Fractional Calculus in Russia at the End of XIX Century

Rogosin, Sergei; Dubatovskaya, Maryna

doi:10.3390/math9151736

Cited by 13 publications

(4 citation statements)

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“…The differential and integral operators of arbitrary positive orders form fractional calculus [1][2][3][4][5][6][7]. The study of such operators has a long history [137][138][139][140][141][142]. Another important tool is the calculus of finite-differences of integer and non-integer orders.…”

Section: Exact Finite-difference Of Arbitrary Positive Ordermentioning

confidence: 99%

Exact Finite-Difference Calculus: Beyond Set of Entire Functions

Tarasov

2024

Mathematics

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In this paper, a short review of the calculus of exact finite-differences of integer order is proposed. The finite-difference operators are called the exact finite-differences of integer orders, if these operators satisfy the same characteristic algebraic relations as standard differential operators of the same order on some function space. In this paper, we prove theorem that this property of the exact finite-differences is satisfies for the space of simple entire functions on the real axis (i.e., functions that can be expanded into power series on the real axis). In addition, new results that describe the exact finite-differences beyond the set of entire functions are proposed. A generalized expression of exact finite-differences for non-entire functions is suggested. As an example, the exact finite-differences of the square root function is considered. The use of exact finite-differences for numerical and computer simulations is not discussed in this paper. Exact finite-differences are considered as an algebraic analog of standard derivatives of integer order.

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Section: Exact Finite-difference Of Arbitrary Positive Ordermentioning

confidence: 99%

Exact Finite-Difference Calculus: Beyond Set of Entire Functions

Tarasov

2024

Mathematics

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“…He observed that if f (t) = e st , t ∈ R, the summation was convergent if Re(s) > 0. This formula was the base of the fractional derivative definition used by Grünwald [60] and Letnikov [61,62]. It is important to note that the Liouville definition was assumed to be valid for functions of exponential type and defined on R or C, while Grünwald and Letnikov worked on [t 0 , t] ∈ R.…”

Section: Differences and Fractional Calculusmentioning

confidence: 99%

Discrete-Time Fractional Difference Calculus: Origins, Evolutions, and New Formalisms

Ortigueira

2023

Fractal Fract

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Differences are introduced as outputs of linear systems called differencers, being considered two classes: shift and scale-invariant. Several types are presented, namely: nabla and delta, bilateral, tempered, bilinear, stretching, and shrinking. Both continuous and discrete-time differences are described. ARMA-type systems based on differencers are introduced and exemplified. In passing, the incorrectness of the usual delta difference is shown.

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“…The attraction behind the development of fractional calculus is rooted in the desire to generalize the classical calculus to a wider class of functions. Fractional calculus has applications in various fields, including physics, engineering, finance, and biology [22,39]. In physics, fractional calculus is used to describe the behavior of complex systems, such as viscoelastic materials, fractals, and anomalous diffusion [13,16].…”

Section: Introductionmentioning

confidence: 99%

A Comprehensive Study of Fractional-Order Derivative and Their Interplay with Basic Functions

Pandey,

Phaijoo,

Gurung

2024

J. Nep. Math. Soc.

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Fractional calculus from the nineteenth century to date has gained considerable attention due to its versatile applications in various scientific and engineering domains. This work examines the complex relationship between fractional-order derivative and basic functions, unraveling the profound interplay between mathematics and simulation. In this study, we illustrate the Mittag-Leffler function, Grunwald-Letnikov’s, Riemann-Liouville’s, and Caputo’s fractional derivative and integral are presented with examples of basic functions and their graphical presentations. The purpose of this study is to examine the features of fractional derivatives from the perspective of researchers’ motivations and interests.

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Fractional Calculus in Russia at the End of XIX Century

Abstract: In this survey paper, we analyze the development of Fractional Calculus in Russia at the end of the XIX century, in particular, the results by A. V. Letnikov, N. Ya. Sonine, and P. A. Nekrasov. Some of the discussed results are either unknown or inaccessible.

Cited by 13 publications

References 13 publications

Exact Finite-Difference Calculus: Beyond Set of Entire Functions

Exact Finite-Difference Calculus: Beyond Set of Entire Functions

Discrete-Time Fractional Difference Calculus: Origins, Evolutions, and New Formalisms

A Comprehensive Study of Fractional-Order Derivative and Their Interplay with Basic Functions

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