Investigation of the cumulative diminution process using the Fibonacci method and fractional calculus

F. Buyukkilic, Z. Ok Bayrakdar and D. Demirhan

Physica A: Statistical Mechanics and its Applications, 2016, vol. 444, issue C, 336-344

Abstract: In this study, we investigate the cumulative diminution phenomenon for a physical quantity and a diminution process with a constant acquisition quantity in each step in a viscous medium. We analyze the existence of a dynamical mechanism that underlies the success of fractional calculus ​compared with standard mathematics for describing stochastic processes by ​proposing a Fibonacci approach, where we assume that the complex processes evolves cumulatively in fractal space and discrete time. ​Thus, when the differential–integral order α is attained, this indicates the ​involvement of the viscosity of the medium ​in the evolving process. The future value of the diminishing physical quantity is obtained in terms of the Mittag-Leffler function (MLF) and two rheological laws ​are inferred from the asymptotic limits. Thus, we conclude that the differential–integral calculus of fractional mathematics implicitly embodies the cumulative diminution mechanism ​that occurs in a viscous medium.

Keywords: Fibonacci numbers; Fractional calculus; Mittag-Leffler function; Rate equation (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:444:y:2016:i:c:p:336-344

DOI: 10.1016/j.physa.2015.09.049

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