Generalized fractional derivatives and their applications to mechanical systems

Katsikadelis, John T.

doi:10.1007/s00419-014-0969-0

Cited by 21 publications

(9 citation statements)

References 12 publications

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“…The last years a vast literature concerning the application of fractional calculus in mechanics has been developed, see for instance [3,4,11,12,17,21,29] to mention randomly just a few of them. Here, we shall confine ourselves only to the fractional derivative of order 1/2, which we are going to use in the next sections.…”

Section: The Fractional Derivative Of Order 1/2mentioning

confidence: 99%

On Hamilton's Principle for Discrete and Continuous Media

Kalpakides¹,

Charalambopoulos²

2019

Preprint

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In an attempt to generalize the Hamilton's principle, an action functional is proposed which, unlike the standard version of the principle, accounts properly for all initial data and the possible presence of dissipation. To this end, the convolution is used instead of the L 2 inner product so as to eliminate the undesirable end temporal condition of Hamilton's principle. Also, fractional derivatives are used to account for dissipation and the Dirac delta function is exploited so as the initial velocity to be inherently set into the variational setting. The proposed approach applies in both finite and infinite dimensional systems.

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Section: The Fractional Derivative Of Order 1/2mentioning

confidence: 99%

On Hamilton's Principle for Discrete and Continuous Media

Kalpakides¹,

Charalambopoulos²

2019

Preprint

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“…Most people assume that fractional calculus is an abstract field of mathematics that has very little use and almost no application. Nowadays, various studies have begun to emerge regarding the application of fractional calculus in various fields, such as physics, engineering, chemistry, biology, environment, economics and finance (Debnath, 2003;Kisela, 2008;Dalir and Bashour, 2010;David et al, 2011;Katsikadelis, 2014;Giusti and Colombaro, 2017;Rusyaman et al, 2017;Rusyaman et al, 2018;Sumiati et al, 2018;Sun et al, 2018). Fractional Black-Scholes partial differential equation to determine option prices is one application of fractional calculus in economics and finance.…”

Section: Introductionmentioning

confidence: 99%

Laplace Decomposition Method for Solving Fractional Black-Scholes European Option Pricing Equation

Owoyemi¹,

Sumiati

Rusyaman

2020

ijqrm

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Fractional calculus is related to derivatives and integrals with the order is not an integer. Fractional Black-Scholes partial differential equation to determine the price of European-type call options is an application of fractional calculus in the economic and financial fields. Laplace decomposition method is one of the reliable and effective numerical methods for solving fractional differential equations. Thus, this paper aims to apply the Laplace decomposition method for solving the fractional Black-Scholes equation, where the fractional derivative used is the Caputo sense. Two numerical illustrations are presented in this paper. The results show that the Laplace decomposition method is an efficient, easy and very useful method for finding solutions of fractional Black-Scholes partial differential equations and boundary conditions for European option pricing problems.

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“…e development of fractional-order calculus has a history of more than 300 years, but until recently, the research in this field still focused on pure mathematical theory. In recent years, the research and application of fractional calculus in the mechanical engineering field has gradually increased [1], especially to accurately characterize the true constitutive relationship of viscoelastic materials [2,3] and controlling engineering [4,5].…”

Section: Introductionmentioning

confidence: 99%

Threshold for Chaos of a Duffing Oscillator with Fractional‐Order Derivative

Xing

Chen

Chang

et al. 2019

Shock and Vibration

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In this paper, the necessary condition for the chaotic motion of a Duffing oscillator with the fractional-order derivative under harmonic excitation is investigated. The necessary condition for the chaos in the sense of Smale horseshoes is established based on the Melnikov method, and then the chaotic threshold curve is obtained. The largest Lyapunov exponents are provided, and some other typical numerical simulation results, including the time histories, frequency spectrograms, phase portraits, and Poincare maps, are presented and compared. From the analysis of the numerical simulation results, it could be found that, near the chaotic threshold curve, the system generates chaos via the period-doubling bifurcation, from single periodic motion to period-2 motion and period-4 motion to chaotic motion. The effects of fractional-order parameters, the stiffness coefficient, and the damping coefficient on the threshold value of the chaotic motion are analytically discussed. The results show that the coefficient of the fractional-order derivative has great effect on the threshold value of the chaotic motion, while the order of the fractional-order derivative has less. The analysis results reveal some new phenomena, and it could be useful for designing or controlling dynamic systems with the fractional-order derivative.

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Generalized fractional derivatives and their applications to mechanical systems

Cited by 21 publications

References 12 publications

On Hamilton's Principle for Discrete and Continuous Media

On Hamilton's Principle for Discrete and Continuous Media

Laplace Decomposition Method for Solving Fractional Black-Scholes European Option Pricing Equation

Threshold for Chaos of a Duffing Oscillator with Fractional‐Order Derivative

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