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Fractal Fract
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Recent Developments and Applications of Fractional Differential Equations in Mathematical...
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Recent Developments and Applications of Fractional Differential Equations 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 (31 July 2024) | Viewed by 8984

Special Issue Editors

Prof. Dr. Ahmet Bekir

E-Mail Website
Guest Editor
Independent Researcher, Neighbourhood of Akcaglan, Eskisehir, Turkey
Interests: soliton theory; integrability and exact solutions of partial differential equations in mathematical physics; semi-analytical and approximate solutions of nonlinear evolution equations applied sciences and engineering; ODEs; PDEs; fractional differential equations; integral equations and analytical methods
Prof. Dr. Adem Cevikel

E-Mail Website
Guest Editor
Department of Mathematics, Yildiz Technical University, Davutpasa Campus, 34100 Istanbul, Turkey
Interests: exact solutions of partial differential equations in mathematical physics; semi-analytical and approximate solutions of nonlinear evolution equations applied sciences and engineering; game theory and fuzzy mathematics
Dr. Emad Zahran

E-Mail Website
Guest Editor
Department of Mathematical and Physical Engineering, Faculty of Engineering, Benha University, 13511 Shubra, Egypt
Interests: numerical analysis; numerical methods and exact solutions of partial differential equations in mathematical physics, semi-analytical and approximate solutions of nonlinear evolution equations applied sciences and engineering

Special Issue Information

Dear Colleagues,

In recent years, fractional differential equations have been extensively used in mathematical models of interesting and important phenomena observed in science and technology. In recent decades, many powerful methods to construct exact and numerical solutions of fractional differential equations have been established and developed, which has led to one of the most exciting advances of nonlinear science and theoretical physics. These relatively new methods proved to be fully synchronized with the complexities of the physical problems. The investigation of exact solutions for nonlinear evolution equations also plays an important role in the study of nonlinear physical phenomena.

This Special Issue aims to combine contributions across a variety of exact, analytical, and numerical solutions of fractional differential equations and invite authors to submit original research and/or domain reviews in various methods. This issue will become an international forum for researchers to present the most recent research and ideas about fractional differential equations using different methods. Original research that reflects the recent theoretical advances and experimental results as well as new topics are invited on all aspects of object tracking.

Potential topics include, but are not limited to:

  • New definitions and theories in fractional calculus.
  • Fractional mathematical models in applied mathematics.
  • Fractional differential/integral equations.
  • Exact solutions of fractional differential equations.
  • Numerical methods for fractional differential equations.
  • Existence, uniqueness, and regularity of solutions.
  • Analysis of convergence and stability.
  • Applications to science and engineering.
  • Further equations in physics and applied mathematics.

Prof. Dr. Ahmet Bekir
Prof. Dr. Adem Cengiz Cevikel
Dr. Emad H.M. Zahran
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
  • fractional order derivatives
  • fractional dynamical systems
  • exact solutions
  • numerical analysis
  • soliton theory
  • traveling wave solutions
  • analytical methods
  • numerical methods
  • mathematical modeling

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  • Greater discoverability: Special Issues support the reach and impact of scientific research. Articles in Special Issues are more discoverable and cited more frequently.
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  • External promotion: Articles in Special Issues are often promoted through the journal's social media, increasing their visibility.
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Further information on MDPI's Special Issue polices can be found here.

Published Papers (7 papers)

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Research

24 pages, 12404 KiB  
Article
Inverse Scattering Integrability and Fractional Soliton Solutions of a Variable-Coefficient Fractional-Order KdV-Type Equation
by Sheng Zhang, Hongwei Li and Bo Xu
Fractal Fract. 2024, 8(9), 520; https://doi.org/10.3390/fractalfract8090520 - 31 Aug 2024
Viewed by 830
Abstract
In the field of nonlinear mathematical physics, Ablowitz et al.’s algorithm has recently made significant progress in the inverse scattering transform (IST) of fractional-order nonlinear evolution equations (fNLEEs). However, the solved fNLEEs are all constant-coefficient models. In this study, we establish a fractional-order [...] Read more.
In the field of nonlinear mathematical physics, Ablowitz et al.’s algorithm has recently made significant progress in the inverse scattering transform (IST) of fractional-order nonlinear evolution equations (fNLEEs). However, the solved fNLEEs are all constant-coefficient models. In this study, we establish a fractional-order KdV (fKdV)-type equation with variable coefficients and show that the IST is capable of solving the variable-coefficient fKdV (vcfKdV)-type equation. Firstly, according to Ablowitz et al.’s fractional-order algorithm and the anomalous dispersion relation, we derive the vcfKdV-type equation contained in a new class of integrable fNLEEs, which can be used to describe the dispersion transport in fractal media. Secondly, we reconstruct the potential function based on the time-dependent scattering data, and rewrite the explicit form of the vcfKdV-type equation using the completeness of eigenfunctions. Thirdly, under the assumption of reflectionless potential, we obtain an explicit expression for the fractional n-soliton solution of the vcfKdV-type equation. Finally, as specific examples, we study the spatial structures of the obtained fractional one- and two-soliton solutions. We find that the fractional soliton solutions and their linear, X-shaped, parabolic, sine/cosine, and semi-sine/semi-cosine trajectories formed on the coordinate plane have power–law dependence on discrete spectral parameters and are also affected by variable coefficients, which may have research value for the related hyperdispersion transport in fractional-order nonlinear media. Full article
(This article belongs to the Special Issue Recent Developments and Applications of Fractional Differential Equations in Mathematical Physics)
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18 pages, 750 KiB  
Article
Analysis of the(3+1)-Dimensional Fractional Kadomtsev–Petviashvili–Boussinesq Equation: Solitary, Bright, Singular, and Dark Solitons
by Aly R. Seadway, Asghar Ali, Ahmet Bekir and Adem C. Cevikel
Fractal Fract. 2024, 8(9), 515; https://doi.org/10.3390/fractalfract8090515 - 30 Aug 2024
Viewed by 703
Abstract
We looked at the (3+1)-dimensional fractional Kadomtsev–Petviashvili–Boussinesq (KP-B) equation, which comes up in fluid dynamics, plasma physics, physics, and superfluids, as well as when connecting the optical model and hydrodynamic domains. Furthermore, unlike the Kadomtsev–Petviashvili equation (KPE), which permits the modeling of waves [...] Read more.
We looked at the (3+1)-dimensional fractional Kadomtsev–Petviashvili–Boussinesq (KP-B) equation, which comes up in fluid dynamics, plasma physics, physics, and superfluids, as well as when connecting the optical model and hydrodynamic domains. Furthermore, unlike the Kadomtsev–Petviashvili equation (KPE), which permits the modeling of waves traveling in both directions, the zero-mass assumption, which is required for many scientific applications, is not required by the KP-B equation. In several applications in engineering and physics, taking these features into account allows researchers to acquire more precise conclusions, particularly in studies pertaining to the dynamics of water waves. The foremost purpose of this manuscript is to establish diverse solutions in the form of exponential, trigonometric, hyperbolic, and rational functions of the (3+1)-dimensional fractional (KP-B) via the application of four analytical methods. This KP-B model has fruitful applications in fluid dynamics and plasma physics. Additionally, in order to better explain the potential and physical behavior of the equation, the relevant models of the findings are visually indicated, and 2-dimensional (2D) and 3-dimensional (3D) graphics are drawn. Full article
(This article belongs to the Special Issue Recent Developments and Applications of Fractional Differential Equations in Mathematical Physics)
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21 pages, 1723 KiB  
Article
Exploring Solitons Solutions of a (3+1)-Dimensional Fractional mKdV-ZK Equation
by Amjad E. Hamza, Osman Osman, Muhammad Umair Sarwar, Khaled Aldwoah, Hicham Saber and Manel Hleili
Fractal Fract. 2024, 8(9), 498; https://doi.org/10.3390/fractalfract8090498 - 24 Aug 2024
Viewed by 688
Abstract
This study presents the application of the ϕ6 model expansion technique to find exact solutions for the (3+1)-dimensional space-time fractional modified KdV-Zakharov-Kuznetsov equation under Jumarie’s modified Riemann–Liouville derivative (JMRLD). The suggested method captures dark, periodic, traveling, and singular soliton solutions, providing deep [...] Read more.
This study presents the application of the ϕ6 model expansion technique to find exact solutions for the (3+1)-dimensional space-time fractional modified KdV-Zakharov-Kuznetsov equation under Jumarie’s modified Riemann–Liouville derivative (JMRLD). The suggested method captures dark, periodic, traveling, and singular soliton solutions, providing deep insights into wave behavior. Clear graphics demonstrate that the solutions are greatly affected by changes in the fractional order, deepening our understanding and revealing the hidden dynamics of wave propagation. The considered equation has several applications in fluid dynamics, plasma physics, and nonlinear optics. Full article
(This article belongs to the Special Issue Recent Developments and Applications of Fractional Differential Equations in Mathematical Physics)
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23 pages, 1448 KiB  
Article
Comparative Analysis of the Chaotic Behavior of a Five-Dimensional Fractional Hyperchaotic System with Constant and Variable Order
by Awatif Muflih Alqahtani, Arun Chaudhary, Ravi Shanker Dubey and Shivani Sharma
Fractal Fract. 2024, 8(7), 421; https://doi.org/10.3390/fractalfract8070421 - 18 Jul 2024
Viewed by 793
Abstract
A five-dimensional hyperchaotic system is a dynamical system with five state variables that exhibits chaotic behavior in multiple directions. In this work, we incorporated a 5D hyperchaotic system with constant- and variable-order Caputo and the Caputo–Fabrizio fractional derivatives. These fractional 5D hyperchaotic systems [...] Read more.
A five-dimensional hyperchaotic system is a dynamical system with five state variables that exhibits chaotic behavior in multiple directions. In this work, we incorporated a 5D hyperchaotic system with constant- and variable-order Caputo and the Caputo–Fabrizio fractional derivatives. These fractional 5D hyperchaotic systems are solved numerically. Through simulations, the chaotic behavior of these fractional-order hyperchaotic systems is analyzed and a comparison between constant- and variable-order fractional hyperchaotic systems is presented. Full article
(This article belongs to the Special Issue Recent Developments and Applications of Fractional Differential Equations in Mathematical Physics)
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22 pages, 3001 KiB  
Article
Distinctive Shape Functions of Fractional Differential Quadrature for Solving Two-Dimensional Space Fractional Diffusion Problems
by Abdelfattah Mustafa, Ola Ragb, Mohamed Salah, Reda S. Salama and Mokhtar Mohamed
Fractal Fract. 2023, 7(9), 668; https://doi.org/10.3390/fractalfract7090668 - 4 Sep 2023
Cited by 1 | Viewed by 1114
Abstract
The aim of this study is to utilize a differential quadrature method with various kernels, such as Lagrange interpolation and discrete singular convolution, to tackle problems related to the Riesz fractional diffusion equation and the Riesz fractional advection–dispersion equation. The governing equation for [...] Read more.
The aim of this study is to utilize a differential quadrature method with various kernels, such as Lagrange interpolation and discrete singular convolution, to tackle problems related to the Riesz fractional diffusion equation and the Riesz fractional advection–dispersion equation. The governing equation for convection and diffusion depends on both spatial and transient factors. By using the block marching technique, we transform these equations into an algebraic system using differential quadrature methods and the Caputo-type fractional operator. Next, we develop a MATLAB program that generates code capable of solving the fractional convection–diffusion equation in (1+2) dimensions for each shape function. Our goal is to ensure that our methods are reliable, accurate, efficient, and capable of convergence. To achieve this, we conduct two experiments, comparing the numerical and graphical results with both analytical and numerical solutions. Additionally, we evaluate the accuracy of our findings using the L∞ error. Our tests show that the differential quadrature method, which relies mainly on the discrete singular convolution shape function, is a highly effective numerical approach for fractional convective diffusion problems. It offers superior accuracy, faster convergence, and greater reliability than other techniques. Furthermore, we study the impact of fractional order derivatives, velocity, and positive diffusion parameters on the results. Full article
(This article belongs to the Special Issue Recent Developments and Applications of Fractional Differential Equations in Mathematical Physics)
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12 pages, 6591 KiB  
Article
A Numerical Solution of Generalized Caputo Fractional Initial Value Problems
by Rania Saadeh, Mohamed A. Abdoon, Ahmad Qazza and Mohammed Berir
Fractal Fract. 2023, 7(4), 332; https://doi.org/10.3390/fractalfract7040332 - 17 Apr 2023
Cited by 30 | Viewed by 2186
Abstract
In this article, the numerical adaptive predictor corrector (Apc-ABM) method is presented to solve generalized Caputo fractional initial value problems. The Apc-ABM method was utilized to establish approximate series solutions. The presented technique is considered to be an extension to the original Adams–Bashforth–Moulton [...] Read more.
In this article, the numerical adaptive predictor corrector (Apc-ABM) method is presented to solve generalized Caputo fractional initial value problems. The Apc-ABM method was utilized to establish approximate series solutions. The presented technique is considered to be an extension to the original Adams–Bashforth–Moulton approach. Numerical simulations and figures are presented and discussed, in order to show the efficiency of the proposed method. In the future, we anticipate that the provided generalized Caputo fractional derivative and the suggested method will be utilized to create and simulate a wide variety of generalized Caputo-type fractional models. We have included examples to demonstrate the accuracy of the present method. Full article
(This article belongs to the Special Issue Recent Developments and Applications of Fractional Differential Equations in Mathematical Physics)
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13 pages, 1596 KiB  
Article
Soliton Solutions of Fractional Stochastic Kraenkel–Manna–Merle Equations in Ferromagnetic Materials
by Wael W. Mohammed, M. El-Morshedy, Clemente Cesarano and Farah M. Al-Askar
Fractal Fract. 2023, 7(4), 328; https://doi.org/10.3390/fractalfract7040328 - 14 Apr 2023
Cited by 14 | Viewed by 1638
Abstract
In this study, we take into account the fractional stochastic Kraenkel–Manna–Merle system (FSKMMS). The mapping approach may be used to produce various type of stochastic fractional solutions, such as elliptic, hyperbolic, and trigonometric functions. Solutions to the Kraenkel–Manna–Merle system equation, which explains the [...] Read more.
In this study, we take into account the fractional stochastic Kraenkel–Manna–Merle system (FSKMMS). The mapping approach may be used to produce various type of stochastic fractional solutions, such as elliptic, hyperbolic, and trigonometric functions. Solutions to the Kraenkel–Manna–Merle system equation, which explains the propagation of a magnetic field in a zero-conductivity ferromagnet, may provide insight into a variety of fascinating scientific phenomena. Moreover, we construct a variety of 3D and 2D graphics in MATLAB to illustrate the influence of the stochastic term and the conformable derivative on the exact solutions of the FSKMMS. Full article
(This article belongs to the Special Issue Recent Developments and Applications of Fractional Differential Equations in Mathematical Physics)
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