IEEE Systems, Man and Cybernetics Magazine - July 2018 - 25

independently by Isaac Newton and Leibniz [12]. However,
the history of fractional calculus--and integer-order differential calculus--started with the important concept of the
geometrization of time, attributable to John Barrows [13],
who was Newton's predecessor in the Lucasian chair of
mathematics at Cambridge University. This allowed Newton
and his followers to describe colliding bodies and gravitational attraction as curves in space and time, and, suddenly,
all geometric tools and methods became applicable to the
study of dynamic processes. Also, there was an important
difference in notation: while Newton was using dots to indicate differentiation (xo , xp , f), a notation still surviving in
physics textbooks, Leibniz was using letters to denote the
order of differentiation ^d n x/dt n, n = 1, 2, f h, which immediately provoked L'Hospital (30 September 1695) to ask
his famous question about the meaning of the operation
when n = 1/2 [14].
This idea was further elaborated by Leonhard Euler
when he employed fractional-order operators to formulate
and solve a fractional-order differential equation using
what we now call beta and gamma functions [14]. Next followed Niels Henrik Abel's solution to the tautochrone
problem: find the curve in space such that a body sliding
on it without friction will reach the end in the same time
regardless of its initial position [14]. Abel expressed his
answer [15] in the form of a fractional-order integral equation (Figure 3). The tautochrone equation was first given
by Abel in 1823 as

#

0

x

sl (h) dh
(s - h) a = } (x),

whose solution he expressed as differentiation with a negative fractional order:

relative time, calling the latter duration, a term popularized
by Henri Bergson [17]. Wiener, in the first part of his 1948
book [3] on cybernetics, calls this a distinction between
Newtonian time and Bergsonian time. Leibniz clearly understood these ideas when he described space as "an order of
coexistences" and time as "an order of successions," concepts that closely parallel the modern meaning of the fractional-order operators in space and time (Figure 4). Such
fractional-order operators seem ideal for modeling real systems as irreversible dynamical processes that evolve
according to their own individual time scales.

s

t

Figure 3. the tautochrone problem defines a family

of processes that have the same time scale but
different arc lengths, i.e., different length scales. this
diagram can be viewed as an animation at [42].
(Image courtesy of Wikipedia.)

d - a } (x)
1
s (x) = C (1 - a) dx - a .
0 βU
0 zx zβ

t

α
0Dt U

h

α

Bn

- a2

Em - a 2 En

0 βU = F
0 zx zβ
Rmβ unm = fnm
"

α
0Dt U

"

This notation indicates that Abel realized that he had
inverted the fractional derivative that appears on the lefthand side of the tautochrone equation.
The notation for fractional-order derivatives and integrals has also gone through stages of development and
application, from the left-sided and right-sided fractional
derivatives introduction by Joseph Liouville in 1832 [14] to
the integral form used today that reflects the fact that the
value of the fractional derivative depends on both the
order of the operator and the interval over which the operation is evaluated. In addition, applications in physics have
led to the symmetric, or double-sided, fractional-order
Riesz derivative in space and to the Caputo fractionalorder time derivative that allows the classical formulation
of initial conditions [16].
Leibniz once said, "I hold space, and also time, to be
something purely relative. Space is an order of coexistences
and time is an order of successions" [12]. Both Newton and
Leibniz, the inventors of differential calculus, considered what
we may designate as ideal absolute time and measurable

Figure 4. the discretization and solution of the

diffusion-wave equation with fractional-order
derivatives with respect to time and space is an
illustration of the mutual links between space
(coexistences) and time (successions) and of the
feedback loop in the algorithm of the numerical
solution of fractional-order equations. this figure
image is from the toolbox "matrix approach to
Discretization of ODEs and PDEs of arbitrary real
Order," matLab Central File Exchange ID 22071 [43].
(Image courtesy of Wikipedia.)

Ju ly 2018

IEEE SyStEmS, man, & CybErnEtICS magazInE

25



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