 = limdt→0

{f 1(t0)} − {f 0(t1)}

dt =

d{f (t)}0...1dt

= tan α. (3.3)

Based on formula (3.2), GLs+ for l = 1 and (η > 0) ∈ R is expressedby:

D1 ηtl t0{f (t)}0...1 = limdt→0

{f 1(t0)} − η{f 0(t1)}

(dt)η =

d{f (t)}0...1(dt)η

. (3.4)

An additional magnitude is introduced to the left upper section of Davis’ notation in (3.4), l value of formula (3.2), which is also the equiva-lent of the complete order derivative (2.6) for η > 0 order of the derivative(1 stands for the first derivative, 2 for the second, etc.).

Let a functional sequence of two elements:

{v(t)}0...1 = {v0(t1), v1(t0)} (3.5)

represent velocity measurements of a moving point executed at two consec-utive instants t1 and t0 at every time interval dt (2.4).

Let a stand for instantaneous acceleration:

a = limdt→0

{v1(t0)} − {v0(t1)}

dt . (3.6)

The acceleration formula (3.6) is identical with the equation describingGLs+ for (η = 1) and (l = 1), (3.3).

The acceleration (3.6) is determined on the basis of measurements of variable velocity, whose value depends on the instant of measurement. Thevalue of acceleration is thus dependent on the precision of determining theinstant of measurement. Since that instant is determined as a multiple


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164 R. Cioć

of the time interval dt, the accuracy of its determination and thereby in-directly the accuracy of acceleration measurement depends on a precisedetermination of dt.

Let ∆T stand for the measurement error of dt. It is interpreted as anabsolute error and added as positive or negative to dt. Assuming ∆T istaken into consideration at the start of dt, the acceleration can be formu-lated as follows:

a∆T = limdt→0

{v1(t0)} − {v0

t1 + (±∆T )

}

dt + (±∆T ) . (3.7)

Let (dt)η stand for variation of dt considering ∆T :

(dt)η

= dt + (±∆T ). (3.8)The impact of ∆T on dt and velocity measurement is illustrated in Fig.

3.1.

Fig. 3.1: Geometrical interpretation of Grünwald-Letnikov differintegralof positive order functional sequence (GLs+)

The velocities at t1 + ∆T and t1 − ∆T are not known as the mea-surements are carried out every dt. Therefore, acceleration including ∆T cannot be defined. Given similarities b etween (3.6) and (3.3) as well as(3.7) and (3.4), velocities in desired time points can be estimated by meansof a GL+ order. These similarities imply:

{v0(t1 + (±∆T )} = η{v0(t1)} ∼= v

t1 + (±∆T )

, (3.9)

where η is a parameter (as well as the order of GLs+) estimating the velocityv at t1 + (±∆T ) and derived from (3.8):

η = logdt

dt + (±∆T )

. (3.10)

Figure 3.2 shows the dependence of η on the percentage value of | ±∆T |relative to dt.


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Fig. 3.2: Dependence of η on |±∆T |/dt

Substituting (3.8) to (3.7) and considering (3.9), an instantaneous ac-celeration formula is obtained which corresponds to the fractional orderdifferintegral GLs+, (3.4):

a∆T

= limdt→0

{v1(t0)}−{v0(t1+|∆T |)dt+|∆T | = tan β

limdt→0

{v1(t0)}−{v0(t1−|∆T |)dt−|∆T | = tan γ

= limdt→0

{v1(t0)}−η{v0(t1)}(dt)η

∀η>1 ∧

v0(t1) < v1(t0)

∀η v1(t0)

≡ D1 ηtl t0{f (t)}0...1 .Analyzing the procedure illustrated with the dependencies (3.1) through

(3.11), one can conclude: the Grünwald-Letnikov positive order differinte-gral (3.4) of a functional sequence (3.5) describing velocity of a point inmotion is interpreted as acceleration of the same point, determined on thebasis of measurements of its velocity read every time interval dt subjectto a measurement error of ∆T , where the order of the Grünwald-Letnikovdifferintegral is related to dt and ∆T by means of (3.8).

4. Physical and geometrical interpretationof Grünwald-Letnikov differintegral of a negative order

functional sequence (GLs-)

For n


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By generalizing (2.6) with the aid of (4.1) to orders (−η


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= limdt→0

(dt)ηl

m=0

{f l−m(tm)} ≡

t0 τ l−(dt)η

{f (t)}0...l(dt)η

=

t0 τ 1

{f l(t0)}(dt)η +

τ 1 τ 2

{f l−1(t1)}(dt)η

+ · · · +

τ l τ l−(dt)η

{f 0(tl)}(dt)η , (4.5)

where: τ 1 =

t0 − (dt)η

, τ 2 =

τ 1 − (dt)η

, τ 3 =

τ 2 − (dt)η

, . . .Fig. 4.3 compares graphical representations of GLs- orders (−η = −1)

and (−1 < −η) for l = 2 and dt


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The path (4.7) is determined by measurements of velocity, whose valuedepends on the instant of measurement and thereby on precision of deter-mining that instant. Since that instant is determined as a multiple of thetime interval dt, accuracy of its determination and thus accuracy of pathdetermination depend on a precise determination of dt.

Let ∆T stand for measurement error of dt. ∆T is assumed to havea constant sign and value for each element of the sequence ( 4.6). ∆T isinterpreted as an absolute error and added as positive or negative to dt.

Including ∆T in (4.7) produces:

S ∆T = limdt→0

dt + (±∆T )

lm=0

{vl−m

tm + (±∆T )

}

=

t0 τ 1

{vl(t0)}(dt ± ∆T ) +

τ 1 τ 2

{vl−1(t1)}(dt ± ∆T )

+ · · · +

τ l τ l+1

{v0(t1)}(dt ± ∆T ), (4.8)

where:

τ 1 = t0 − (dt ± ∆T ), τ 2 = τ 1 − (dt ± ∆T ),. . . , τ l+1 = τ l − (dt ± ∆T ).

As quadrature rules (for which value of the function is constant in therange of integration step) are applied to determine the path, it is assumedthat:

vl−m

tm(±∆T )

= vl−m(tm). (4.9)

Let (dt)η stand for an individual interval of measurement time (identicalwith the range of integration step) considering ∆T like in the equation (3.8).

Considering 4.9 and the measurement time interval 3.8, S ∆T becomes:

S ∆T = limdt→0

(dt)ηl

m=0

{vl−m(tm)} ≡ D−η

tl t0 {v(t)}0...l . (4.10)

The path described by (4.10) corresponds to negative order Grünwald-Letnikov differintegral (GLs-), (4.5).


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Fig. 4.4: Geometrical interpretation of negative order Grünwald-Letnikovdifferintegral as a path measurement

The path calculations (4.10) will show maximum accuracy if the numberof elementary paths in the time interval under discussion tends to infinity(l → ∞) and if the time division tends towards 0 (dt → 0). In actual


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170 R. Cioć

measurements, b oth the number of elementary paths and time division arefinite, therefore, the following is assumed:

(0 < dt 1 (∆T < 0), a minimum path S min will be determined, amaximum path S max will be determined for η 0), while a pathS 0 liable to an error ∆T = 0 will be determined for η = 1:

S min = limdt→0∆T1

(dt)ηl

m=0

{vl−m(tm)}, (4.12)

S max = limdt→0∆T>0

(dt + ∆T )l

m=0

{vl−m(tm)} = limdt→0η


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[2] S. Das, Functional Fractional Calculus for System Identification and Controls . Springer-Verlag, Berlin Heidelberg (2008).

[3] J.F. Gómez-Aguilar, R. Razo-Hernández, D. Granados-Lieberman,A physical interpretation of fractional calculus in observables terms:Analysis of the fractional time constant and the transitory response.Revista Mexicana de F́ısica

60 (2014), 32–38.[4] R. Herrmann, Towards a geometric interpretation of general-ized fractional integrals – Erd́elyi-Kober type integrals on RN ,as an example. Fract. Calc. Appl. Anal. 17, No 2 (2014), 361–370;DOI: 10.2478/s13540-014-0174-4;http://www.degruyter.com/view/j/fca.2014.17.issue-2/

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Faculty of Transport and Electrical Engineering Kazimierz Pulaski University of Technology and Humanities in Radom

Malczewskiego Str. 29

Radom 26-600, POLAND Received: January 16, 2015

e-mail: [email protected] Revised: November 20, 2015

Please cite to this paper as published in:

Fract. Calc. Appl. Anal., Vol. 19, No 1 (2016), pp. 161–172,DOI: 10.1515/fca-2016-0009


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