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: 31 May 2024 | Viewed by 3539

Special Issue Editors


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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
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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

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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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Published Papers (3 papers)

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Research

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 1259
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
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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 1 | Viewed by 1101
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
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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 834
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<tT,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
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