Journal of Applied Mathematics

Volume 2018, Article ID 1260240, 7 pages

https://doi.org/10.1155/2018/1260240

## A Note on Caputo’s Derivative Operator Interpretation in Economy

^{1}School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600 Selangor Darul Ehsan, Malaysia^{2}Center for Language and Foundation Studies, A’ Sharqiyah University, Post Box No. 42, Post Code No. 400 Ibra, Oman^{3}College of Applied and Health Sciences, A’ Sharqiyah University, Post Box No. 42, Post Code No. 400 Ibra, Oman

Correspondence should be addressed to Maslina Darus; ym.ude.mku@anilsam

Received 29 May 2018; Revised 26 August 2018; Accepted 5 September 2018; Published 1 October 2018

Academic Editor: Igor Andrianov

Copyright © 2018 Hameed Ur Rehman et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We propound the economic idea in terms of fractional derivatives, which involves the modified Caputo’s fractional derivative operator. The suggested economic interpretation is based on a generalization of average count and marginal value of economic indicators. We use the concepts of which analyses the economic performance with the presence of memory. The reaction of economic agents due to recurrence identical alteration is minimized by using the modified Caputo’s derivative operator of order instead of integer order derivative . The two sides of Caputo’s derivative are expressed by a brief time-line. The degree of attenuation is further depressed by involving the modified Caputo’s operator.

#### 1. Introduction

Fractional calculus is the field of Mathematical study that grows out of the traditional definition of calculus integral and derivative operators as the same way of fractional exponents is an outgrowth of exponents of integer value. The birth of fractional calculus occurred in a letter from G. F. A. de L’Hospital to G. W Leibniz in 1695 posing a possible question “what if the order of derivative such that ”, in his reply he wrote “it will lead to a paradox, from which one day useful consequences will be drawn”; for more details, refer to a book by M. Sen [1, p. 1] and an article by Machado [2]. The quoted conversation of the two famous mathematicians evoked interest among the researchers and the theory of noninteger orders was further developed, generalized, and formulated by famous mathematicians such as Riemann, Liouville, L Euler, Letnikov, Grunwald, Marchuad, Weyl, Riesz, Caputo, Abel, and others; see, for example, [3–6]. Fractional derivatives and fractional integrals have vast applications in the field of physics, mechanics, engineering, and biology [7, 8]. In recent progress the fractional derivatives and fractional integrals are applied to narrate the financial process [9–11] and economic process with memory [12]. Fractional derivatives and fractional integration have been interpreted in many ways such as geometric interpretation [13–16], informatic interpretation [17], and economic interpretation [18]. Some of the modern definitions of fractional derivative and fractional integrals are enlisted in the next section.

#### 2. Modern Definitions of Fractional Calculus

*Definition 1. *Riemann defined the fractional integral of order as

*Definition 2. *Riemann-Liouville defined the fractional derivative of order as

*Definition 3 (see [20]). *Abel’s integral equation of order is given by

*Remark 4. *It is notable that Abel was the first one to use fractional calculus for and solved the famous tautochrone problem [20, 21]. For more details about the solution of problem and animated cycloid curve [19] see Figure 1. The total time of fall for iswhich is in fact the application of fractional integral of order , where the equation of cycloid is