Research

18 pages, 1842 KiB

Open AccessArticle

On a Generalization of One-Dimensional Kinetics

by Vladimir V. Uchaikin, Renat T. Sibatov and Dmitry N. Bezbatko

Mathematics 2021, 9(11), 1264; https://doi.org/10.3390/math9111264 - 31 May 2021

Cited by 1 | Viewed by 1713

One-dimensional random walks with a constant velocity between scattering are considered. The exact solution is expressed in terms of multiple convolutions of path-distributions assumed to be different for positive and negative directions of the walk axis. Several special cases are considered when the [...] Read more.

One-dimensional random walks with a constant velocity between scattering are considered. The exact solution is expressed in terms of multiple convolutions of path-distributions assumed to be different for positive and negative directions of the walk axis. Several special cases are considered when the convolutions are expressed in explicit form. As a particular case, the solution of A. S. Monin for a symmetric random walk with exponential path distribution and its generalization to the asymmetric case are obtained. Solution of fractional telegraph equation with the fractional material derivative is presented. Asymptotic behavior of its solution for an asymmetric case is provided. Full article

(This article belongs to the Special Issue Fractional Calculus in Anomalous Transport Theory)

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14 pages, 299 KiB

Open AccessArticle

Factorization à la Dirac Applied to Some Equations of Classical Physics

by Zine El Abiddine Fellah, Erick Ogam, Mohamed Fellah and Claude Depollier

Mathematics 2021, 9(8), 899; https://doi.org/10.3390/math9080899 - 18 Apr 2021

Viewed by 1573

Abstract

In this paper, we present an application of Dirac’s factorization method to three types of the partial differential equations, i.e., the wave equation, the scattering equation, and the telegrapher’s equation. This method gives results that contribute to a better understanding of physical phenomena [...] Read more.

In this paper, we present an application of Dirac’s factorization method to three types of the partial differential equations, i.e., the wave equation, the scattering equation, and the telegrapher’s equation. This method gives results that contribute to a better understanding of physical phenomena by generalizing the Euler and constituent equations. Its application to the wave equation shows that it is indeed a factorization method, since it gives d’Alembert’s solutions in a more general framework. In the case of the diffusion equation, a fractional differential equation has been established that has already been highlighted by other authors in particular cases, but by indirect methods. Dirac’s method brings several new results in the case of the telegraphers’ equation corresponding to the propagation of an acoustic wave in a dissipative fluid. On the one hand, its formalism facilitates the temporal interpretation of phenomena, in particular the density and compressibility of the fluid become temporal operators, which can be “seen” as susceptibilities of the fluid. On the other hand, a consequence of this temporal modeling is the highlighting in Euler’s equation of a term similar to the one that was introduced by Boussinesq and Basset in the equation of the motion of a solid sphere in a unsteady fluid. Full article

(This article belongs to the Special Issue Fractional Calculus in Anomalous Transport Theory)

15 pages, 3079 KiB

Open AccessArticle

Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and Applications

by Yong Zhang, Dongbao Zhou, Wei Wei, Jonathan M. Frame, Hongguang Sun, Alexander Y. Sun and Xingyuan Chen

Mathematics 2021, 9(7), 790; https://doi.org/10.3390/math9070790 - 06 Apr 2021

Cited by 4 | Viewed by 2021

Abstract

Fractional calculus-based differential equations were found by previous studies to be promising tools in simulating local-scale anomalous diffusion for pollutants transport in natural geological media (geomedia), but efficient models are still needed for simulating anomalous transport over a broad spectrum of scales. This [...] Read more.

Fractional calculus-based differential equations were found by previous studies to be promising tools in simulating local-scale anomalous diffusion for pollutants transport in natural geological media (geomedia), but efficient models are still needed for simulating anomalous transport over a broad spectrum of scales. This study proposed a hierarchical framework of fractional advection-dispersion equations (FADEs) for modeling pollutants moving in the river corridor at a full spectrum of scales. Applications showed that the fixed-index FADE could model bed sediment and manganese transport in streams at the geomorphologic unit scale, whereas the variable-index FADE well fitted bedload snapshots at the reach scale with spatially varying indices. Further analyses revealed that the selection of the FADEs depended on the scale, type of the geomedium (i.e., riverbed, aquifer, or soil), and the type of available observation dataset (i.e., the tracer snapshot or breakthrough curve (BTC)). When the pollutant BTC was used, a single-index FADE with scale-dependent parameters could fit the data by upscaling anomalous transport without mapping the sub-grid, intermediate multi-index anomalous diffusion. Pollutant transport in geomedia, therefore, may exhibit complex anomalous scaling in space (and/or time), and the identification of the FADE’s index for the reach-scale anomalous transport, which links the geomorphologic unit and watershed scales, is the core for reliable applications of fractional calculus in hydrology. Full article

(This article belongs to the Special Issue Fractional Calculus in Anomalous Transport Theory)

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11 pages, 913 KiB

Open AccessArticle

Nucleation Controlled by Non-Fickian Fractional Diffusion

by Vyacheslav Svetukhin

Mathematics 2021, 9(7), 740; https://doi.org/10.3390/math9070740 - 31 Mar 2021

Viewed by 1471

Abstract

Kinetic models of aggregation and dissolution of clusters in disordered heterogeneous materials based on subdiffusive equations containing fractional derivatives are studied. Using the generalized fractional Fick law and fractional Fokker–Planck equation for impurity diffusion with localization, we consider modifications of the classical models [...] Read more.

Kinetic models of aggregation and dissolution of clusters in disordered heterogeneous materials based on subdiffusive equations containing fractional derivatives are studied. Using the generalized fractional Fick law and fractional Fokker–Planck equation for impurity diffusion with localization, we consider modifications of the classical models of Ham, Aaron–Kotler, and Lifshitz–Slezov for nucleation and decomposition of solid solutions. The asymptotic time dependencies of supersaturation degree, average cluster size, and other characteristics at the stages of subdiffusion-limited nucleation and coalescence are calculated and analyzed. Full article

(This article belongs to the Special Issue Fractional Calculus in Anomalous Transport Theory)

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13 pages, 1033 KiB

Open AccessArticle

Fractal Stochastic Processes on Thin Cantor-Like Sets

by Alireza Khalili Golmankhaneh and Renat Timergalievich Sibatov

Mathematics 2021, 9(6), 613; https://doi.org/10.3390/math9060613 - 15 Mar 2021

Cited by 19 | Viewed by 2051

Abstract

We review the basics of fractal calculus, define fractal Fourier transformation on thin Cantor-like sets and introduce fractal versions of Brownian motion and fractional Brownian motion. Fractional Brownian motion on thin Cantor-like sets is defined with the use of non-local fractal derivatives. The [...] Read more.

We review the basics of fractal calculus, define fractal Fourier transformation on thin Cantor-like sets and introduce fractal versions of Brownian motion and fractional Brownian motion. Fractional Brownian motion on thin Cantor-like sets is defined with the use of non-local fractal derivatives. The fractal Hurst exponent is suggested, and its relation with the order of non-local fractal derivatives is established. We relate the Gangal fractal derivative defined on a one-dimensional stochastic fractal to the fractional derivative after an averaging procedure over the ensemble of random realizations. That means the fractal derivative is the progenitor of the fractional derivative, which arises if we deal with a certain stochastic fractal. Full article

(This article belongs to the Special Issue Fractional Calculus in Anomalous Transport Theory)

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13 pages, 328 KiB

Open AccessArticle

Non-Debye Relaxations: Two Types of Memories and Their Stieltjes Character

by Katarzyna Górska and Andrzej Horzela

Mathematics 2021, 9(5), 477; https://doi.org/10.3390/math9050477 - 26 Feb 2021

Cited by 6 | Viewed by 1218

Abstract

In this paper, we show that spectral functions relevant for commonly used models of the non-Debye relaxation are related to the Stieltjes functions supported on the positive semi-axis. Using only this property, it can be shown that the response and relaxation functions are [...] Read more.

In this paper, we show that spectral functions relevant for commonly used models of the non-Debye relaxation are related to the Stieltjes functions supported on the positive semi-axis. Using only this property, it can be shown that the response and relaxation functions are non-negative. They are connected to each other and obey the time evolution provided by integral equations involving the memory function

M (t)

, which is the Stieltjes function as well. This fact is also due to the Stieltjes character of the spectral function. Stochastic processes-based approach to the relaxation phenomena gives the possibility to identify the memory function

M (t)

with the Laplace (Lévy) exponent of some infinitely divisible stochastic processes and to introduce its partner memory

k (t)

. Both memories are related by the Sonine equation and lead to equivalent evolution equations which may be freely interchanged in dependence of our knowledge on memories governing the process. Full article

(This article belongs to the Special Issue Fractional Calculus in Anomalous Transport Theory)

24 pages, 596 KiB

Open AccessEditor’s ChoiceArticle

Diffusion–Advection Equations on a Comb: Resetting and Random Search

by Trifce Sandev, Viktor Domazetoski, Alexander Iomin and Ljupco Kocarev

Mathematics 2021, 9(3), 221; https://doi.org/10.3390/math9030221 - 22 Jan 2021

Cited by 15 | Viewed by 2447

Abstract

This review addresses issues of various drift–diffusion and inhomogeneous advection problems with and without resetting on comblike structures. Both a Brownian diffusion search with drift and an inhomogeneous advection search on the comb structures are analyzed. The analytical results are verified by numerical [...] Read more.

This review addresses issues of various drift–diffusion and inhomogeneous advection problems with and without resetting on comblike structures. Both a Brownian diffusion search with drift and an inhomogeneous advection search on the comb structures are analyzed. The analytical results are verified by numerical simulations in terms of coupled Langevin equations for the comb structure. The subordination approach is one of the main technical methods used here, and we demonstrated how it can be effective in the study of various random search problems with and without resetting. Full article

(This article belongs to the Special Issue Fractional Calculus in Anomalous Transport Theory)

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10 pages, 248 KiB

Open AccessArticle

Higher-Order Symmetries of a Time-Fractional Anomalous Diffusion Equation

by Rafail K. Gazizov and Stanislav Yu. Lukashchuk

Mathematics 2021, 9(3), 216; https://doi.org/10.3390/math9030216 - 21 Jan 2021

Cited by 7 | Viewed by 1518

Abstract

Higher-order symmetries are constructed for a linear anomalous diffusion equation with the Riemann–Liouville time-fractional derivative of order

α \in (0, 1) \cup (1, 2)

. It is proved that the equation in question has infinite sequences of [...] Read more.

Higher-order symmetries are constructed for a linear anomalous diffusion equation with the Riemann–Liouville time-fractional derivative of order

α \in (0, 1) \cup (1, 2)

. It is proved that the equation in question has infinite sequences of nontrivial higher-order symmetries that are generated by two local recursion operators. It is also shown that some of the obtained higher-order symmetries can be rewritten as fractional-order symmetries, and corresponding fractional-order recursion operators are presented. The proposed approach for finding higher-order symmetries is applicable for a wide class of linear fractional differential equations. Full article

(This article belongs to the Special Issue Fractional Calculus in Anomalous Transport Theory)

14 pages, 771 KiB

Open AccessArticle

Fractal Generalization of the Scher–Montroll Model for Anomalous Transit-Time Dispersion in Disordered Solids

by Renat T. Sibatov

Mathematics 2020, 8(11), 1991; https://doi.org/10.3390/math8111991 - 08 Nov 2020

Cited by 3 | Viewed by 1667

Abstract

The Scher–Montroll model successfully describes subdiffusive photocurrents in homogeneously disordered semiconductors. The present paper generalizes this model to the case of fractal spatial disorder (self-similar random distribution of localized states) under the conditions of the time-of-flight experiment. Within the fractal model, we calculate [...] Read more.

The Scher–Montroll model successfully describes subdiffusive photocurrents in homogeneously disordered semiconductors. The present paper generalizes this model to the case of fractal spatial disorder (self-similar random distribution of localized states) under the conditions of the time-of-flight experiment. Within the fractal model, we calculate charge carrier densities and transient current for different cases, solving the corresponding fractional-order equations of dispersive transport. Photocurrent response after injection of non-equilibrium carriers by the short laser pulse is expressed via fractional stable distributions. For the simplest case of one-sided instantaneous jumps (tunneling) between neighboring localized states, the dispersive transport equation contains fractional Riemann–Liouville derivatives on time and longitudinal coordinate. We discuss the role of back-scattering, spatial correlations induced by quenching of disorder, and spatiotemporal non-locality produced by the fractal trap distribution and the finite velocity of motion between localized states. We derive expressions for the photocurrent and transit time that allow us to determine the fractal dimension of the distribution of traps and the dispersion parameter from the time-of-flight measurements. Full article

(This article belongs to the Special Issue Fractional Calculus in Anomalous Transport Theory)

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35 pages, 525 KiB

Open AccessFeature PaperArticle

Two Forms of the Integral Representations of the Mittag-Leffler Function

by Viacheslav Saenko

Mathematics 2020, 8(7), 1101; https://doi.org/10.3390/math8071101 - 05 Jul 2020

Cited by 2 | Viewed by 1845

Abstract

The integral representation of the two-parameter Mittag-Leffler function

E_{ρ, μ} (z)

is considered in the paper that expresses its value in terms of the contour integral. For this integral representation, the transition is made from integration over a complex [...] Read more.

The integral representation of the two-parameter Mittag-Leffler function

E_{ρ, μ} (z)

is considered in the paper that expresses its value in terms of the contour integral. For this integral representation, the transition is made from integration over a complex variable to integration over real variables. It is shown that as a result of such a transition, the integral representation of the function

E_{ρ, μ} (z)

has two forms: the representation “A” and “B”. Each of these representations has its advantages and drawbacks. In the paper, the corresponding theorems are formulated and proved, and the advantages and disadvantages of each of the obtained representations are discussed. Full article

(This article belongs to the Special Issue Fractional Calculus in Anomalous Transport Theory)

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