Connecting Caputo-Fabrizio Fractional Derivatives Without Singular Kernel and Proportional Derivatives

Connecting Caputo-Fabrizio Fractional Derivatives Without Singular Kernel and Proportional Derivatives Concordia College, Moorhead, Minnesota USA 7 July 2018

Intro

Caputo and Fabrizio (2015, 2017) introduced a new fractional time derivative without singular kernel given by Dt α f (t) = 1 t f (τ)e α 1 α (t τ) dτ. 1 α a If α = 0, we recover D 0 t f (t) = t a f (τ)dτ = f (t) f (a). If α = 1, then D 1 t f (t) = f (t). Their related fractional time integral is ait α f (t) = 1 t α a f (τ)e 1 α α (t τ) dτ.

For α [0, 1], we will define a proportional derivative via D α f = αf + (1 α)f. This talk will lead to a proportional integral of f by ai α t f (t) := 1 α t Note that we then have the relationships a f (τ)e 1 α α (t τ) dτ. (1.1) D α t f (t) = a I 1 α t f (t) and a I α t f (t) = a I α t f (t).

(Naive) Motivation In control theory (see Ding, et al), a proportional-derivative (PD) controller takes the form where u = controller output t = time κ d = derivative gain κ p = proportional gain u(t) = κ d d dt E(t) + κ pe(t), E = error between state and process variables

PD-Controller: Postpone Onset of Hopf Bifurcation d u(t) = κ d dt E(t) + κ pe(t) For an example of a recent use of PD-controllers, let x denote a network congestion window and y be the queuing delay. A model proposed by Darvishi and Mohammadinejad (2016) is ( ) b x(t)y(t τ) ẋ(t) = a + κ d E x (t) + κ p E x (t) ẏ(t) = 1 c d + y(t τ) ( x(t) d + y(t τ) c (d + y(t τ)) ) 2 + κ d E y (t) + κ p E y (t), where E x (t) = x(t) x and E y (t) = y(t) ȳ.

Proportional α-derivative Definition u(t) = κ d d dt E(t) + κ pe(t) For α [0, 1], the differential operator D α is defined via D α f (t) = αf (t) + (1 α)f (t) provided the function f is differentiable at t, where f := d dt f. This will be called the proportional α-derivative.

Example using the α-derivative Example For α [0, 1], consider the α-derivative of the function f (t) = t 2 at α = n 8, n Z, 0 n 8. Then D α f (t) = αf (t) + (1 α)f (t) becomes which looks like... D n/8 f (t) = nt 4 + 8 n 8 t2,

Example of the α-derivative (cont.) 8 6 t 2 4 2-2 -1 1 2 3-2 2t -4 Figure: In this graph, we can see that as α increases from 0 to 1, the curve t 2 bends toward 2t.

Basic α-derivatives Lemma Let D α f (t) = αf (t) + (1 α)f (t), where α [0, 1]. Then 1 D α [af + bg] = ad α [f ] + bd α [g] for all a, b R; 2 D α [c] = c(1 α) for all constants c R; 3 D α [fg] = fd α [g] + gd α [f ] (1 α)fg; 4 D α [f /g] = gdα [f ] fd α [g] g 2 + (1 α) f g.

Lines and MVT

The α-constant: e 0 (t, s) Definition (The α-exponential Function) For α (0, 1] and s t, define the α-exponential function as Then In particular, for p 0, t p(τ) (1 α) e p (t, s) := e s α dτ. D α e p (t, s) = p(t)e p (t, s). 1 α e 0 (t, s) = e ( α )(t s), and D α e 0 (t, s) = 0.

Properties of the α-exponential The exponential function e p (t, s) has the following properties: (i) e p (t, t) 1. (ii) e p (t, s)e p (s, r) = e p (t, r). (iii) 1 e p(t,s) = e p(s, t).

Visualizing the α-constants: ce 0 (t, s) 2 1-1 1 2 3 4-1 -2 Figure: For α = 0.5, various constant functions.

α-lines Definition Using α-exponentials and our derivative, we define a horizontal α-line to be a function f with slope D α f = 0, and thus of the form 1 α f (t) = ce 0 (t, s) = ce ( α )(t s). A general α-line is a function of the form f (t) = c 1 e 0 (t, s) + c 2 e 0 (t, s)t. These are known as geodesics, curves with zero acceleration.

Parallel Lines Below, given the line y 1 and a point p not on y 1, there exists exactly one line y 2 intersecting p and parallel to y 1. 1.0 y 1 0.5 y 2-1 1 2 3 4 5 p -0.5

α-secant and α-tangent Lines Theorem Let α (0, 1] and h 1 (t, a) := t a α. Two important geodesics are the α-secant line for a function f from a to b given by σ(t) := e 0 (t, a)f (a) + h 1 (t, a) e 0(t, b)f (b) e 0 (t, a)f (a), h 1 (b, a) and the α-tangent line for a function f differentiable at a given by l(t) := e 0 (t, a)f (a) + h 1 (t, a)e 0 (t, a)d α f (a).

Visualizing α-derivatives Tangent Lines 30 20 10 0-5 0 5 Figure: For α = 2/3, f (t) = t 2 on its slope field.

Tangent Function: Extrema for the α-derivative 0.0-0.5-1.0-1.5-2.0-2.5-1.5-1.0-0.5 0.0 0.5 1.0 1.5 Figure: For α = 0.275, f (t) = tan t on its slope field.

Tangent Function: Critical Points Explained tan (t) α= tan (t) - sec (t) 2 1 3 α 1 4 1 6 1 12 - π 2-3 π 8 - π 4 - π 8 t Figure: For α [0, 1], the zeros of D α [tan t] illustrated for α as a function of t.

Rolle s Theorem Note that if the α-secant line for a function f is an α-line with slope = 0 from (a, f (a)) to (b, f (b)), then f (a) = e 0 (a, b)f (b). Theorem Let α (0, 1]. If the function f is continuous on [a, b] and differentiable on (a, b), with f (a) = e 0 (a, b)f (b), then there exists at least one number c (a, b) such that D α f (c) = 0.

Rolle s Theorem Example -5-4 -3-2 -1 c b -1-2 -3 σ(t) a f(t)=t l(t) α = 1 2-4 -5

Mean Value Theorem Theorem Let α (0, 1]. If the function f is continuous on [a, b] and differentiable on (a, b), then there exists at least one number c (a, b) such that where h 1 (b, a) = b a α. D α f (c) = e 0(c, b)f (b) e 0 (c, a)f (a), h 1 (b, a)

Picture of MVT Example for α = 0.5 1.15 1.10 1.05 1.00 0.95 a σ(t) f(t)=1 c b 0.90 0.85 l(t) 0.80 α = 1 2 0.0 0.2 0.4 0.6 0.8 1.0

FTC

Definite α-integral Definition Let α [0, 1]. Then the definite α-integral from a to b of an integrable function f is ai α t f (τ) := 1 α t a f (τ)e 1 α α (t τ) dτ.

The Fundamental Theorem of Calculus Theorem (Fundamental Theorem of Calculus) Let α [0, 1]. (FTC I) If f is integrable, then D α [ a I α t f (τ)] = f (t). (FTC II) If f : [a, b] R is differentiable on [a, b] and f is integrable on [a, b], then ai α b Dα [f (t)] = f (b) f (a)e 0 (b, a).

Proof of FTC II: Let P be any partition of [a, b], P = {t 0, t 1,, t n }, and recall that h 1 (t, a) = t a α. By MVT applied to f on [t i 1, t i ], there exist c i (t i 1, t i ) such that or equivalently D α f (c i ) = e 0(c i, t i )f (t i ) e 0 (c i, t i 1 )f (t i 1 ), h 1 (t i, t i 1 ) D α [f (c i )]e 0 (b, c i )h 1 (t i, t i 1 ) = e 0 (b, t i )f (t i ) e 0 (b, t i 1 )f (t i 1 ). After forming the Riemann-Stieltjes sum n S(P, f, µ) = D α [f (c i )]e 0 (b, c i ) (µ(t i ) µ(t i 1 )) i=1 i=1 with µ(t) := t a α = h 1(t, a), we see that n S(P, f, µ) = e 0 (b, t i )f (t i ) e 0 (b, t i 1 )f (t i 1 ) = f (b) e 0 (b, a)f (a).

Since the partition P was arbitrary, f (b) e 0 (b, a)f (a) = = b a b a b D α [f (t)]e 0 (b, t)dµ(t) D α [f (t)]e 0 (b, t)µ (t)dt = 1 D α [f (t)]e 0 (b, t)dt α a = a Ib α Dα [f (t)]. This completes the proof of FTC II.

α Extrema

α-increasing and α-decreasing Definition Let α (0, 1]. A function f is strictly α-increasing on an interval I if f (t 1 ) < e 0 (t 1, t 2 )f (t 2 ), whenever t 1 < t 2, t 1, t 2 I, and is strictly α-decreasing if e 0 (t 2, t 1 )f (t 1 ) > f (t 2 ), whenever t 1 < t 2, t 1, t 2 I. Theorem (α-increasing/α-decreasing Test) Letting α (0, 1], suppose that D α f exists on some interval I. 1 If D α f (t) > 0 for all t I, then f is strictly α-increasing on I. 2 If D α f (t) < 0 for all t I, then f is strictly α-decreasing on I.

α-increasing/α-decreasing Example 4 2 0-2 -4 f(t)=t -4-2 0 2 4 α = 1 2

Critical Point and α-max/α-min Definition (Critical Point) Let α [0, 1]. A function f has a critical point at t 0 if D α f (t 0 ) = 0 or D α f (t 0 ) does not exist. Definition (Max/Min) Let α (0, 1]. A function f has 1 a local α-max at t 0 if f (t 0 ) e 0 (t 0, t)f (t); 2 a local α-min at t 0 if f (t 0 ) e 0 (t 0, t)f (t) for all t near t 0.

First α-derivative Test Theorem Letting α (0, 1], suppose that t 0 is a critical point of f. 1 If D α f changes from + to at t 0, then f has a local α-max at t 0. 2 If D α f changes from to + at t 0, then f has a local α-min at t 0. 3 If D α f does not change sign at t 0, then f has neither an α-max nor α-min at t 0.

Concavity Test Theorem Letting α (0, 1], suppose D α D α f exists on some interval I. 1 If D α D α f (t) > 0 t I, then the graph of f is concave upward on I. 2 If D α D α f (t) < 0 t I, then the graph of f is concave downward on I. Here concave upward means the curve y = f (t) lies above all of its α-tangents l(t) on I, and concave downward means the curve y = f (t) lies below all of its α-tangents l(t) on I.

Second α-derivative Test Theorem Letting α (0, 1], suppose D α D α f is continuous near t 0. 1 If D α f (t 0 ) = 0 and D α D α f (t 0 ) > 0, then f has a local α-min at t 0. 2 If D α f (t 0 ) = 0 and D α D α f (t 0 ) < 0, then f has a local α-max at t 0.

Figure of α-max/α-min for f (t) = t 2 5 4 α-max 3 2 1 0 α-min -1 α = 1 2-3 -2-1 0 1 2 3

α-minimum and α-maximum Values of sin kt Theorem Let α (0, 1], and assume k is any real, positive number and n is an integer. The function sin kt has an α-max at ( 1 ( ) ) kα k arctan + nπ 1 α k, kα, n odd, (k 2 + 1)α 2 2α + 1 and an α-min at ( 1 ( ) ) kα k arctan + nπ 1 α k, kα, n even. (k 2 + 1)α 2 2α + 1

New Development Recently (February, 2018), V. E. Tarasov published a paper, No nonlocality, No fractional derivative specifically mentioning the Caputo-Fabrizio derivative and saying that it is not a fractional derivative.

Acknowledgements This research was begun with Darin Ulness of the Concordia College-Moorhead chemistry department. It continued with Concordia students Grace Bryan and Laura LeGare in the summer of 2016, and was supported by an NSF STEP grant (DUE 0969568).

Citations Bryan and LeGare, The calculus of proportional α-derivatives, Rose-Hulman Undergraduate Math. J., Vol. 18: Iss. 1, Article 2 (2017). Caputo and Fabrizio, A new definition of fractional derivative without singular kernel, Progr. Fract. Differ. Appl. 1, No. 2, 73 85 (2015)., 3D Memory Constitutive Equations for Plastic Media, J. Engineering Mechanics, Vol. 143, Issue 5 (May 2017). Ding, Zhang, Cao, Wang, and Liang, Bifurcation control of complex networks model via PD controller, Neurocomputing Vol. 175, pp. 1 9 (2016). V. E. Tarasov, No nonlocality. No fractional derivative, Commun Nonlinear Sci Numer Simulat 62 (2018) 157 163.

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