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Fractal Fract
Special Issues
Spectral Methods for Fractional Functional Models
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Spectral Methods for Fractional Functional Models

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 5176

Special Issue Editors

Dr. Kamyar Hosseini

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Guest Editor
Department of Mathematics, Near East University TRNC, Mersin 10, Turkey
Interests: analytical methods; numerical methods; fractional differential equations; wave propagation; mathematical physics; nonlinear partial differential equations
Special Issues, Collections and Topics in MDPI journals
Dr. Khadijeh Sadri

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Guest Editor
Department of Mathematics, Near East University TRNC, Mersin 10, Turkey
Interests: numerical analysis; fractional integral equations; fractional partial differential equation; mathematical models
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Prof. Dr. Evren Hınçal

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Guest Editor
Department of Mathematics, Near East University TRNC, Mersin 10, Turkey
Interests: applied mathematics; statistics; epidemiology
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Prof. Dr. Soheil Salahshour

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Guest Editor
Faculty of Engineering and Natural Sciences, Istanbul Okan University, 34959 Istanbul, Turkey
Interests: fuzzy systems; fractional modelling; optical solitons; applied artificial intelligence
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

It is a well-established fact that many powerful tools, such as partial differential equations, integral equations, and integro-differential equations, have been used to model a wide variety of nonlinear phenomena, ranging from nonlinear optics to plasma physics, circuit theory, and biology. Although the usefulness of such useful tools in modelling nonlinear phenomena is undeniable, researchers have faced issues whereby these tools do not have the necessary efficiency in providing an accurate model with which to describe nonlinear phenomena. Today, such tools, combined with fractional operators, provide effective methods for describing nonlinear phenomena, which have been the subject of much research. Such problems can be handled with a wide range of useful methods including finite difference methods, radial basis function methods, and spectral methods (collocation, Galerkin, and Tau). The key goal of the current Special Issue is to present the latest research on the solutions to the above problems involving fractional operators using spectral methods. Original research and review articles are highly welcomed. Potential topics include, but are not limited to, the following areas:

  • Spectral Methods for Fractional Partial Differential Equations
  • Spectral Methods for Fractional Integral Equations
  • Spectral Methods for Integro-Differential Equations Involving Fractional Operators
  • Spectral Methods for Systems of Fractional Differential Equations

Dr. Kamyar Hosseini
Dr. Khadijeh Sadri
Prof. Dr. Evren Hınçal
Prof. Dr. Soheil Salahshour
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

  • spectral methods for fractional partial differential equations
  • spectral methods for fractional integral equations
  • spectral methods for integro-differential equations involving fractional operators
  • spectral methods for systems of fractional differential equations

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Published Papers (4 papers)

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Research

19 pages, 1158 KiB  
Article
A Novel Semi-Analytical Scheme to Deal with Fractional Partial Differential Equations (PDEs) of Variable-Order
by Samad Kheybari, Farzaneh Alizadeh, Mohammad Taghi Darvishi, Kamyar Hosseini and Evren Hincal
Fractal Fract. 2024, 8(7), 425; https://doi.org/10.3390/fractalfract8070425 - 20 Jul 2024
Viewed by 714
Abstract
This article introduces a new numerical algorithm dedicated to solving the most general form of variable-order fractional partial differential models. Both the time and spatial order of derivatives are considered as non-constant values. A combination of the shifted Chebyshev polynomials is used to [...] Read more.
This article introduces a new numerical algorithm dedicated to solving the most general form of variable-order fractional partial differential models. Both the time and spatial order of derivatives are considered as non-constant values. A combination of the shifted Chebyshev polynomials is used to approximate the solution of such equations. The coefficients of this combination are considered a function of time, and they are obtained using the collocation method. The theoretical aspects of the method are investigated, and then by solving some problems, the efficiency of the method is presented. Full article
(This article belongs to the Special Issue Spectral Methods for Fractional Functional Models)
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16 pages, 328 KiB  
Article
Non-Polynomial Collocation Spectral Scheme for Systems of Nonlinear Caputo–Hadamard Differential Equations
by Mahmoud A. Zaky, Ibrahem G. Ameen, Mohammed Babatin, Ali Akgül, Magda Hammad and António M. Lopes
Fractal Fract. 2024, 8(5), 262; https://doi.org/10.3390/fractalfract8050262 - 27 Apr 2024
Cited by 2 | Viewed by 887
Abstract
In this paper, we provide a collocation spectral scheme for systems of nonlinear Caputo–Hadamard differential equations. Since the Caputo–Hadamard operators contain logarithmic kernels, their solutions can not be well approximated using the usual spectral methods that are classical polynomial-based schemes. Hence, we construct [...] Read more.
In this paper, we provide a collocation spectral scheme for systems of nonlinear Caputo–Hadamard differential equations. Since the Caputo–Hadamard operators contain logarithmic kernels, their solutions can not be well approximated using the usual spectral methods that are classical polynomial-based schemes. Hence, we construct a non-polynomial spectral collocation scheme, describe its effective implementation, and derive its convergence analysis in both L2 and L∞. In addition, we provide numerical results to support our theoretical analysis. Full article
(This article belongs to the Special Issue Spectral Methods for Fractional Functional Models)
14 pages, 4262 KiB  
Article
Predicting the Remaining Useful Life of Turbofan Engines Using Fractional Lévy Stable Motion with Long-Range Dependence
by Deyu Qi, Zijiang Zhu, Fengmin Yao, Wanqing Song, Aleksey Kudreyko, Piercarlo Cattani and Francesco Villecco
Fractal Fract. 2024, 8(1), 55; https://doi.org/10.3390/fractalfract8010055 - 15 Jan 2024
Cited by 1 | Viewed by 1529
Abstract
Remaining useful life prediction guarantees a reliable and safe operation of turbofan engines. Long-range dependence (LRD) and heavy-tailed characteristics of degradation modeling make this method advantageous for the prediction of RUL. In this study, we propose fractional Lévy stable motion for degradation modeling. [...] Read more.
Remaining useful life prediction guarantees a reliable and safe operation of turbofan engines. Long-range dependence (LRD) and heavy-tailed characteristics of degradation modeling make this method advantageous for the prediction of RUL. In this study, we propose fractional Lévy stable motion for degradation modeling. First, we define fractional Lévy stable motion simulation algorithms. Then, we demonstrate the LRD and heavy-tailed property of fLsm to provide support for the model. The proposed method is validated with the C-MAPSS dataset obtained from the turbofan engine. Principle components analysis (PCA) is conducted to extract sources of variance. Experimental data show that the predictive model based on fLsm with exponential drift exhibits superior accuracy relative to the existing methods. Full article
(This article belongs to the Special Issue Spectral Methods for Fractional Functional Models)
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16 pages, 878 KiB  
Article
Study on Abundant Dust-Ion-Acoustic Solitary Wave Solutions of a (3+1)-Dimensional Extended Zakharov–Kuznetsov Dynamical Model in a Magnetized Plasma and Its Linear Stability
by Muhammad Arshad, Aly R. Seadawy, Muhammad Tanveer and Faisal Yasin
Fractal Fract. 2023, 7(9), 691; https://doi.org/10.3390/fractalfract7090691 - 18 Sep 2023
Cited by 12 | Viewed by 1230
Abstract
This article examines how shocks and three-dimensional nonlinear dust-ion-acoustic waves propagate across uniform magnetized electron–positron–ion plasmas. The two-variable (G′/G,1/G)-expansion and generalized exp(−ϕ(ξ))-expansion techniques are presented to construct [...] Read more.
This article examines how shocks and three-dimensional nonlinear dust-ion-acoustic waves propagate across uniform magnetized electron–positron–ion plasmas. The two-variable (G′/G,1/G)-expansion and generalized exp(−ϕ(ξ))-expansion techniques are presented to construct the ion-acoustic wave results of a (3+1)-dimensional extended Zakharov–Kuznetsov (eZK) model. As a result, the novel soliton and other wave solutions in a variety of forms, including kink- and anti-kink-type breather waves, dark and bright solitons, kink solitons, and multi-peak solitons, etc., are attained. With the help of software, the solitary wave results (that signify the electrostatic potential field), electric and magnetic fields, and quantum statistical pressures are also constructed. These solutions have numerous applications in various areas of physics and other areas of applied sciences. Graphical representations of some of the obtained results, and the electric and magnetic fields as well as the electrostatic field potential are also presented. These results demonstrate the effectiveness of the presented techniques, which will also be useful in solving many other nonlinear models that arise in mathematical physics and several other applied sciences fields. Full article
(This article belongs to the Special Issue Spectral Methods for Fractional Functional Models)
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