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

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.