En este trabajo, presentamos algunos apuntes históricos del cálculo fraccionario Local, y destacamos algunas propiedades y aplicaciones de estas nuevas herramientas matemicasIn this paper, we present some historical notes to Generalized Calculus, sometimes called Local Fractional Calculus, and highlight some properties and applications of these new mathematical tool

Guzman, Paulo M.

Lugo, Luciano M

Nápoles V, Juan E.

Vivas Cortez, Miguel

English

In this paper, we present some historical notes to Generalized Calculus, sometimes called Local Fractional Calculus, and highlight some properties and applications of these new mathematical toolsEn este trabajo, presentamos algunos apuntes históricos del cálculo fraccionario Local, y destacamos algunas propiedades y aplicaciones de estas nuevas herramientas matemica

Repositorio de Revistas Digitales de la Universidad del Atlántico

FRACTIONAL CALCULUS: HISTORICAL NOTESAPUNTES HISTÓRICOS DEL CÁLCULO FRACCIONARIOMiguel Vivas Cortezmjvivas@puce.edu.ecPontificia Universidad católica del Ecuador, Facultad de Ciencias Exactas y Naturales, Escuela de Ciencias Físicas y Matemática, Sede Quito, EcuadorPaulo M. Guzmanpaulomatiasguzman@hotmail.comUNNE, Facultad de Ciencias AgrariasLuciano M Lugolmlmb@yahoo.com.arUNNE, FaCENA, Ave. Libertad 5450, Corrientes 3400, ArgentinaJuan E. Nápoles V.jnapoles@exa.unne.edu.arUNNE, FaCENA, Ave. Libertad 5450, Corrientes 3400, ArgentinaRecibido: 07/02/2021 Aceptado: 09/10/2021ResumenEn este trabajo, presentamos algunos apuntes historicos al Cálculo Generalizado, aveces llamado Calculo Fraccionario Local, y destacamos algunas propiedades y aplicaciones de estas nuevas herramientas matemáticas.Palabras claves:Integrales y derivadas fraccionarias, derivadas generalizadas, cálculo fraccionalAbstractIn this paper, we present some historical notes to Generalized Calculus, sometimes called Local Fractional Calculus, and highlight some properties and applications of these new mathematical tools.Keywords:Fractional derivatives e integral, generalized derivative, fractional calculusVIII  | Nº 2, (2021)Esta publicación está bajo una licencia Creative Commons Reconocimiento-NoComercial 4.021 IntroductionFractional calculus studies problems with derivatives and integrals of real or complex order. As a field purely mathematical, the theory offractional calculus was proposed for the first time in the 17th century and since then many renowned scientists worked on this subject, including Euler, Laplace, Fourier, Abel, Liouville and Riemann (See [26]).In the last 5 decades, we have witnessed the development of new operators, differential and integral, which include both fractional and generalized. The latter, in general, are defined as local derivatives and generate integral operators that may or may not be fractional. To date, the study of this area has attracted the attention of many researchers, not only in Pure Mathematics, but also in many fields of applied science. Between its own theoretical development and the multiplicity of applications, the field has grown rapidly in recent years, in such a way that a single definition of fractional derivative or integral does not exist, or at least is not unanimously accepted. in [5] the idea of a fairly complete classification of the known operators of fractional calculus is suggested and justified, but on the other hand, in [4] indicate several reasons why new operators appear in relation to the development of applications of theoretical results. These fractional calculus operators, for example, Hadamard, Riemann-Liouville (RL), Weil, Erdelyi-Kober, Katugampola and others have been defined by different mathematicians. All these operators were generated from the classical local derivatives.The history of differential operators from Newton to Caputo, both local and global, is given in [2] (Chap- ter 1). Here is the definition of a local derivative with a new parameter, which has a large number of applica- tions. More importantly, section 1.4 concludes: “Therefore, we can conclude that both the Riemann-Liouville operator and the Riemann-Liouville operator Caputo are not derivatives, and therefore they are not fractional derivatives, but fractional operators”. We are of in agreement with the result [41] that says “the local frac- tional operator is not a fractional derivative”(page 24). As mentioned above, these tools are new and have demonstrated their potential and usefulness in solving phenomena and process modeling problems in various fields of science and technology (see [3]). Many different types of fractional operators have been proposed in the literature, here we show that several of these different notions of derivatives can be considered particular cases of our definition and, even more relevant, that it is possible to establish a direct relationship between derivatives global (classical) and local, the latter not very accepted by the mathematical community, under two arguments: its local character and compliance with the Leibniz Rule. To facilitate the understanding of the scope of our definition, we present the best known definitions of integral operators and their correspon- ding differential operators (for more details you can consult [6]). Without much difficulty, we can extend these definitions, for any higher order.We assume that the reader is familiar with the classical definition of the Riemann Integral, so we will not present it.In this work, we present an outline of the history of the Generalized Calculus, pointing out some proper- ties and possible applications.32. The Origins of Local Fractional CalculusIn [20] ([21] y [22]) the following fractional derivative was introduced:Definition 2.1. Given a functionexists and is finite, then it is said that the local fractional derivative fraccional of(LFD) of order q, in x = y, exists.This derivative was used for the study of certain attractors of dynamic systems, which are examples of occurrence of curves and continuous surfaces, but highly irregular and non-differentiable. Frequently these graphs are fractal sets and the ordinary calculation is inadequate to characterize them. In this article the notion of local ”fractional derivative ” is developed, suitably modifying the concepts of fractional calculus. They also introduce the concept of local fractional integral based on the previous one, as we know it today. To understand the fractal behavior of functions, Parvate and Gangal (see [38]) introduce the derivative fractal as follows:where the right member is the limit through the fractal F points. It is necessary to mention some ideas.Definition 2.2. Let a be a fix and arbitrary real number . The integral of the step function          of order     for a set F está is given by:also we have,Definition 2.3. The mass function               can be written as follows(See [51] and [52]):Meanwhile, Adda y Cresson propoused a local fractional derivative as follows:with                    being the Riemann-Liouville fractional operactor.Chen gave the fractal derivative notion in [7]:4obtained from (5) ifHe proposed a new fractal derivative useful in engineering applications (see [15]):having in account the relationwhich is the Gao, Yang and Khan unified notation of a local fractional derivative (see [10]). To close this dissertation, we want to note the Yang’s local fractional derivative (ver [53] and [54]), defined as follows.where              is a fractal measure (see [54]) andAvariation was introduced in [55]:3. Stage 2014-2019In [19]] (see also [1]) it is defined, in a very simple way, a new local fractional derivative called ”confor- mable ”depending on a certain incremental quotient as the classical derivativeSo, for a function                       the conformable fractional derivative of order                                 was defined byand the conformable fractional derivative in 0 was defined bySimilarly, in a work of the same year (cf. [17]) it was defined another conformable fractional derivative.Definition 3.1. Let                                   be a function. The fractional dereivative of f , of order     con                is defined by mean the expressionnaturally, if             exists in a certain (0, a) with           it is defined the fractional derivative of order α in 0 as5In [16] it is introduced a new mode by generalizing the fractional derivative as follows: let                be a function and     a real number, then the fractional derivative may be considered as In [40] it was defined a fractional derivative for real valued functions of several variable taking the base of the Khalil fractional derivative aforementioned.In the work [42] it was introduced a truncated fractional derivative M for functions     -differentiables, on the base of the Mittag-Lefler of one parameter as follows:for functions                                         The fractional derivative in 0 is defined as the aforementioned. This fractional derivative generalize those of defined in [19], [17] y [43].In the work of [12] it is open a new approach when it was defined a new local fractional derivative, conformable in this case, as follows.Definition 3.2. Given a function                       then the N-derivada of f , of order            is defined by67whenever the limit exists, it will be called the derivative of f at t or the generalized derivative of f at t and, briefly , also it is said that f is p-differentiable at t. In the case of an closed interval, we define the p-derivative in the terminal points as the corresponding lateral derivative . From this definition , the derivative of order      is constructed for a function as follows:where the ordinary derivative is recovered if     = 1 . It is clear that if f is differentiable at t, then                                               There is no restriction for the sign of the function p neither its partial derivativeTherefore, from its origins, the derivative notion is “local”, contrary to the global property of the integral operator, so , they are not inverse operators in strict sense. It is common to find references to instants, points, specific magnitudes and not intervals. Classical notions of fractional derivatives did not take this fact into account and and an operator that is not local was built, therefore, since their conception, classical fractional derivatives are another nature. As we have said, it is impossible to compare them, so Tarasov’s statements should be reformulated as follows: “There is no locality, there is no differential operator”.Noncompliance with the product rule was always considered to be a feature of fractional (global) deri- vatives. However, a new local derivative can be constructed that violates Leibniz’s Rule, so violation of this rule cannot be a necessary condition for a given operator to be a fractional derivative, back to (12). It is clear that the violation of this rule does not depend (at least not only) on the incremental quotient, but on a factor that we can add to the incremented function, from which we would obtain the non-symmetry of the product rule.Having in account [57] we can write form (12) the following derivative8910The detail that we want to highlight about this derivative and that is very characteristic, is the fact that in a derivative of order α there are successive derivatives “within said order” and not as in the entire case, that it must be fulfilled that the successive derivatives are of higher order than the first.5. Some open problemsRegarding what has already been established regarding generalized conformable and non-conformable derivatives, there are currently some open problems, or at least not found in the literature, such as:- Representation of the fourier transform- Engineering applications using generalized Fourier transforms by means of conformable and non- conformable integral and/or differential operators- Development of partial fractional differential equations for the different conformable and non-conformable fractional derivatives- Comparative studies between these derivatives and their applicationsStudies of these derivatives in generalized vector spaces- In the field of integral inequalities there are some works that relate this topic with the Riemman- Liouville fractional integral and another much more generalized one such as that of R. K. Raina [8, 46, 47, 48, 49, 50]. Since this area is very large, much remains to be done by applying conformable and non-conformable fractional derivatives.6. ConclusionesIn this informative article, a historical summary has been presented, starting with the origins of the fractal derivative and then with a stage between 2014 and the present, regarding generalized fractional derivatives, conformable and not conformable. Additionally, its origins, characteristics, properties and applications have been established, as well as the way in which one and the other are deduced in some cases.11Referencias1213

Fractional Calculus: Historical Notes Apuntes Históricos del Cálculo Fraccionario

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Fractional Calculus: Historical Notes Apuntes Históricos del Cálculo Fraccionario

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