Investigation of cumulative growth process via Fibonacci method and fractional calculus

摘要

In this study, cumulative growth of a physical quantity with Fibonacci method and fractional calculus is handled. The development of the growth process is described in terms of Fibonacci numbers, Mittag–Leffler and exponential functions. A compound growth process with the contribution of a constant quantity is also discussed. For the accumulation of residual quantity, equilibrium and lessening cases are discussed. To the best of our knowledge; compound growth process is solved for the first time in the framework of fractional calculus. In this sense, differintegral order of fractional calculus α has been achieved a physical content. It is emphasized that, in the basis of qualification of the fractional calculus for describing genuine complex physical systems with respect to ordinary descriptions is the cumulative growth mechanism with Fibonacci method. It is concluded that compound diminution and growth process mechanisms can be taken as a basis for the comprehension of derivative and integral operations in fractional calculus.