Fractal and Fractional

Journal Browser

► Journal Browser

Special Issue Information

Dear Colleagues,

The Special Issue is devoted to the application of fractional order evolutionary differential equations to describe dissipative systems and processes. Generally, a dissipative system is understood as any open (non-conservative) system that is not in thermodynamic equilibrium. Dissipative processes include various irreversible thermodynamic processes; mass, electrical and heat transfer; mechanical motion of damped systems; chemical reactions; the radiation and absorption of electromagnetic waves, etc. Particular attention will be paid to initial and boundary value problems for partial differential equations of fractional order, which are the basis for mathematical models of dissipative systems and processes.

Prof. Dr. Arsen V. Pskhu
Prof. Dr. Roman Ivanovich Parovik
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

diffusion wave equations
direct and backward problems
problems without initial conditions
mathematical modeling of microseisms
Sel’kov fractional dynamical system
distributed order fractional diffusion
memory density
Hurst exponents
fractional Riccati equation
variable order fractional derivatives
regular and chaotic regimes
Dubovsky's fractional dynamic system.

Related Special Issue

Published Papers (3 papers)

Download All Papers

Research

12 pages, 933 KiB

Open AccessArticle

Fractional Criticality Theory and Its Application in Seismology

by Boris Shevtsov and Olga Sheremetyeva

Fractal Fract. 2023, 7(12), 890; https://doi.org/10.3390/fractalfract7120890 - 18 Dec 2023

Cited by 1 | Viewed by 1235

Abstract

To understand how the temporal non-locality («memory») properties of a process affect its critical regimes, the power-law compound and time-fractional Poisson process is presented as a universal hereditary model of criticality. Seismicity is considered as an application of the theory of criticality. On the basis of the proposed hereditarian criticality model, the critical regimes of seismicity are investigated. It is shown that the seismic process has the property of «memory» (non-locality over time) and statistical time-dependence of events. With a decrease in the fractional exponent of the Poisson process, the relaxation slows down, which can be associated with the hardening of the medium and the accumulation of elastic energy. Delayed relaxation is accompanied by an abnormal increase in fluctuations, which is caused by the non-local correlations of random events over time. According to the found criticality indices, the seismic process is in subcritical regimes for the zero and first moments and in supercritical regimes for the second statistical moment of events’ reoccurrence frequencies distribution. The supercritical regimes indicate the instability of the deformation changes that can go into a non-stationary regime of a seismic process. Full article

(This article belongs to the Special Issue Fractional Evolutionary Equations and Modeling of Dissipative Processes, 2nd Edition)

► Show Figures

Figure 1

15 pages, 2087 KiB

Open AccessArticle

Bioheat Transfer with Thermal Memory and Moving Thermal Shocks

by Nehad Ali Shah, Bander Almutairi, Dumitru Vieru, Beomseon Lee and Jae Dong Chung

Fractal Fract. 2023, 7(8), 629; https://doi.org/10.3390/fractalfract7080629 - 18 Aug 2023

Cited by 1 | Viewed by 1048

Abstract

This article investigates the effects of thermal memory and the moving line thermal shock on heat transfer in biological tissues by employing a generalized form of the Pennes equation. The mathematical model is built upon a novel time-fractional generalized Fourier’s law, wherein the thermal flux is influenced not only by the temperature gradient but also by its historical behavior. Fractionalization of the heat flow via a fractional integral operator leads to modeling of the finite speed of the heat wave. Moreover, the thermal source generates a linear thermal shock at every instant in a specified position of the tissue. The analytical solution in the Laplace domain for the temperature of the generalized model, respectively the analytical solution in the real domain for the ordinary model, are determined using the Laplace transform. The influence of the thermal memory parameter on the heat transfer is analyzed through numerical simulations and graphic representations. Full article

(This article belongs to the Special Issue Fractional Evolutionary Equations and Modeling of Dissipative Processes, 2nd Edition)

► Show Figures

Figure 1

17 pages, 344 KiB

Open AccessArticle

Fractional Telegraph Equation with the Caputo Derivative

by Ravshan Ashurov and Rajapboy Saparbayev

Fractal Fract. 2023, 7(6), 483; https://doi.org/10.3390/fractalfract7060483 - 17 Jun 2023

Cited by 1 | Viewed by 793

Abstract

The Cauchy problem for the telegraph equation

{(D_{t}^{ρ})}^{2} u (t) + 2 α D_{t}^{ρ} u (t) + A u (t) = f (t)

( [...] Read more.

The Cauchy problem for the telegraph equation

{(D_{t}^{ρ})}^{2} u (t) + 2 α D_{t}^{ρ} u (t) + A u (t) = f (t)

(

0 < t \leq T, 0 < ρ < 1

α > 0

), with the Caputo derivative is considered. Here, A is a selfadjoint positive operator, acting in a Hilbert space, H;

D_{t}

is the Caputo fractional derivative. Conditions are found for the initial functions and the right side of the equation that guarantee both the existence and uniqueness of the solution of the Cauchy problem. It should be emphasized that these conditions turned out to be less restrictive than expected in a well-known paper by R. Cascaval et al. where a similar problem for a homogeneous equation with some restriction on the spectrum of the operator, A, was considered. We also prove stability estimates important for the application. Full article

(This article belongs to the Special Issue Fractional Evolutionary Equations and Modeling of Dissipative Processes, 2nd Edition)

Journal Menu

Journal Browser

Fractional Evolutionary Equations and Modeling of Dissipative Processes, 2nd Edition

Share This Special Issue

Special Issue Editors

Special Issue Information

Keywords

Related Special Issue

Published Papers (3 papers)

Research

Further Information

Guidelines

MDPI Initiatives

Follow MDPI