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

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: closed (31 May 2024) | Viewed by 5429

Special Issue Editors

Prof. Dr. Arsen V. Pskhu

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Guest Editor
Institute of Applied Mathematics and Automation, Kabardino-Balkarian Scientific Center of Russian Academy of Sciences, 89-A Shortanov Street, 360000 Nalchik, Russia
Interests: fractional calculus; fractional differential equation; mathematical modeling
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Roman Ivanovich Parovik

E-Mail Website
Guest Editor
Physical Processes Modeling Laboratory, Institute of Cosmophysical Research and Radio Wave Propagation, Far Eastern Branch of the Russian Academy of Sciences, Kamchatskiy Kray, 684034 Paratunka, Russia
Interests: fractional calculus; fractional oscillators; fractional dynamics; numerical methods; mathematical modeling
Special Issues, Collections and Topics in MDPI journals

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.

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Related Special Issue

Published Papers (4 papers)

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Research

12 pages, 660 KiB  
Article
Non-Local Problems for the Fractional Order Diffusion Equation and the Degenerate Hyperbolic Equation
by Menglibay Ruziev, Roman Parovik, Rakhimjon Zunnunov and Nargiza Yuldasheva
Fractal Fract. 2024, 8(9), 538; https://doi.org/10.3390/fractalfract8090538 - 16 Sep 2024
Viewed by 661
Abstract
This research explores nonlocal problems associated with fractional diffusion equations and degenerate hyperbolic equations featuring singular coefficients in their lower-order terms. The uniqueness of the solution is established using the energy integral method, while the existence of the solution is equivalently reduced to [...] Read more.
This research explores nonlocal problems associated with fractional diffusion equations and degenerate hyperbolic equations featuring singular coefficients in their lower-order terms. The uniqueness of the solution is established using the energy integral method, while the existence of the solution is equivalently reduced to solving Volterra integral equations of the second kind and a fractional differential equation. The study focuses on a mixed domain where the parabolic section aligns with the upper half-plane, and the hyperbolic section is bounded by two characteristics of the equation under consideration and a segment of the x-axis. By utilizing the solution representation of the fractional-order diffusion equation, a primary functional relationship is derived between the traces of the sought function on the x-axis segment from the parabolic part of the mixed domain. An explicit solution form for the modified Cauchy problem in the hyperbolic section of the mixed domain is presented. This solution, combined with the problem’s boundary condition, yields a fundamental functional relationship between the traces of the unknown function, mapped to the interval of the equation’s degeneration line. Through the conjugation condition of the problem, an equation with fractional derivatives is obtained by eliminating one unknown function from two functional relationships. The solution to this equation is explicitly formulated. For a specific solution of the proposed problem, visualizations are provided for various orders of the fractional derivative. The analysis demonstrates that the derivative order influences both the intensity of the diffusion (or subdiffusion) process and the shape of the wave front. Full article
(This article belongs to the Special Issue Fractional Evolutionary Equations and Modeling of Dissipative Processes, 2nd Edition)
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12 pages, 933 KiB  
Article
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 1557
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 [...] Read more.
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)
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15 pages, 2087 KiB  
Article
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 2 | Viewed by 1507
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 [...] Read more.
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)
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17 pages, 344 KiB  
Article
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 1198
Abstract
The Cauchy problem for the telegraph equation (Dtρ)2u(t)+2αDtρu(t)+Au(t)=f(t) ( [...] Read more.
The Cauchy problem for the telegraph equation (Dtρ)2u(t)+2αDtρu(t)+Au(t)=f(t) (0<t≤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; Dt 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)
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