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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

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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

Published Papers (2 papers)

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Research

14 pages, 4262 KiB

Open AccessArticle

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

Viewed by 1032

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. 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

Open AccessArticle

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 1 | Viewed by 866

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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