Characteristic Roots of a Class of Fractional Oscillators

Li, Ming; Lim, Sungbin; Cattani, Carlo; Scalia, Massimo

doi:10.1155/2013/853925

Cited by 15 publications

(15 citation statements)

References 64 publications

(58 reference statements)

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“…Then, it is easy to see that there exists infinitely many characteristic roots in the above, also see Li et al [18]. A contribution in this work in representing characteristic roots of three classes of fractional oscillators is that they are expressed analytically.…”

Section: There Exists Infinity Of Natural Frequencies Of a Fractionalmentioning

confidence: 81%

“…In addition to keep fractional properties of fractional oscillators with its equivalences, for instance, the characteristic roots of a fractional oscillator being infinitely large as explained by Li et al [18] and Duan et al [39], based on the proposed equivalent oscillators, we also reveal other properties of fractional oscillators, which may be very difficult, if not impossible, to be described directly from the point of view of fractional differential equations, such as the equivalent, i.e., intrinsic, masses m eqj , equivalent dampings c eqj , equivalent natural frequencies ω eqn,j and ω eqd,j (j = 1, 2, 3) of fractional oscillators, which are nonlinear with the power laws in terms of oscillation frequency ω as stated in Sections 4 and 5.…”

Section: Discussionmentioning

confidence: 99%

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Three Classes of Fractional Oscillators

2018

Symmetry

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This article addresses three classes of fractional oscillators named Class I, II and III. It is known that the solutions to fractional oscillators of Class I type are represented by the Mittag-Leffler functions. However, closed form solutions to fractional oscillators in Classes II and III are unknown. In this article, we present a theory of equivalent systems with respect to three classes of fractional oscillators. In methodology, we first transform fractional oscillators with constant coefficients to be linear 2-order oscillators with variable coefficients (variable mass and damping). Then, we derive the closed form solutions to three classes of fractional oscillators using elementary functions. The present theory of equivalent oscillators consists of the main highlights as follows. (1) Proposing three equivalent 2-order oscillation equations corresponding to three classes of fractional oscillators; (2) Presenting the closed form expressions of equivalent mass, equivalent damping, equivalent natural frequencies, equivalent damping ratio for each class of fractional oscillators; (3) Putting forward the closed form formulas of responses (free, impulse, unit step, frequency, sinusoidal) to each class of fractional oscillators; (4) Revealing the power laws of equivalent mass and equivalent damping for each class of fractional oscillators in terms of oscillation frequency; (5) Giving analytic expressions of the logarithmic decrements of three classes of fractional oscillators; (6) Representing the closed form representations of some of the generalized Mittag-Leffler functions with elementary functions. The present results suggest a novel theory of fractional oscillators. This may facilitate the application of the theory of fractional oscillators to practice.

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Section: There Exists Infinity Of Natural Frequencies Of a Fractionalmentioning

confidence: 81%

Section: Discussionmentioning

confidence: 99%

Three Classes of Fractional Oscillators

2018

Symmetry

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“…In this section, we consider a fractional generalization of the damped oscillator equation, involving Erdélyi-Kober-type integrals. There are many investigations nowadays about fractional damped oscillators involving Caputo derivatives, in the framework of the fractional mechanics (see for example [10,21,22] and references therein). Here we consider a special form of the fractional damped equation with time-varying coefficients, containing Erdélyi-Kober-type integrals, and we assume the elastic term to be time dependent.…”

Section: Damped Fractional Oscillator Involving Erdélyi-kober-type Inmentioning

confidence: 99%

Fractional Relaxation and Fractional Oscillation Models Involving Erdélyi-Kober Integrals

Concezzi

Garra

Spigler

2015

FCAA

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We consider fractional relaxation and fractional oscillation equations involving Erdélyi-Kober integrals. In terms of Riemann-Liouville integrals, the equations we analyze can be understood as equations with timevarying coefficients. Replacing Riemann-Liouville integrals with Erdélyi-Kober-type integrals in certain fractional oscillation models, we obtain some more general integro-differential equations. The corresponding Cauchytype problems can be solved numerically, and, in some cases analytically, in terms of Saigo-Kilbas Mittag-Leffler functions. The numerical results are obtained by a treatment similar to that developed by K. Diethelm and N.J. Ford to solve the Bagley-Torvik equation. Novel results about the numerical approach to the fractional damped oscillator equation with time-varying coefficients are also presented.MSC 2010 : Primary 34C26; 26A33; 65L05; Secondary 326A33; 26A48; 34A08; 33E12

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“…Recently, fractional-order PMD (FOPMD) has been studied in image denoising [6][7][8][9]. The fractional derivative can be seen as the generalization of the integer-order derivative [10][11][12][13]. FOPMD whose fractional-order is , 0 ≤ ≤ 2 is a "natural interpolation" between PMD and fourth-order PDEs.…”

Section: Advances In Mathematical Physicsmentioning

confidence: 99%

External Fractional-Order Gradient Vector Perona-Malik Diffusion for Sinogram Restoration of Low-Dosed X-Ray Computed Tomography

2013

Advances in Mathematical Physics

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Existing fractional-order Perona-Malik Diffusion (FOPMD) algorithms are defined as fully spatial fractional-order derivatives (FSFODs). However, we argue that FSFOD is not the best way for diffusion since different parts of spatial derivative play different roles in Perona-Malik diffusion (PMD) and derivative orders should be decided according to their roles. Therefore, we adopt a novel fractional-order diffusion scheme, named external fractional-order gradient vector Perona-Malik diffusion (EFOGV-PMD), by only replacing integer-order derivatives of "external" gradient vector to their fractional-order counterparts while keeping integerorder derivatives of gradient vector for diffusion coefficients since the ability of edge indicator for 1-order derivative is demonstrated both in theory and applications. Here "external" indicates the spatial derivatives except for the derivatives used in diffusion coefficients. In order to demonstrate the power of the proposed scheme, some real sinograms of low-dosed computed tomography (LDCT) are used to compare the different performances. These schemes include PMD, regularized PMD (RPMD), and FOPMD. Experimental results show that the new scheme has good ability in edge preserving, is convergent quickly, has good stability for iteration number, and can avoid artifacts, dark resulting images, and speckle effect.

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Characteristic Roots of a Class of Fractional Oscillators

Cited by 15 publications

References 64 publications

Three Classes of Fractional Oscillators

Three Classes of Fractional Oscillators

Fractional Relaxation and Fractional Oscillation Models Involving Erdélyi-Kober Integrals

External Fractional-Order Gradient Vector Perona-Malik Diffusion for Sinogram Restoration of Low-Dosed X-Ray Computed Tomography

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