Investigation of cumulative growth process via Fibonacci method and fractional calculus
Introduction
In ordinary approach, it is intimately accepted that the physical phenomenon evolve in a continuous Euclidean space and time with a Markovian memory. In this standard framework, deterministic mechanic is sufficient in describing dynamical systems. But the real processes of stochastic systems evolve in a fractal medium and having discrete time intervals as well as a non-Markovian memory. In this context, however, Hamiltonian mechanics and ordinary calculus are insufficient in describing stochastic systems. In this sense, fractional calculus is very successful in describing stochastic systems [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33]. On the other hand, the elements of fractional mathematics are given with different definitions, regardless of their applicability, consequently this situation creates difficulties for research workers. For instance, the operation of differintegral is not expressed rigorously and uniquely. This uncertainty constitutes an obstruction for those who want to use fractional calculus.
In this study, the existence of the fundamental physical mechanism on which the differintegral of fractional calculus is based on is uncovered with Fibonacci approach and compound growth process. For instance, it is concluded that from the solutions of rate equation using fractional calculus, the order α of differintegral can be interpreted as a parameter that controls the evolution of stochastic processes. The growth process that is outlined in this study develops in a fractal medium in a discrete time and space with a non-Markovian memory. Thus, a mathematical bridge is established between the present and future values of the quantity which is under investigation. In this regard one of the functions of the fractional calculus namely Mittag–Leffler (M–L) function is achieved by cumulative growth process should be noted. It is expected that, to an extent, the inadequacies of the standard approach are overcome [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50].
In order to justify the vibrancy of field of this research, recent articles which are within the framework of social science should be mentioned. The quantitative methods of physical sciences are also applicable to social sciences. Cumulative growth processes are ubiquitous in everyday life and it is a relevant set up to study social events as well. In this content, compound growth process plays a crucial role for the following examples: Evolution of the most common English words and phrases over the centuries within this frame work [51]. The quest concerning to the development of physical discoveries across over the two centuries are given in [52]. It is reported that the emergence and diminution of scientific paradigm are governed by robust principles of self-organization. Languages can be treated as living creatures. With the use of quantitative methods of science and analyzing the usage of words in several languages, authors of Ref. [53] came to the conclusion that, languages expands and contract within allometric proportions which is communicated in their concomitant paper.
As a conclusion it is communicated that those dynamical very different entities that involve cumulative growth and cumulative diminutions processes are those events which self organize in the progress of time. Within this ensemble, following examples namely, preferential attachment in wealth, preferential attachment in network sizes, cumulative advantage, carrier longevity, corporation sizes, population of cities, avalanches, forest-fire size, efficacy in technological systems, etc., can be recited [54].
This study is arranged as follows: In Section 1, the well known rate equation is handled for a physical quantity A and the solution for growth process is investigated. It is pointed out that the solution which is done by standard calculus is carried out for a regular growth which evolves in a continuous Euclidian space and continuous time, with a Markovian memory [21], [26]. In Section 2, the solution of rate equation is presented for the growth process which evolves in fractal space, with a discrete time and a non-Markovian memory, a cumulative growth by Fibonacci approach is used [34], [35], [38], [43], [44]. A relation between the present value and the future value of the quantity A is established. In Section 3, to the best of our knowledge, it is the first time that the fractional growth process is handled and solved within the framework of fractional calculus. By comparing the solution of cumulative method and the solution of fractional calculus, a physical explanation/content is given to the order of differintegral, namely α [21], [22], [25], [26], [34], [35], [39], [42], [55], [59]. In Section 4, a cumulative growth process with an initial value A0, and with a constant contribution value of P in each step is handled. It is observed that when time is fixed for the cumulative growth description of the processes, the universal Fibonacci numbers are the main elements of description.
Section snippets
Cumulative growth is revisited
The events in nature and human behaviors which are under study, compoundly evolve in a space and time. In this section, a cumulative growth process in a free medium is introduced. In nature, as the examples of the development of the complex stochastic systems, earthquakes, fire, avalanche, cancerous cells, germ colonies, meteorological events, etc. can be seen very commonly. On the other hand, as examples of the human activities; stock exchange index, interest income of the capital movements,
Cumulative growth in fractal medium
In this section, let us discuss the cumulative growth by Fibonacci approach in a process where time is discrete and space is fractal. Fibonacci method can be summarized as follows: the elements of an ensemble which possess a hierarchy, for example, a growing sequence, the elements of the set are not independent from each other. This interdependency can be described by a very simple mathematical relation. In this regard the recurrence relation for the Fibonacci numbers can be used. A Fibonacci
Discussion of the cumulative growth process within the framework of fractional calculus
The rate equation namely Eq. (3) given in Section 1, can be generalized to fractional differential equation form where is a fractional differential operator, α is the order of fractional differentiation such that 0 ≤ α ≤ 1. Since fractional derivative is written in Eq. (20) with the order of α, in order to maintain the correct dimensionality of the concomitant equation, λα is taken instead of λ. Afterwards, Eq. (20) can be written as an integral equation
Cumulative growth process with a constant contribution quantity
In this section, the cumulative growth process is handled with the contribution of a constant quantity to exhibit the importance of Fibonacci approach and cumulative growth process with regard to regular growth. For this purpose, let us handle again the process that we mentioned in Section 2. Assuming that the quantity P is added to the initial value A0 in each following steps and P is naturally subjected to same cumulative growth process. After time let us indicate the future value of A0
Results and discussions
In this work, the simple cumulative growth, and also cumulative growth with a constant contribution process are handled with Fibonacci method, by taking into account that the process is progressing in a fractal space and discrete time medium and with a non-Markovian memory. In this regard, the relation between the future and the present values of the physical quantity A which is under investigation is established. It is observed that after sufficiently large steps M–L function of fractional
Acknowledgment
We express our sincere thanks to the anonymous reviewer for his contribution to enlarge our vision on the applications in social sciences of the topic by bringing to our attention to relevant references.
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