Rules for Fractional-Dynamic Generalizations: Difficulties of Constructing Fractional Dynamic Models

Abstract

This article is a review of problems and difficulties arising in the construction of fractional-dynamic analogs of standard models by using fractional calculus. These fractional generalizations allow us to take into account the effects of memory and non-locality, distributed lag, and scaling. We formulate rules (principles) for constructing fractional generalizations of standard models, which were described by differential equations of integer order. Important requirements to building fractional generalization of dynamical models (the rules for “fractional-dynamic generalizers”) are represented as the derivability principle, the multiplicity principle, the solvability and correspondence principles, and the interpretability principle. The characteristic properties of fractional derivatives of non-integer order are the violation of standard rules and properties that are fulfilled for derivatives of integer order. These non-standard mathematical properties allow us to describe non-standard processes and phenomena associated with non-locality and memory. However, these non-standard properties lead to restrictions in the sequential and self-consistent construction of fractional generalizations of standard models. In this article, we give examples of problems arising due to the non-standard properties of fractional derivatives in construction of fractional generalizations of standard dynamic models in economics.

Fractional Derivatives and Integrals: What Are They Needed For?

“…which is used in the Gorenflo-Luchko-Mainardi (GLM) operator [23][24][25] with some parameters γ ∈ R and β > 0 of the kernel K(t, τ) (for details see Equations (1) and (12) in [24], and Equations (4) and (39) in [25]). This operator is also known as the left-sided Caputo-type modification of the Erdelyi-Kober fractional derivative (see Equation (12) in [24] (p. 362)). Please note that the GLM operator was introduced for the first time in [23] in connection with the scale-invariant solutions of the time-fractional diffusion-wave equation (see Equation 58on [23] (p. 188)).…”

“…Remark 6. Given the above, we can state that the operator with kernel, which satisfies the conditions (12) and (13), cannot be interpreted as fractional derivative of non-integer order for positive integer values of n. The correct interpretation of this operator is integer order derivative with the continuously distributed lag [29]. As a basis for the definition of this operator, which is integer order operators, we can use expression (14) with conditions (15) instead of Equation 1with conditions (12) and (13).…”

“…We also do not plan to delve into the "linguistic" question of which operators might be called fractional and which are not. The first author has already formulated their point of view on this issue in articles [8][9][10][11][12]. There are also many important contributions to this discussion (for example, see [13][14][15][16]).…”

Dirac particle with memory: Proper time non-locality

“…These non-standard mathematical properties allow us to describe non-standard processes and phenomena associated with non-locality and memory [20,21]. On the other hand, these non-standard properties lead to difficulties [37] in sequential constructing a fractional generalizations of standard models. In article [37], we show how problems arise when building fractional generalizations of standard models.…”

“…On the other hand, these non-standard properties lead to difficulties [37] in sequential constructing a fractional generalizations of standard models. In article [37], we show how problems arise when building fractional generalizations of standard models.…”