Generalized Memory: Fractional Calculus Approach

Tarasov, Vasily E.

doi:10.3390/fractalfract2040023

Cited by 68 publications

(44 citation statements)

References 44 publications

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“…Let us briefly describe possible directions for application of the proposed approach. a) We should note the power-law kernels function can be used to consider an approximation of the generalized memory functions [57]. Using the generalized Taylor series in the Trujillo-Rivero-Bonilla form for the memory function, we proved [57] that the equations with memory functions can be represented through the Riemann-Liouville fractional integrals and the Caputo fractional derivatives of non-integer orders for wide class of the kernels.…”

Section: Discussionmentioning

confidence: 96%

“…a) We should note the power-law kernels function can be used to consider an approximation of the generalized memory functions [57]. Using the generalized Taylor series in the Trujillo-Rivero-Bonilla form for the memory function, we proved [57] that the equations with memory functions can be represented through the Riemann-Liouville fractional integrals and the Caputo fractional derivatives of non-integer orders for wide class of the kernels. We can also note that the Abel-type fractional integral operator with Kummer function in the kernel (see Equation (37.1) in [1] (p. 731), and [32]) can be represented as an infinite series of the Riemann-Liouville fractional integrals.…”

Section: Discussionmentioning

confidence: 96%

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Fractional Derivatives and Integrals: What Are They Needed For?

Tarasova

2020

Mathematics

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The question raised in the title of the article is not philosophical. We do not expect general answers of the form “to describe the reality surrounding us”. The question should actually be formulated as a mathematical problem of applied mathematics, a task for new research. This question should be answered in mathematically rigorous statements about the interrelations between the properties of the operator’s kernels and the types of phenomena. This article is devoted to a discussion of the question of what is fractional operator from the point of view of not pure mathematics, but applied mathematics. The imposed restrictions on the kernel of the fractional operator should actually be divided by types of phenomena, in addition to the principles of self-consistency of mathematical theory. In applications of fractional calculus, we have a fundamental question about conditions of kernels of fractional operator of non-integer orders that allow us to describe a particular type of phenomenon. It is necessary to obtain exact correspondences between sets of properties of kernel and type of phenomena. In this paper, we discuss the properties of kernels of fractional operators to distinguish the following types of phenomena: fading memory (forgetting) and power-law frequency dispersion, spatial non-locality and power-law spatial dispersion, distributed lag (time delay), distributed scaling (dilation), depreciation, and aging.

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Section: Discussionmentioning

confidence: 96%

Section: Discussionmentioning

confidence: 96%

Fractional Derivatives and Integrals: What Are They Needed For?

Tarasova

2020

Mathematics

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“…The state space is used to represent the sequence of points (the fractional state space portrait, FSSP, and pseudo phase plane, PPP) corresponding to the states over time. The fractional calculus approach has been used to describe the concept of memory itself for economic processes in [39,40,[110][111][112][113][114][115][116][117], and to define basic concepts of economic processes with memory and non-locality in works .…”

Section: Deterministic Chaos Stage (Approach)mentioning

confidence: 99%

On History of Mathematical Economics: Application of Fractional Calculus

Tarasov

2019

Mathematics

168

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Modern economics was born in the Marginal revolution and the Keynesian revolution. These revolutions led to the emergence of fundamental concepts and methods in economic theory, which allow the use of differential and integral calculus to describe economic phenomena, effects, and processes. At the present moment the new revolution, which can be called “Memory revolution”, is actually taking place in modern economics. This revolution is intended to “cure amnesia” of modern economic theory, which is caused by the use of differential and integral operators of integer orders. In economics, the description of economic processes should take into account that the behavior of economic agents may depend on the history of previous changes in economy. The main mathematical tool designed to “cure amnesia” in economics is fractional calculus that is a theory of integrals, derivatives, sums, and differences of non-integer orders. This paper contains a brief review of the history of applications of fractional calculus in modern mathematical economics and economic theory. The first stage of the Memory Revolution in economics is associated with the works published in 1966 and 1980 by Clive W. J. Granger, who received the Nobel Memorial Prize in Economic Sciences in 2003. We divide the history of the application of fractional calculus in economics into the following five stages of development (approaches): ARFIMA; fractional Brownian motion; econophysics; deterministic chaos; mathematical economics. The modern stage (mathematical economics) of the Memory revolution is intended to include in the modern economic theory new economic concepts and notions that allow us to take into account the presence of memory in economic processes. The current stage actually absorbs the Granger approach based on ARFIMA models that used only the Granger–Joyeux–Hosking fractional differencing and integrating, which really are the well-known Grunwald–Letnikov fractional differences. The modern stage can also absorb other approaches by formulation of new economic notions, concepts, effects, phenomena, and principles. Some comments on possible future directions for development of the fractional mathematical economics are proposed.

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“…The aim of the present paper is to solve the fractional-order a Klein-Gordon and Gas Dynamics equations. In fact, most engineering and physical phenomena are modeled correctly by using fractional Differential Equations (FDEs) [1][2][3]. The FDEs have many applications in science and engineering such as earthquake model [4], signal control system [5], wave models [6], finance models [7] and so on.…”

Section: Introductionmentioning

confidence: 99%

Analytical Solutions of Fractional Klein-Gordon and Gas Dynamics Equations, via the (G′/G)-Expansion Method

2019

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In this article, the ( G ′ / G ) -expansion method is used for the analytical solutions of fractional-order Klein-Gordon and Gas Dynamics equations. The fractional derivatives are defined in the term of Jumarie’s operator. The proposed method is based on certain variable transformation, which transforms the given problems into ordinary differential equations. The solution of resultant ordinary differential equation can be expressed by a polynomial in ( G ′ / G ) , where G = G ( ξ ) satisfies a second order linear ordinary differential equation. In this paper, ( G ′ / G ) -expansion method will represent, the travelling wave solutions of fractional-order Klein-Gordon and Gas Dynamics equations in the term of trigonometric, hyperbolic and rational functions.

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Generalized Memory: Fractional Calculus Approach

Cited by 68 publications

References 44 publications

Fractional Derivatives and Integrals: What Are They Needed For?

Fractional Derivatives and Integrals: What Are They Needed For?

On History of Mathematical Economics: Application of Fractional Calculus

Analytical Solutions of Fractional Klein-Gordon and Gas Dynamics Equations, via the (G′/G)-Expansion Method

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