A comparison study of steady-state vibrations with single fractional-order and distributed-order derivatives

AbstractWe conduct a detailed study and comparison for the one-degree-of-freedom steady-state vibrations under harmonic driving with a single fractional-order derivative and a distributed-order derivative. For each of the two vibration systems, we consider the stiffness contribution factor and damping contribution factor of the term of fractional derivatives, the amplitude and the phase difference for the response. The effects of driving frequency on these response quantities are discussed. Also the influences of the orderαof the fractional derivative and the parameterγparameterizing the weight function in the distributed-order derivative are analyzed. Two cases display similar response behaviors, but the stiffness contribution factor and damping contribution factor of the distributed-order derivative are almost monotonic change with the parameterγ, not exactly like the case of single fractional-order derivative for the orderα. The case of the distributed-order derivative provides us more options for the weight function and parameters.

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Digital Fractional Order Savitzky-Golay Differentiator and Its Application

Volume 3: 2011 ASME/IEEE International Conference on Mechatronic and Embedded Systems and Applications, Parts A and B ◽

10.1115/detc2011-47864 ◽

2011 ◽

Cited By ~ 3

Author(s):

Dali Chen ◽

Dingyu Xue ◽

YangQuan Chen

Keyword(s):

Fractional Order ◽

Least Square ◽

One Dimension ◽

Order Derivative ◽

Noisy Signal ◽

Fractional Order Derivative ◽

Least Square Fitting ◽

Image Enhancing ◽

The One ◽

Derivatives Of

Firstly the one-dimension digital fractional order Savitzky-Golay differentiator (1-D DFOSGD), which generalizes the Savitzky-Golay filter from the integer order to the fractional order, is proposed to estimate the fractional order derivative of the noisy signal. The polynomial least square fitting technology and the Riemann-Liouville fractional order derivative definition are used to ensure robust and accuracy. Experiments demonstrate that 1-D DFOSGD can estimate the fractional order derivatives of both ideal signal and noisy signal accurately. Secondly, the two-dimension DFOSGD is obtained from 1-D DFOSGD by defining a group of direction operators, and a new image enhancing method based on 2-D DFOSGD is presented. Experiments demonstrate that 2-D DFOSGD has very good performance on image enhancement.

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A new general fractional derivative Goldstein-Kac-type telegraph equation

Thermal Science ◽

10.2298/tsci2006893c ◽

2020 ◽

Vol 24 (6 Part B) ◽

pp. 3893-3898

Author(s):

Ping Cui ◽

Yi-Ying Feng ◽

Jian-Gen Liu ◽

Lu-Lu Geng

Keyword(s):

Fractional Order ◽

Telegraph Equation ◽

Order Derivative ◽

Mathematical Tool ◽

Fractional Order Derivative ◽

Derivative Formula ◽

The Laplace Transform ◽

The One ◽

First Time ◽

Derivatives Of

In this paper, we consider the Riemann-Liouville-type general fractional derivatives of the non-singular kernel of the one-parametric Lorenzo-Hartley function. A new general fractional-order-derivative Goldstein-Kac-type telegraph equation is proposed for the first time. The analytical solution of the considered model with the graphs is obtained with the aid of the Laplace transform. The general fractional-order-derivative formula is as a new mathematical tool proposed to model the anomalous behaviors in complex and power-law phenomena.

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Geometric interpretation of fractional-order derivative

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2016-0062 ◽

2016 ◽

Vol 19 (5) ◽

Cited By ~ 6

Author(s):

Vasily E. Tarasov

Keyword(s):

Differential Geometry ◽

Fractional Order ◽

Infinite Series ◽

Fractional Derivatives ◽

Geometric Interpretation ◽

Order Derivative ◽

Jet Bundles ◽

Fractional Order Derivative ◽

Integer Order ◽

Derivatives Of

AbstractA new geometric interpretation of the Riemann-Liouville and Caputo derivatives of non-integer orders is proposed. The suggested geometric interpretation of the fractional derivatives is based on modern differential geometry and the geometry of jet bundles. We formulate a geometric interpretation of the fractional-order derivatives by using the concept of the infinite jets of functions. For this interpretation, we use a representation of the fractional-order derivatives by infinite series with integer-order derivatives. We demonstrate that the derivatives of non-integer orders connected with infinite jets of special type. The suggested infinite jets are considered as a reconstruction from standard jets with respect to order.

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Vibration Systems with Fractional-Order and Distributed-Order Derivatives Characterizing Viscoinertia

Fractal and Fractional ◽

10.3390/fractalfract5030067 ◽

2021 ◽

Vol 5 (3) ◽

pp. 67

Author(s):

Jun-Sheng Duan ◽

Di-Chen Hu

Keyword(s):

Weight Function ◽

Fractional Order ◽

Fractional Derivatives ◽

Vibration System ◽

Equivalent Damping ◽

Vibration Model ◽

Equivalent Mass ◽

Weyl Fractional Derivative ◽

Distributed Order ◽

Frequency Phase

We considered forced harmonic vibration systems with the Liouville–Weyl fractional derivative where the order is between 1 and 2 and with a distributed-order derivative where the Liouville–Weyl fractional derivatives are integrated on the interval [1,2] with respect to the order. Both types of derivatives enhance the viscosity and inertia of the system and contribute to damping and mass, respectively. Hence, such types of derivatives characterize the viscoinertia and represent an “inerter-pot” element. For such vibration systems, we derived the equivalent damping and equivalent mass and gave the equivalent integer-order vibration systems. Particularly, for the distributed-order vibration model where the weight function was taken as an exponential function that involved a parameter, we gave detailed analyses for the weight function, the damping contribution, and the mass contribution. Frequency–amplitude curves and frequency–phase curves were plotted for various coefficients and parameters for the comparison of the two types of vibration models. In the distributed-order vibration system, the weight function of the order enables us to simultaneously involve different orders, whilst the fractional-order model has a single order. Thus, the distributed-order vibration model is more general and flexible than the fractional vibration system.

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Subharmonic Resonance of Van Der Pol Oscillator with Fractional-Order Derivative

Mathematical Problems in Engineering ◽

10.1155/2014/738087 ◽

2014 ◽

Vol 2014 ◽

pp. 1-17 ◽

Cited By ~ 4

Author(s):

Yongjun Shen ◽

Peng Wei ◽

Chuanyi Sui ◽

Shaopu Yang

Keyword(s):

Steady State ◽

Fractional Order ◽

Averaging Method ◽

Frequency Equation ◽

Lyapunov Theory ◽

Subharmonic Resonance ◽

Order Derivative ◽

Van Der Pol ◽

Fractional Order Derivative ◽

Equivalent Linear

The subharmonic resonance of van der Pol (VDP) oscillator with fractional-order derivative is studied by the averaging method. At first, the first-order approximate solutions are obtained by the averaging method. Then the definitions of equivalent linear damping coefficient (ELDC) and equivalent linear stiffness coefficient (ELSC) for subharmonic resonance are established, and the effects of the fractional-order parameters on the ELDC, the ELSC, and the dynamical characteristics of system are also analysed. Moreover, the amplitude-frequency equation and phase-frequency equation of steady-state solution for subharmonic resonance are established. The corresponding stability condition is presented based on Lyapunov theory, and the existence condition for subharmonic resonance (ECSR) is also obtained. At last, the comparisons of the fractional-order and the traditional integer-order VDP oscillator are fulfilled by the numerical simulation. The effects of the parameters in fractional-order derivative on the steady-state amplitude, the amplitude-frequency curves, and the system stability are also studied.

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Novel Approaches for Getting the Solution of the Fractional Black–Scholes Equation Described by Mittag-Leffler Fractional Derivative

Discrete Dynamics in Nature and Society ◽

10.1155/2020/8047347 ◽

2020 ◽

Vol 2020 ◽

pp. 1-11

Author(s):

Ndolane Sene ◽

Babacar Sène ◽

Seydou Nourou Ndiaye ◽

Awa Traoré

Keyword(s):

Analytical Solution ◽

Fractional Derivative ◽

Fractional Order ◽

Numerical Scheme ◽

Fractional Derivatives ◽

Fractional Integration ◽

Order Derivative ◽

Graphical Representations ◽

Fractional Order Derivative ◽

Black Scholes

The value of an option plays an important role in finance. In this paper, we use the Black–Scholes equation, which is described by the nonsingular fractional-order derivative, to determine the value of an option. We propose both a numerical scheme and an analytical solution. Recent studies in fractional calculus have included new fractional derivatives with exponential kernels and Mittag-Leffler kernels. These derivatives have been found to be applicable in many real-world problems. As fractional derivatives without nonsingular kernels, we use a Caputo–Fabrizio fractional derivative and a Mittag-Leffler fractional derivative. Furthermore, we use the Adams–Bashforth numerical scheme and fractional integration to obtain the numerical scheme and the analytical solution, and we provide graphical representations to illustrate these methods. The graphical representations prove that the Adams–Bashforth approach is helpful in getting the approximate solution for the fractional Black–Scholes equation. Finally, we investigate the volatility of the proposed model and discuss the use of the model in finance. We mainly notice in our results that the fractional-order derivative plays a regulator role in the diffusion process of the Black–Scholes equation.

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Leibniz Rule and Fractional Derivatives of Power Functions

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4031364 ◽

2015 ◽

Vol 11 (3) ◽

Cited By ~ 18

Author(s):

Vasily E. Tarasov

Keyword(s):

Power Function ◽

Fractional Order ◽

Special Form ◽

Fractional Derivatives ◽

Algebraic Approach ◽

Leibniz Rule ◽

Order Derivative ◽

Power Functions ◽

Fractional Order Derivative ◽

Derivatives Of

In this paper, we prove that unviolated simple Leibniz rule and equation for fractional-order derivative of power function cannot hold together for derivatives of orders α≠1. To prove this statement, we use an algebraic approach, where special form of fractional-order derivatives is not applied.

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GENERALIZED THEORY OF MAGNETO-THERMO-VISCOELASTIC SPHERICAL CAVITY PROBLEM UNDER FRACTIONAL ORDER DERIVATIVE: STATE SPACE APPROACH

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.9.11.86 ◽

2020 ◽

Vol 9 (11) ◽

pp. 9769-9780

Author(s):

S.G. Khavale ◽

K.R. Gaikwad

Keyword(s):

State Space ◽

Fractional Order ◽

Order Derivative ◽

Laplace Inversion ◽

State Space Approach ◽

Spherical Region ◽

Fractional Order Derivative ◽

Numerical Laplace Inversion ◽

Mathcad Software ◽

Space Approach

This paper is dealing the modified Ohm's law with the temperature gradient of generalized theory of magneto-thermo-viscoelastic for a thermally, isotropic and electrically infinite material with a spherical region using fractional order derivative. The general solution obtained from Laplace transform, numerical Laplace inversion and state space approach. The temperature, displacement and stresses are obtained and represented graphically with the help of Mathcad software.

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Novel hyperbolic and exponential ansatz methods to the fractional fifth-order Korteweg–de Vries equations

Advances in Difference Equations ◽

10.1186/s13662-020-03087-w ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Choonkil Park ◽

R. I. Nuruddeen ◽

Khalid K. Ali ◽

Lawal Muhammad ◽

M. S. Osman ◽

...

Keyword(s):

Fractional Derivative ◽

Fractional Order ◽

Mathematical Physics ◽

Order Derivative ◽

Conformable Fractional Derivative ◽

Fractional Order Derivative ◽

De Vries ◽

Fifth Order ◽

2D And 3D

Abstract This paper aims to investigate the class of fifth-order Korteweg–de Vries equations by devising suitable novel hyperbolic and exponential ansatze. The class under consideration is endowed with a time-fractional order derivative defined in the conformable fractional derivative sense. We realize various solitons and solutions of these equations. The fractional behavior of the solutions is studied comprehensively by using 2D and 3D graphs. The results demonstrate that the methods mentioned here are more effective in solving problems in mathematical physics and other branches of science.

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