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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