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Fractional Derivative Order Determination from Harmonic Oscillator Damping Factor

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Fractional Derivative Order Determination from Harmonic Oscillator Damping Factor Fractional derivative order determination from harmonic oscillator damping factor. Luis Felipe Alves da Silva1 ∗, Valdiney Rodrigues Pedrozo Júnior 1. João Vítor Batista Ferreira 1. 1 Institute of Physics, Federal University of Mato Grosso do Sul. Campo Grande, MS, Brazil February 10, 2020 This article analysis differential equations which represents damped and fractional oscillators. First, it is shown that prior to using physical quantities in fractional calculus, it is imperative that they are turned dimensionless. Afterwards, approximated expressions that relate the two equations parameters for the case that the fractional order is close to an integer number are presented. Following, a numerical regression is made using power series expansion, and, also from fractional calculus, the fact that both equations cannot be equivalent is concluded. In the end, from the numerical regression data, the analytical approximated expressions that relate the two equations’ parameters are refined. Keywords: fractional calculus, numerical resolution, computer algebra system, fractional oscillator. arXiv:2002.02479v1 [physics.class-ph] 6 Feb 2020 ∗Electronic Mail Adress: [email protected] 1 1 Introduction 2 Fractional Calculus and Classical Phys- Fractional calculus is still a novelty to many Physics undergrad- ical Models uate students, despite excellent articles that make it easier to On this section, we briefly comment about the fractional calcu- understand [1]. Many physical phenomena are represented by lus and its use on representing a damped mass-spring system. first (α = 1) or second (α = 2) order linear differential equa- tions with constant coefficients. Non-integer order differential equations usage, for example α = 1:82, is studied by the area 2.1 Fractional Derivative known as fractional calculus. Initially restricted to the pure Fractional calculus is still a novelty to many Physics under- mathematics field, fractional differential equations are also be- graduate students. Good texts can be found in [7, 8]. The ing used to study physical phenomena. Diffusion processes, non-integer order derivative history starts in the beginnings of of viscoelasticity and others, can be described by non-integer XVII traditional calculus. It was at that time that Gottfried order differential equations. [2, 3, 4]. The computer algebra Leibniz speculated about an α = 1=2 order derivative applied systems tools (CAS) ease the burden of fractional calculus to the function f (x) = x. For a more intuitive understanding concept usage. about integer order calculus operators it is common to use a The use of these tools is changing the way physicists ap- geometrical visualization and interpretation. In fractional calcu- proach the study of natural phenomena. As an example, this lus, this is a problem due to the absence of an easy geometrical article analysis the damped harmonic oscillator case. This phe- interpretation for it. However, this challenge should not be nomena can, apparently, be described either by an integer order considered as insurmountable, like the interpretation of the differential equation or by a fractional differential equation [5]. complex number i exemplifies. The conceiving of a number This duplicity in its description leads to conjecture the existence with the i2 = −1 property is hard, but with time the complex of an analytical equivalence between those two expressions. number theory has been developed, today being a fundamental The proof of this equivalence existence is difficult to obtain by piece in quantum mechanics and electromagnetism studies. analytical mathematics. Caputo fractional derivative. If power series and numerical calculus are used instead of Definitions for Caputo fractional derivative have been ob- analytical mathematics, it is possible to come to a conclusion. tained from the thesis [9]. The real order α ≥ 0 derivative of a This alternative procedure is only practical using computational function ξ(τ) defined on the interval 0 < τ < 1, ξ and τ being algebra systems to manipulate series with many terms. This dimensionless variables, is article uses a power series with sixty terms to represent a func- tion which is a fractional differential equation solution. The dαξ(τ) dnξ ≡ Jn−αf g ; (1) solution achieved implies the two differential equations have dτα dτn no analytical equivalence. In this article, the CoCalc programming environment has dnξ 1 Z τ dnξ(ν) Jn−αf g ≡ τ − ν (n−α)−1 dν ; been used as the computational algebra system. It allows both n ( ) n (2) dτ Γ(n − α) 0 dν analytical as numerical manipulation, as well as a good function where n is integer, positive and n − 1 < α ≤ n. If α = n, the graphic visualization. CoCalc, previously known as SageMath- integer order calculus is recovered. The usual definition for Cloud [6], is an on-line computational platform that offers free gamma function Γ is used [10]. access under some conditions and there is no need for instal- This way, Caputo definition for the α order fractional deriva- lation on the user computer. CoCalc is available in many pro- tive of a function ξ(τ) can be understood as the integral trans- gramming languages, Python being the most used and there are form of an integer order derivative, Eq.1 many auxiliary documents on the internet about the software. Assuming that the integration inverse operation is a deriva- The concepts involved on this article are shown on section tive, in a similar way, but not identically, to Cauchy integral 2, being the fractional derivative and the mass-spring system. theorem 1 [11] the following notation equivalence can be made The section 3 presents the results obtained in this work, which n d ≡ −n shows the easiness and practicality in using computational dτ J . tools to solve physical complicated problems. This section also Generally, Caputo fractional derivative does not satisfy the analyses the differential equations that represents the damped commutative property, but satisfies the following relation oscillator and the fractional oscillator. First, it is shown that ! dα+βξ(τ) dα dβξ(τ) dmξ(τ) prior to using physical quantities in fractional calculus, it is α+β = α β = m ; (3) imperative that they are turned dimensionless. Afterwards, dτ dτ dτ dτ expressions that relate the two equations parameters for the m being an integer and positive number such that m = α + β case that the fractional derivative order is close to the integer (β real and positive). This property is demonstrated on page derivative order are presented. Then, with the use of a power 56 of Ref.[12]. This is interesting because the fractional order series expansion it is possible to come to a conclusion that the derivative becomes a natural extension of the integer order two equations cannot be equivalent. Lastly, using data from derivative and the integer order derivative can be defined as a numerical regression, the analytical expressions that relate the composition of fractional order derivatives. parameters from both equations are refined. A discussion and On physics, the rate of change is an important concept. For the conclusion are made on the section 4 of this paper. example, velocity is a rate of change of a particle position as a function of time. Mathematically, this corresponds to an 1For Jα operator order α > 0. 2 order one derivative: dx=dt. According to the equations 1 and is d2 x 2, the integer order rate of change concept can be applied to + !2 x = 0 ; (4) the fractional derivative: the fractional derivative of x(t), as a dt2 0 p function of t, is the integer order rate of change followed by an where !0 ≡ k=m is the oscillation angular frequency. The integral transform of the convolution type [13]. simple harmonic oscillator solution is x(t) and can be deter- Caputo definition brings two advantages. The first is that it mined by mathematical methods like Laplace transform, makes a fractional derivative of a constant to always be equals zero, which makes sense to Physics. The second is that deriving x˙(0) x(t) = Acos(!0t + ') ≡ x(0)cos(!0t) + sin(!0t) ; (5) ξ(τ) initially by α and then by β the same result is obtained both !0 by this process and by the α + β order derivative. It is generally where ', A, x(0) and x˙(0) are respectively the phase constant, agreed to use Caputo definition for functions that have time oscillation amplitude, initial position and initial velocity. variations and known boundary conditions, which is the case On physical situations closer to macroscopic reality, me- for a harmonic oscillator. chanical energy dissipations are always present due to drag or Fractional derivative analytical manipulation can become fluid viscosity where the movement may take place. Represent- very difficult if the ξ(τ) function is not elementary. In this ing the damping as a velocity proportional force F = −ρdx=dt, article, we show that it is possible to overcome this problem then by Newton second law, the fractional damped harmonic by expanding the ξ(τ) function in a polynomial and by using oscillator differential equation becomes computers to treat polynomials with a large number of terms. Some works [3, 14, 5] presents generalized differential d2 x(t) dx + γ + !2 x(t) = 0 ; (6) equations (α real and positive) that models classical problems dt2 dt 0 while using fractional calculus. Since those phenomena are already described by integer order differential equations, they where we call the γ = ρ/m parameter as damping coefficient. become important studies that analyze the equivalence between For the underdamped case γ < 2!0, the solution is the two approaches, the fractional and the integer order. This x(t) = Ae−(γ=2)tcos(!t + ') ; (7) article contributes to this analysis. 2 2 2 in which ! = !0 − (γ=2) . 2.2 Spring-Mass System Fractional Oscillator. This article assumes as the definition for fractional har- The spring-mass oscillator system is one of the most simple monic oscillator the expression built from the not damped models in classical mechanics.
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