scholarly journals On fractal space-time and fractional calculus

2016 ◽  
Vol 20 (3) ◽  
pp. 773-777 ◽  
Author(s):  
Yue Hu ◽  
Ji-Huan He
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Physical Meaning ◽  
Solution Process ◽  
Space Time ◽  
Continuum Models ◽  
Nano Structure ◽  
Fractional Complex Transform ◽  
Fractal Space ◽  
Whitham Equation

This paper gives an explanation of fractional calculus in fractal space-time. On observable scales, continuum models can be used, however, when the scale tends to a smaller threshold, a fractional model has to be adopted to describe phenomena in micro/nano structure. A time-fractional Fornberg-Whitham equation is used as an example to elucidate the physical meaning of the fractional order, and its solution process is given by the fractional complex transform.

RIEMANN-CHRISTOFFEL TENSOR IN DIFFERENTIAL GEOMETRY OF FRACTIONAL ORDER APPLICATION TO FRACTAL SPACE-TIME

Fractals ◽  
2013 ◽  
Vol 21 (01) ◽  
pp. 1350004 ◽  
Author(s):  
GUY JUMARIE
Keyword(s):  
Differential Geometry ◽  
Fractional Order ◽  
Differential Calculus ◽  
Differentiable Manifold ◽  
Space Time ◽  
Differentiable Functions ◽  
Fractal Derivative ◽  
Fractal Space ◽  
Fractional Differential ◽  
The One

By using fractional differences, one recently proposed an alternative to the formulation of fractional differential calculus, of which the main characteristics is a new fractional Taylor series and its companion Rolle's formula which apply to non-differentiable functions. The key is that now we have at hand a differential increment of fractional order which can be manipulated exactly like in the standard Leibniz differential calculus. Briefly the fractional derivative is the quotient of fractional increments. It has been proposed that this calculus can be used to construct a differential geometry on manifold of fractional order. The present paper, on the one hand, refines the framework, and on the other hand, contributes some new results related to arc length of fractional curves, area on fractional differentiable manifold, covariant fractal derivative, Riemann-Christoffel tensor of fractional order, fractional differential equations of fractional geodesic, strip modeling of fractal space time and its relation with Lorentz transformation. The relation with Nottale's fractal space-time theory then appears in quite a natural way.

COMMENTS ON "RIEMANN–CHRISTOFFEL TENSOR IN DIFFERENTIAL GEOMETRY OF FRACTIONAL ORDER APPLICATION TO FRACTAL SPACE-TIME", [FRACTALS 21 (2013) 1350004]

Fractals ◽  
2015 ◽  
Vol 23 (02) ◽  
pp. 1575001 ◽  
Author(s):  
VASILY E. TARASOV
Keyword(s):  
Differential Geometry ◽  
Fractional Derivative ◽  
Power Function ◽  
Fractional Order ◽  
Fractional Derivatives ◽  
Space Time ◽  
Leibniz Rule ◽  
Integer Order ◽  
Fractal Space ◽  
Derivatives Of

We prove that main properties represented by Eq. (4.2) for fractional derivative of power function and the non-fractional Leibniz rule in the form (4.3) of the considered paper, cannot hold together for derivatives of non-integer order. As a result, we prove that the usual Leibniz rule (4.3) cannot hold for fractional derivatives.

Mkhitar Djrbashian and his contribution to Fractional Calculus

2020 ◽  
Vol 23 (6) ◽  
pp. 1797-1809
Author(s):  
Sergei Rogosin ◽  
Maryna Dubatovskaya
Keyword(s):  
Cauchy Problem ◽  
Fractional Calculus ◽  
Fractional Order ◽  
Approximation Theory ◽  
Fractional Derivatives ◽  
Complex Analysis ◽  
Integral Transforms ◽  
Modern Theory ◽  
Survey Paper ◽  
The Cauchy Problem

Abstract This survey paper is devoted to the description of the results by M.M. Djrbashian related to the modern theory of Fractional Calculus. M.M. Djrbashian (1918-1994) is a well-known expert in complex analysis, harmonic analysis and approximation theory. Anyway, his contributions to fractional calculus, to boundary value problems for fractional order operators, to the investigation of properties of the Queen function of Fractional Calculus (the Mittag-Leffler function), to integral transforms’ theory has to be understood on a better level. Unfortunately, most of his works are not enough popular as in that time were published in Russian. The aim of this survey is to fill in the gap in the clear recognition of M.M. Djrbashian’s results in these areas. For same purpose, we decided also to translate in English one of his basic papers [21] of 1968 (joint with A.B. Nersesian, “Fractional derivatives and the Cauchy problem for differential equations of fractional order”), and were invited by the “FCAA” editors to publish its re-edited version in this same issue of the journal.

The bouncing ball and the Grünwald-Letnikov definition of fractional derivative

2021 ◽  
Vol 24 (4) ◽  
pp. 1003-1014
Author(s):  
J. A. Tenreiro Machado
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Restitution Coefficient ◽  
Classical Physics ◽  
Bouncing Ball ◽  
Mechanical Experiment ◽  
Fractional Models ◽  
Definition Of

Abstract This paper proposes a conceptual experiment embedding the model of a bouncing ball and the Grünwald-Letnikov (GL) formulation for derivative of fractional order. The impacts of the ball with the surface are modeled by means of a restitution coefficient related to the coefficients of the GL fractional derivative. The results are straightforward to interpret under the light of the classical physics. The mechanical experiment leads to a physical perspective and allows a straightforward visualization. This strategy provides not only a motivational introduction to students of the fractional calculus, but also triggers possible discussion with regard to the use of fractional models in mechanics.

FRACTALS AND THE QUANTUM THEORY OF SPACETIME

1989 ◽  
Vol 04 (19) ◽  
pp. 5047-5117 ◽  
Author(s):  
LAURENT NOTTALE
Keyword(s):  
Fractal Dimension ◽  
Heavy Ion ◽  
Space Time ◽  
Curvilinear Coordinates ◽  
Fractal Structures ◽  
Ion Collisions ◽  
Principle Of Relativity ◽  
First Results ◽  
Fractal Space ◽  
Dimension 2

We review in this paper the first results obtained in an attempt at understanding quantum space-time based on a new extension of the principle of relativity and on the geometrical concept of fractals. We present methods for dealing with the nondifferentiability and the infinities of fractals, as a first step towards the definition and intrinsic description of a fractal space. After having recalled that the Heisenberg relations imply a transition of spatial coordinates of a particle to fractal dimension 2 about the de Broglie length λ = ħ/p, it is suggested that a similar transition occurs for temporal coordinates about the de Broglie time τ = ħ/E. We then investigate the hypothesis that the microstructure of space-time is of fractal nature, and that the observed properties of the quantum world at a given resolution result from the smoothing of curvilinear coordinates of such a spacetime projected into classical spacetime. Along this road, we successively study the link of fractal dimension 2 to spin, we give first hints on the expected behavior of families of fractal geodesics, and we exhibit a general class of fractal structures which is assumed to yield a lowest order description of the quantum vacuum. The links between the new approach and both special and general relativity are touched upon. We finally suggest that the anomalous peaks recently observed in the spectra of positrons from supercritical heavy ion collisions may be understood in this context.

scholarly journals Nonlinear Viscoelastic-Plastic Creep Model of Rock Based on Fractional Calculus

2022 ◽  
Vol 2022 ◽  
pp. 1-7
Author(s):  
Erjian Wei ◽  
Bin Hu ◽  
Jing Li ◽  
Kai Cui ◽  
Zhen Zhang ◽  
...  
Keyword(s):  
Fractional Calculus ◽  
Fractional Order ◽  
Engineering Application ◽  
Creep Model ◽  
Nonlinear Viscoelastic ◽  
Rock Creep ◽  
Order Change ◽  
Creep Constitutive Model ◽  
Plastic Creep

A rock creep constitutive model is the core content of rock rheological mechanics theory and is of great significance for studying the long-term stability of engineering. Most of the creep models constructed in previous studies have complex types and many parameters. Based on fractional calculus theory, this paper explores the creep curve characteristics of the creep elements with the fractional order change, constructs a nonlinear viscoelastic-plastic creep model of rock based on fractional calculus, and deduces the creep constitutive equation. By using a user-defined function fitting tool of the Origin software and the Levenberg–Marquardt optimization algorithm, the creep test data are fitted and compared. The fitting curve is in good agreement with the experimental data, which shows the rationality and applicability of the proposed nonlinear viscoelastic-plastic creep model. Through sensitivity analysis of the fractional order β2 and viscoelastic coefficient ξ2, the influence of these creep parameters on rock creep is clarified. The research results show that the nonlinear viscoelastic-plastic creep model of rock based on fractional calculus constructed in this paper can well describe the creep characteristics of rock, and this model has certain theoretical significance and engineering application value for long-term engineering stability research.

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