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Fractal Fract
Special Issues
Application of Fractional-Calculus in Physical Systems
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Application of Fractional-Calculus in Physical Systems

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

Deadline for manuscript submissions: closed (10 April 2023) | Viewed by 10215

Special Issue Editors

Prof. Dr. Salah Mahmoud Boulaaras

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Guest Editor
Department of Mathematics, College of Sciences and Arts in ArRass, Qassim University, Buraydah 51452, Saudi Arabia
Interests: mathematical analysis; parabolic variational inequalities; Hamilton–Jacobi–Bellman equations; numerical methods for PDEs
Special Issues, Collections and Topics in MDPI journals
Dr. Viet-Thanh Pham

E-Mail Website
Guest Editor
Nonlinear Systems and Applications, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, Ho Chi Minh City, Vietnam
Interests: electronics; information and communication technology; electronic engineering; signal processing; electronics and communication;
Dr. Rashid Jan

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, China
Interests: fractional-calculus; mathematical modeling; numerical methods; mathematical biology; dynamical behaviour; chaos and bifurcation

Special Issue Information

Dear Colleagues:

It is well known that fractional calculus has numerous applications in engineering, science, and technology. The dynamics of challenging physical systems are closely connected to fractional calculus. Due to their non-local nature, fractional operators can more accurately and systematically represent a variety of natural phenomena. Fractional order differential equations may correctly control a wide variety of mathematical and physical models. It follows that the conclusions for the fractional mathematical model are more accurate and broader since the classical models are specific examples of the fractional order mathematical models. Fractional calculus also offers a number of techniques for resolving nonlinear models, integro-differential equations, and differential, integral, and integral-differential equations in mathematical physics. Fractional calculus on the complex plane has received a lot of attention in the last 10 years. The connection between fractional calculus and other mathematical and physical disciplines may open up new study directions and lead to new discoveries and applications. The purpose of this Special Issue is to bring together top academicians and researchers from a variety of engineering disciplines including applied mathematicians and physics, moreover, to provide them a forum to present their creative research. The fundamental focus of the articles includes theoretical, analytical, and numerical approaches with cutting-edge mathematical modeling and new advancements in differential and integral equations of arbitrary order originating in physical systems. Fractional calculus and its application for physical systems is a topic of extensive theoretical and analytical research around the world. Recent contributions to this essentially interdisciplinary field from theoretical, analytical, numerical, and computational perspectives are the focus of this Special Issue. This Special Issue collects original research work on recent developments in fractional calculus including:

  • Fractional calculus in physical systems;
  • Fractional differential equations;
  • Modeling and simulation;
  • Fractional dynamical system;
  • Fractional control theory;
  • Numerical methods;
  • Fractional calculus and chaos;
  • Non-locality and memory effects;
  • Non-locality in physical systems;
  • Modeling biological phenomena;
  • Non-locality in epidemic models;
  • Theoretical and computational analysis.

Prof. Dr. Salah Mahmoud Boulaaras
Dr. Viet-Thanh Pham
Dr. Rashid Jan
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
  • physical systems
  • mathematical modeling
  • theoretical and computational analysis
  • epidemic models
  • numerical analysis

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Published Papers (6 papers)

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Research

11 pages, 403 KiB  
Article
Fractional Model of Multiple Trapping with Charge Leakage: Transient Photoconductivity and Transit–Time Dispersion
by Fadila Serdouk, Abdelmalek Boumali and Renat T. Sibatov
Fractal Fract. 2023, 7(3), 243; https://doi.org/10.3390/fractalfract7030243 - 8 Mar 2023
Cited by 3 | Viewed by 1271
Abstract
The model of multiple trapping into energy-distributed states is a successful tool to describe the transport of nonequilibrium charge carriers in amorphous semiconductors. Under certain conditions, the model leads to anomalous diffusion equations that contain time fractional derivatives. From this perspective, the multiple-trapping [...] Read more.
The model of multiple trapping into energy-distributed states is a successful tool to describe the transport of nonequilibrium charge carriers in amorphous semiconductors. Under certain conditions, the model leads to anomalous diffusion equations that contain time fractional derivatives. From this perspective, the multiple-trapping model can be used to interpret fractional transport equations, formulate initial and boundary conditions for them, and to construct numerical methods for solving fractional kinetic equations. Here, we shortly review the application of fractional multiple-trapping equations to problems of transient photoconductivity relaxation and transit–time dispersion in the time-of-flight experiment and discuss the connection of the multiple-trapping model with generalized fractional kinetic equations. Different types of charge leakage are discussed. The tempered fractional relaxation is obtained for recombination via localized states and distributed order equations arise for the non-exponential density of states presented as a weighted mixture of exponential functions. Analytical solutions for photocurrent decay in transient photoconductivity and time-of-flight experiments are provided for several simplified situations. Full article
(This article belongs to the Special Issue Application of Fractional-Calculus in Physical Systems)
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11 pages, 1210 KiB  
Article
Using Laplace Residual Power Series Method in Solving Coupled Fractional Neutron Diffusion Equations with Delayed Neutrons System
by Mohammed Shqair, Ibrahim Ghabar and Aliaa Burqan
Fractal Fract. 2023, 7(3), 219; https://doi.org/10.3390/fractalfract7030219 - 27 Feb 2023
Cited by 8 | Viewed by 1610
Abstract
In this paper, a system of coupled fractional neutron diffusion equations with delayed neutrons was solved efficiently by using a combination of residual power series and Laplace transform techniques, and the anomalous diffusion was considered by taking the non-Gaussian case with different values [...] Read more.
In this paper, a system of coupled fractional neutron diffusion equations with delayed neutrons was solved efficiently by using a combination of residual power series and Laplace transform techniques, and the anomalous diffusion was considered by taking the non-Gaussian case with different values of fractional parameter α. The Laplace residual power series method (LRPSM) does not require differentiation, conversion, or discretization for the assumed conditions, so the approach is simple and suitable for solving higher-order fractional differential equations. To assure the theoretical results, two different neutron flux initial conditions were presented numerically, where the needed Mathematica codes were performed using essential nuclear reactor cross-section data, and the results for different values of times were tabulated and graphically figured out. Finally, it must be noted that the results align with the Adomian decomposition method. Full article
(This article belongs to the Special Issue Application of Fractional-Calculus in Physical Systems)
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15 pages, 763 KiB  
Article
Noise Spectral of GML Noise and GSR Behaviors for FGLE with Random Mass and Random Frequency
by Lini Qiu, Guitian He, Yun Peng, Hui Cheng and Yujie Tang
Fractal Fract. 2023, 7(2), 177; https://doi.org/10.3390/fractalfract7020177 - 10 Feb 2023
Cited by 3 | Viewed by 1238
Abstract
Due to the interest of anomalous diffusion phenomena and their application, our work has widely studied a fractional-order generalized Langevin Equation (FGLE) with a generalized Mittag–Leffler (GML) noise. Significantly, the spectral of GML noise involving three parameters is well addressed. Furthermore, the spectral [...] Read more.
Due to the interest of anomalous diffusion phenomena and their application, our work has widely studied a fractional-order generalized Langevin Equation (FGLE) with a generalized Mittag–Leffler (GML) noise. Significantly, the spectral of GML noise involving three parameters is well addressed. Furthermore, the spectral amplification (SPA) of an FGLE has also been investigated. The generalized stochastic resonance (GSR) phenomenon for FGLE only influenced by GML noise has been found. Furthermore, material GSR for FGLE influenced by two types of noise has been studied. Moreover, it is found that the GSR behaviors of the FGLE could also be induced by the fractional orders of the FGLE. Full article
(This article belongs to the Special Issue Application of Fractional-Calculus in Physical Systems)
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15 pages, 26448 KiB  
Article
A Robust Study of Tumor-Immune Cells Dynamics through Non-Integer Derivative
by Rashid Jan, Salah Boulaaras, Hussain Ahmad, Muhammad Jawad, Sulima Zubair and Mohamed Abdalla
Fractal Fract. 2023, 7(2), 164; https://doi.org/10.3390/fractalfract7020164 - 7 Feb 2023
Cited by 2 | Viewed by 1840
Abstract
It is renowned that the immune reaction in the tumour micro environment is a complex cellular process that requires additional research. Therefore, it is important to interrogate the tracking path behaviour of tumor-immune dynamics to alert policy makers about critical factors of the [...] Read more.
It is renowned that the immune reaction in the tumour micro environment is a complex cellular process that requires additional research. Therefore, it is important to interrogate the tracking path behaviour of tumor-immune dynamics to alert policy makers about critical factors of the system. Here, we use fractional derivative to structure tumor-immune interactions. Furthermore, in our research, we concentrated on the qualitative investigation and time series analysis of tumor-immune cell interactions. The solution routes are examined using a new numerical technique to emphasis the impact of the factors on tumor-immune system. We focused on the behaviour of the system with fluctuation of different values. The most crucial components of the proposed system are identified and policymakers are advised. The outcomes of the present study are the strong predictor of clinical success and the in-out of immune cells in a tumour is also critical to treatment efficacy. As a result, studying the behaviour of tumor-immune cell interactions is important to predict crucial factors for the prevention and management to the health officials. Full article
(This article belongs to the Special Issue Application of Fractional-Calculus in Physical Systems)
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18 pages, 556 KiB  
Article
Investigation of the Ripa Model via NHRS Scheme with Its Wide-Ranging Applications
by H. G. Abdelwahed, Mahmoud A. E. Abdelrahman, A. F. Alsarhana and Kamel Mohamed
Fractal Fract. 2022, 6(12), 745; https://doi.org/10.3390/fractalfract6120745 - 17 Dec 2022
Cited by 6 | Viewed by 1618
Abstract
This paper presents numerical modeling and investigation for the Ripa system. This model is derived from a shallow water model by merging the horizontal temperature gradients. We applied the non-homogeneous Riemann solver (NHRS) method for solving the Ripa model. This scheme contains two [...] Read more.
This paper presents numerical modeling and investigation for the Ripa system. This model is derived from a shallow water model by merging the horizontal temperature gradients. We applied the non-homogeneous Riemann solver (NHRS) method for solving the Ripa model. This scheme contains two stages named predictor and corrector. The first one is made up of a control parameter that is responsible for the numerical diffusion. The second one recuperates the balance conservation equation. One of the main features of the NHRS scheme, it can determine the numerical flux corresponding to the real state of solution in the non-attendance of Riemann solution. Various test cases of physical interest are considered. These case studies display the high resolution of the NHRS scheme and emphasize its ability to produce accurate results for the Ripa model. The presented solutions are very critical in superfluid applications of energy and many others. Finally, the NHRS technique can be used to solve a wide range of additional models in applied research. Full article
(This article belongs to the Special Issue Application of Fractional-Calculus in Physical Systems)
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14 pages, 604 KiB  
Article
A Comparative Study of Responses of Fractional Oscillator to Sinusoidal Excitation in the Weyl and Caputo Senses
by Jun-Sheng Duan, Yu-Jie Lan and Ming Li
Fractal Fract. 2022, 6(12), 692; https://doi.org/10.3390/fractalfract6120692 - 23 Nov 2022
Cited by 2 | Viewed by 1207
Abstract
The fractional oscillator equation with the sinusoidal excitation mx″(t)+bDtαx(t)+kx(t)=Fsin(ωt), [...] Read more.
The fractional oscillator equation with the sinusoidal excitation mx″(t)+bDtαx(t)+kx(t)=Fsin(ωt), m,b,k,ω>0 and 0<α<2 is comparatively considered for the Weyl fractional derivative and the Caputo fractional derivative. In the sense of Weyl, the fractional oscillator equation is solved to be a steady periodic oscillation xW(t). In the sense of Caputo, the fractional oscillator equation is solved and subjected to initial conditions. For the fractional case α∈(0,1)∪(1,2), the response to excitation, S(t), is a superposition of three parts: the steady periodic oscillation xW(t), an exponentially decaying oscillation and a monotone recovery term in negative power law. For the two responses to initial values, S0(t) and S1(t), either of them is a superposition of an exponentially decaying oscillation and a monotone recovery term in negative power law. The monotone recovery terms come from the Hankel integrals which make the fractional case different from the integer-order case. The asymptotic behaviors of the solutions removing the steady periodic response are given for the four cases of the initial values. The Weyl fractional derivative is suitable for a describing steady-state problem, and can directly lead to a steady periodic solution. The Caputo fractional derivative is applied to an initial value problem and the steady component of the solution is just the solution in the corresponding Weyl sense. Full article
(This article belongs to the Special Issue Application of Fractional-Calculus in Physical Systems)
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