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Fractal Fract
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Fractional Calculus and Fractals in Mathematical Physics
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Fractional Calculus and Fractals in Mathematical Physics

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

Deadline for manuscript submissions: closed (30 April 2022) | Viewed by 9460

Special Issue Editors

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
Prof. Dr. ‪Ravshan Radjabovich Ashurov

E-Mail Website
Guest Editor
Head of laboratory, Institute of Mathematics, Academy of Science of Uzbekistan, 100174, Students town, Tashkent, Uzbekistan
Interests: partial and fractional differential equations

Special Issue Information

Dear Colleagues,

Fractional calculus is widely used to describe various non-local dynamic processes and systems. Nonlocality or heredity is the property of dynamic systems to preserve their prehistory. Heredity can be described using derivatives of fractional orders, and the orders of fractional derivatives are responsible for the intensity of the process under study (for example, the processes of anomalous diffusion are known—subdiffusion and superdiffusion). At the same time, the orders of fractional derivatives can be associated with the fractal dimension of the medium, where the process under study takes place, and here, we can go over to the theory of fractals. From the point of view of mathematical modeling, the introduction of derivatives of fractional orders gives a more flexible description of the process under study due to an additional degree of freedom—the order of the fractional derivative. In this Special Issue, we focus on the applications of fractional calculus and fractal theory to mathematical physics and related sciences. Topics that may be featured in the Special Edition may include (but are not limited to):

  1. Application of fractional calculus in the theory of diffusion-wave processes;
  2. Application of fractional calculus to describe fractional-order oscillatory systems;
  3. The use of fractional calculus to describe the transfer processes;
  4. Mathematical methods in the theory of fractals;
  5. Inverse problems of mathematical physics.

Prof. Dr. Roman Ivanovich Parovik
Prof. Dr. ‪Ravshan Radjabovich Ashurov
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

  • Fractional calculus
  • Fractal
  • Mathematical physics
  • Mathematical modeling
  • Fractional dynamics

Published Papers (6 papers)

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Research

13 pages, 1544 KiB  
Article
Novel Precise Solitary Wave Solutions of Two Time Fractional Nonlinear Evolution Models via the MSE Scheme
by Zillur Rahman, Alrazi Abdeljabbar, Harun-Or-Roshid and M. Zulfikar Ali
Fractal Fract. 2022, 6(8), 444; https://doi.org/10.3390/fractalfract6080444 - 17 Aug 2022
Cited by 16 | Viewed by 1378
Abstract
We construct soliton solutions of the complex time fractional Schrodinger model (tFSM), as well as the space–time fractional differential model (stFDM), leading wave spread through electrical transmission lines model (ETLM) in low pass with the help of modified simple equation scheme. The approach [...] Read more.
We construct soliton solutions of the complex time fractional Schrodinger model (tFSM), as well as the space–time fractional differential model (stFDM), leading wave spread through electrical transmission lines model (ETLM) in low pass with the help of modified simple equation scheme. The approach provides us with generalized rational exponential function solutions with some free parameters. A few well-known solitary wave resolutions are derived, starting from the generalized rational solutions selecting specific values of the free constants. The precise solutions acquired via the technique signify that the scheme is comparatively easier to execute and attractive in view of the results. No auxiliary equation is needed to solve any nonlinear fractional models by the scheme. Additionally, we observed that the numerical results are very encouraging for researchers conducting further research on stFDMs in mathematics and physics. Full article
(This article belongs to the Special Issue Fractional Calculus and Fractals in Mathematical Physics)
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11 pages, 685 KiB  
Article
Fractional Model of the Deformation Process
by Olga Sheremetyeva and Boris Shevtsov
Fractal Fract. 2022, 6(7), 372; https://doi.org/10.3390/fractalfract6070372 - 1 Jul 2022
Cited by 4 | Viewed by 1137
Abstract
The article considers the fractional Poisson process as a mathematical model of deformation activity in a seismically active region. The dislocation approach is used to describe five modes of the deformation process. The change in modes is determined by the change in the [...] Read more.
The article considers the fractional Poisson process as a mathematical model of deformation activity in a seismically active region. The dislocation approach is used to describe five modes of the deformation process. The change in modes is determined by the change in the intensity of the event stream, the regrouping of dislocations, and the change in and the appearance of stable connections between dislocations. Modeling of the change of deformation modes is carried out by changing three parameters of the proposed model. The background mode with independent events is described by a standard Poisson process. To describe variations from the background mode of seismic activity, when connections are formed between dislocations, the fractional Poisson process and the Mittag–Leffler function characterizing it are used. An approximation of the empirical cumulative distribution function of waiting time of the foreshocks obtained as a result of processing the seismic catalog data was carried out on the basis of the proposed model. It is shown that the model curves, with an appropriate choice of the Mittag–Leffler function’s parameters, gives results close to the experimental ones and can be allowed to characterize the deformation process in the seismically active region under consideration. Full article
(This article belongs to the Special Issue Fractional Calculus and Fractals in Mathematical Physics)
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26 pages, 1083 KiB  
Article
Fractal Properties of the Magnetic Polarity Scale in the Stochastic Hereditary αω-Dynamo Model
by Gleb Vodinchar  and Lyubov Feshchenko
Fractal Fract. 2022, 6(6), 328; https://doi.org/10.3390/fractalfract6060328 - 13 Jun 2022
Cited by 3 | Viewed by 1246
Abstract
We study some fractal properties of the hereditary αω-dynamo model in the two-mode approximation. The phase variables of the model describe the temporal dynamics of the toroidal and poloidal components of the magnetic field. The hereditary operator of the quenching the [...] Read more.
We study some fractal properties of the hereditary αω-dynamo model in the two-mode approximation. The phase variables of the model describe the temporal dynamics of the toroidal and poloidal components of the magnetic field. The hereditary operator of the quenching the α-effect by field helicity in numerical simulation is determined using the Riemann–Liouville fractional differentiation operator. The model also includes a stochastic term. The structure of this term corresponds to the effect of coherent structures from small-scale magnetic field and velocity modes. A difference scheme and a program code for numerical simulation have been developed and verified. A series of computational experiments with the model has been carried out. The Hausdorff dimension of the polarity scale in the model and the distribution of polarity intervals are calculated. It is shown that the Hausdorff dimension of the polarity scale is less than 1, i.e., this scale is a fractal. The numerical value of the dimension for some values of the control parameters is 0.87, which is consistent with the dimension of the real geomagnetic polarity scale. The distribution histogram of polarity intervals in the model has a pronounced power-law tail, which also agrees with the properties of real polarity scales. Full article
(This article belongs to the Special Issue Fractional Calculus and Fractals in Mathematical Physics)
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19 pages, 348 KiB  
Article
On the Solvability of Some Boundary Value Problems for the Nonlocal Poisson Equation with Boundary Operators of Fractional Order
by Kairat Usmanov, Batirkhan Turmetov and Kulzina Nazarova
Fractal Fract. 2022, 6(6), 308; https://doi.org/10.3390/fractalfract6060308 - 31 May 2022
Cited by 3 | Viewed by 1086
Abstract
In this paper, in the class of smooth functions, integration and differentiation operators connected with fractional conformable derivatives are introduced. The mutual reversibility of these operators is proved, and the properties of these operators in the class of smooth functions are studied. Using [...] Read more.
In this paper, in the class of smooth functions, integration and differentiation operators connected with fractional conformable derivatives are introduced. The mutual reversibility of these operators is proved, and the properties of these operators in the class of smooth functions are studied. Using transformations generalizing involutive transformations, a nonlocal analogue of the Laplace operator is introduced. For the corresponding nonlocal analogue of the Poisson equation, the solvability of some boundary value problems with fractional conformable derivatives is studied. For the problems under consideration, theorems on the existence and uniqueness of solutions are proved. Necessary and sufficient conditions for solvability of the studied problems are obtained, and integral representations of solutions are given. Full article
(This article belongs to the Special Issue Fractional Calculus and Fractals in Mathematical Physics)
8 pages, 270 KiB  
Article
On a Nonlocal Problem for Mixed-Type Equation with Partial Riemann-Liouville Fractional Derivative
by Menglibay Ruziev and Rakhimjon Zunnunov
Fractal Fract. 2022, 6(2), 110; https://doi.org/10.3390/fractalfract6020110 - 14 Feb 2022
Cited by 3 | Viewed by 1157
Abstract
The present paper presents a study on a problem with a fractional integro-differentiation operator in the boundary condition for an equation with a partial Riemann-Liouville fractional derivative. The unique solvability of the problem is proved. In the hyperbolic part of the considered domain, [...] Read more.
The present paper presents a study on a problem with a fractional integro-differentiation operator in the boundary condition for an equation with a partial Riemann-Liouville fractional derivative. The unique solvability of the problem is proved. In the hyperbolic part of the considered domain, the functional equation is solved by the iteration method. The problem is reduced to solving the Volterra integro-differential equation. Full article
(This article belongs to the Special Issue Fractional Calculus and Fractals in Mathematical Physics)
21 pages, 404 KiB  
Article
On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations
by Ravshan Ashurov and Yusuf Fayziev
Fractal Fract. 2022, 6(1), 41; https://doi.org/10.3390/fractalfract6010041 - 12 Jan 2022
Cited by 13 | Viewed by 2226
Abstract
The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0<ρ<1, 0<t≤T), [...] Read more.
The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0<ρ<1, 0<t≤T), u(ξ)=αu(0)+φ (α is a constant and 0<ξ≤T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riemann–Liouville derivative; naturally, in the case of the Riemann–Liouville derivatives, the nonlocal boundary condition should be slightly changed. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant α on the existence of a solution to problems is investigated. Inequalities of coercivity type are obtained and it is shown that these inequalities differ depending on the considered type of fractional derivatives. The inverse problems of determining the right-hand side of the equation and the function φ in the boundary conditions are investigated. Full article
(This article belongs to the Special Issue Fractional Calculus and Fractals in Mathematical Physics)
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