Applications of proportional calculus and a non-Newtonian logistic growth model

Pinto, Manuel; Torres, Ricardo; Campillay-Llanos, William; Guevara-Morales, Felipe

doi:10.22199/issn.0717-6279-2020-06-0090

Cited by 7 publications

(16 citation statements)

References 22 publications

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“…Particularly, Stanley [55], Córdova-Lepe [11], Slavík [10], Pap [44] and Bashirov et al [12], have called the attention of mathematicians to the topic. More recently, non-Newtonian calculi have become a hot topic in economic and finance [56], quantum calculus [54], complex analysis [57][58][59], numerical analysis [2,37,42], inequalities [40,60], biomathematics [7,61] and mathematical education [14].…”

Section: Discussionmentioning

confidence: 99%

“…Here we follow closely this last approach-in particular, the exposition of the non-Newton calculus as found in [7,14,35], because it is appealing to scientists who seek ways to express laws in a scale-free form.…”

Section: Methodsmentioning

confidence: 99%

“…For other choices of α and X, we can get an infinitude of other arithmetics from which Grossman and Katz produced a series of non-Newton calculi, compiled in the seminal book of 1972 [1]. Among all such non-Newton calculi, recently great interest has been focused on the Grossman-Katz calculus obtained when we fix α(x) = e x , α −1 (x) = ln(x) and X = R + for the set of real numbers strictly greater than zero [7,12,14,15]. We shall concentrate here on one option, originally called by Grossman and Katz the geometric/exponential/bigeometric calculus [1,2,13,[34][35][36][37], but from which other different terminology and small variations of the original calculus have grown up in the literature, in particular, the multiplicative calculus [3][4][5]8,12,15,36,[38][39][40], and more recently, the proportional calculus [7,11,14,41], which is essentially the bigeometric calculus of [35].…”

Section: Methodsmentioning

confidence: 99%

“…Among all such non-Newton calculi, recently great interest has been focused on the Grossman-Katz calculus obtained when we fix α(x) = e x , α −1 (x) = ln(x) and X = R + for the set of real numbers strictly greater than zero [7,12,14,15]. We shall concentrate here on one option, originally called by Grossman and Katz the geometric/exponential/bigeometric calculus [1,2,13,[34][35][36][37], but from which other different terminology and small variations of the original calculus have grown up in the literature, in particular, the multiplicative calculus [3][4][5]8,12,15,36,[38][39][40], and more recently, the proportional calculus [7,11,14,41], which is essentially the bigeometric calculus of [35]. Here we follow closely this last approach-in particular, the exposition of the non-Newton calculus as found in [7,14,35], because it is appealing to scientists who seek ways to express laws in a scale-free form.…”

Section: Methodsmentioning

confidence: 99%

“…Recently, it has been shown that non-Newtonian/multiplicative calculi are more suitable than the ordinary Newtonian/additive calculus for some problems, e.g., in actuarial science, finance, economics, biology, demography, pattern recognition in images, signal processing, thermostatistics and quantum information theory [3][4][5][6][7]. This is explained by the fact that while the basis for the standard/additive calculus is the representation of a function as locally linear, the basis of a multiplicative calculus is the representation of a function as locally exponential [1,3,7]. In fact, the usefulness of product integration goes back to Volterra (1860Volterra ( -1940, who introduced in 1887 the notion of a product integral and used it to study solutions of differential equations [8,9].…”

Section: Introductionmentioning

confidence: 99%

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On a Non-Newtonian Calculus of Variations

Torres

2021

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The calculus of variations is a field of mathematical analysis born in 1687 with Newton’s problem of minimal resistance, which is concerned with the maxima or minima of integral functionals. Finding the solution of such problems leads to solving the associated Euler–Lagrange equations. The subject has found many applications over the centuries, e.g., in physics, economics, engineering and biology. Up to this moment, however, the theory of the calculus of variations has been confined to Newton’s approach to calculus. As in many applications negative values of admissible functions are not physically plausible, we propose here to develop an alternative calculus of variations based on the non-Newtonian approach first introduced by Grossman and Katz in the period between 1967 and 1970, which provides a calculus defined, from the very beginning, for positive real numbers only, and it is based on a (non-Newtonian) derivative that permits one to compare relative changes between a dependent positive variable and an independent variable that is also positive. In this way, the non-Newtonian calculus of variations we introduce here provides a natural framework for problems involving functions with positive images. Our main result is a first-order optimality condition of Euler–Lagrange type. The new calculus of variations complements the standard one in a nontrivial/multiplicative way, guaranteeing that the solution remains in the physically admissible positive range. An illustrative example is given.

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