Mathematical Modelling of Real Phenomena Based on Fractional Derivatives

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

Deadline for manuscript submissions: closed (15 June 2023) | Viewed by 9166

Special Issue Editors

Department of Mathematics, Clarkson University, Potsdam, NY 13699, USA
Interests: fractional calculus; numerical methods; mathematical biology; mathematical modeling; dynamic systems; fractional dynamics; fractional differential equations; partial differential equations
1. Department of Computer Science and Mathematics, Lebanese American University, Beirut P.O. Box 13-5053, Lebanon
2. Department of Mathematics, Art and Science Faculty, Siirt University, Siirt 56100, Turkey
3. Department of Mathematics, Mathematics Research Center, Near East University, Near East Boulevard, Mersin 99138, Turkey
Interests: applied mathematics; fractional calculus; numerical methods
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional calculus is an extension of ordinary calculus, in a way where derivatives and integrals are defined for an arbitrary real order. In some phenomena, fractional operators allow for the modeling of better than ordinary derivatives and ordinary integrals, and can represent more efficient systems with high-order dynamics and complex nonlinear phenomena. This is due to two main reasons; first, we have the freedom to choose any order for the derivative and integral operators, and not be restricted to an integer-only order. Secondly, fractional-order derivatives depend not only on local conditions, but also on the past, which is useful when the system has a long-term memory.

This Special Issue is a place for researchers to share novel ideas regarding theories, applications, numerical and analytical methods, and simulations of fractional calculus and fractional differential equations. Topics of interest are defined below, and submissions relating to relevant fields are welcome.

  • Novel numerical and analytical techniques to solve fractional order problems;
  • Computational methods for solving fractional-order differential equations;
  • Fractional-based modeling, control, and analysis of real phenomena in the following;
  • Diseases-related fractional systems;
  • Optimal control for fractional models of HIV/AIDS, TB, and hepatitis infections;
  • Latest advancements in COVID-19 pandemic fractional systems;
  • Control and optimization for fractional systems;
  • Fractional inverse problems;
  • Stochastic processes involving fractional PDEs;
  • Fractional chaotic processes;
  • Fractional inverse problems: modeling and simulation;
  • Fluid dynamics and thermal engineering;
  • Biology, biophysics, and biomathematics;
  • Stochastic fractional dynamic systems;
  • Cancer dynamic fractional systems: optimality and modeling;
  • Stochastic analysis for fractional mathematical models;
  • Applications of fractional problems in science and engineering;
  • Implementation methods and simulations for fractional models;
  • Deterministic and stochastic fractional-order model systems;
  • Fractional engineering problems;
  • Nano applications of fractional modeling;
  • Control problems and model identifications with fractional operators;
  • Fractional diffusion models;
  • Fractional-order chaos-based cryptography applications;
  • Fractional calculus of fluid dynamics;
  • Biomechanical and biomedical applications of fractional calculus;
  • Nano-applications of fractional modeling;
  • Fractional-order financial models and systems;
  • Fractional-order economic models and systems;
  • Analysis and design of fractional-order controls.

Dr. Mohammad Partohaghighi
Prof. Dr. Ali Akgül
Guest Editors

Manuscript Submission Information

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Keywords

  • fractional calculus
  • fractional dynamical systems
  • fractional modeling
  • disease-related fractional problems
  • fractional ordinary equations
  • fractional partial differential equations

Published Papers (8 papers)

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Research

18 pages, 3353 KiB  
Article
Using Non-Standard Finite Difference Scheme to Study Classical and Fractional Order SEIVR Model
by Rahim Ud Din, Khalid Ali Khan, Ahmad Aloqaily, Nabil Mlaiki and Hussam Alrabaiah
Fractal Fract. 2023, 7(7), 552; https://doi.org/10.3390/fractalfract7070552 - 17 Jul 2023
Viewed by 805
Abstract
In this study, we considered a model for novel COVID-19 consisting on five classes, namely S, susceptible; E, exposed; I, infected; V, vaccinated; and R, recovered. We derived the expression for the basic reproductive rate R0 and [...] Read more.
In this study, we considered a model for novel COVID-19 consisting on five classes, namely S, susceptible; E, exposed; I, infected; V, vaccinated; and R, recovered. We derived the expression for the basic reproductive rate R0 and studied disease-free and endemic equilibrium as well as local and global stability. In addition, we extended the nonstandard finite difference scheme to simulate our model using some real data. Moreover, keeping in mind the importance of fractional order derivatives, we also attempted to extend our numerical results for the fractional order model. In this regard, we considered the proposed model under the concept of a fractional order derivative using the Caputo concept. We extended the nonstandard finite difference scheme for fractional order and simulated our results. Moreover, we also compared the numerical scheme with the traditional RK4 both in CPU time as well as graphically. Our results have close resemblance to those of the RK4 method. Also, in the case of the infected class, we compared our simulated results with the real data. Full article
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18 pages, 6370 KiB  
Article
Fractional Temperature-Dependent BEM for Laser Ultrasonic Thermoelastic Propagation Problems of Smart Nanomaterials
by Mohamed Abdelsabour Fahmy
Fractal Fract. 2023, 7(7), 536; https://doi.org/10.3390/fractalfract7070536 - 11 Jul 2023
Viewed by 638
Abstract
The major goal of this work is to present a novel fractional temperature-dependent boundary element model (BEM) for solving thermoelastic wave propagation problems in smart nanomaterials. The computing performance of the suggested methodology was demonstrated by using stable communication avoiding S-step—generalized minimal residual [...] Read more.
The major goal of this work is to present a novel fractional temperature-dependent boundary element model (BEM) for solving thermoelastic wave propagation problems in smart nanomaterials. The computing performance of the suggested methodology was demonstrated by using stable communication avoiding S-step—generalized minimal residual method (SCAS-GMRES) to solve discretized linear BEM systems. The benefits of SCAS-GMRES are investigated and compared to those of other iterative techniques. The numerical results are graphed to demonstrate the influence of fractional, piezoelectric, and length scale characteristics on total force-stresses. These findings also demonstrate that the BEM methodology is practical, feasible, effective, and has superiority over domain methods. The results of the present paper help to develop the industrial uses of smart nanomaterials. Full article
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13 pages, 2990 KiB  
Article
Characteristics of New Stochastic Solitonic Solutions for the Chiral Type of Nonlinear Schrödinger Equation
by H. G. Abdelwahed, A. F. Alsarhana, E. K. El-Shewy and Mahmoud A. E. Abdelrahman
Fractal Fract. 2023, 7(6), 461; https://doi.org/10.3390/fractalfract7060461 - 05 Jun 2023
Cited by 3 | Viewed by 926
Abstract
The Wiener process was used to explore the (2 + 1)-dimensional chiral nonlinear Schrödinger equation (CNLSE). This model outlines the energy characteristics of quantum physics’ fractional Hall effect edge states. The sine-Gordon expansion technique (SGET) was implemented to extract stochastic solutions for the [...] Read more.
The Wiener process was used to explore the (2 + 1)-dimensional chiral nonlinear Schrödinger equation (CNLSE). This model outlines the energy characteristics of quantum physics’ fractional Hall effect edge states. The sine-Gordon expansion technique (SGET) was implemented to extract stochastic solutions for the CNLSE through multiplicative noise effects. This method accurately described a variety of solitary behaviors, including bright solitons, dark periodic envelopes, solitonic forms, and dissipative and dissipative–soliton-like waves, showing how the solutions changed as the values of the studied system’s physical parameters were changed. The stochastic parameter was shown to affect the damping, growth, and conversion effects on the bright (dark) envelope and shock-forced oscillatory wave energy, amplitudes, and frequencies. In addition, the intensity of noise resulted in enormous periodic envelope stochastic structures and shock-forced oscillatory behaviors. The proposed technique is applicable to various energy equations in the nonlinear applied sciences. Full article
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13 pages, 702 KiB  
Article
Implementation of Analytical Techniques for the Solution of Nonlinear Fractional Order Sawada–Kotera–Ito Equation
by Rasool Shah, Fatemah Mofarreh, ElSayed M. Tag and Nivin A. Ghamry
Fractal Fract. 2023, 7(4), 299; https://doi.org/10.3390/fractalfract7040299 - 29 Mar 2023
Viewed by 836
Abstract
This article uses the Yang transform decomposition method and the homotopy perturbation transform method to study the seventh-order time-fractional Sawada–Kotera–Ito equation. The fractional derivative is taken into account in the Caputo sense. We used the Yang transform with the Adomian decomposition process and [...] Read more.
This article uses the Yang transform decomposition method and the homotopy perturbation transform method to study the seventh-order time-fractional Sawada–Kotera–Ito equation. The fractional derivative is taken into account in the Caputo sense. We used the Yang transform with the Adomian decomposition process and homotopy perturbation procedure on the time-fractional Sawada–Kotera–Ito problem to obtain the solution. We looked at a single case and contrasted it with the actual result to validate the methodologies. These techniques create recurrence relations representing the proposed problem’s solution. We then produced graphical representations that allowed us to visually check all of the outcomes in the proposed case for various fractional order values. The results of applying the current methodologies revealed strong connections to the precise resolution of the problem under investigation. The present study also illustrates error analysis. The numerical results obtained using the suggested techniques show that the methods are both simple and have excellent computational merit. Full article
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17 pages, 1004 KiB  
Article
The Fractional Analysis of a Nonlinear mKdV Equation with Caputo Operator
by Haifa A. Alyousef, Rasool Shah, Nehad Ali Shah, Jae Dong Chung, Sherif M. E. Ismaeel and Samir A. El-Tantawy
Fractal Fract. 2023, 7(3), 259; https://doi.org/10.3390/fractalfract7030259 - 14 Mar 2023
Cited by 5 | Viewed by 1065
Abstract
In this study, we aim to provide reliable methods for the initial value problem of the fractional modified Korteweg–de Vries (mKdV) equations. Fractional differential equations are essential for more precise simulation of numerous processes. The hybrid Yang transformation decomposition method (YTDM) and Yang [...] Read more.
In this study, we aim to provide reliable methods for the initial value problem of the fractional modified Korteweg–de Vries (mKdV) equations. Fractional differential equations are essential for more precise simulation of numerous processes. The hybrid Yang transformation decomposition method (YTDM) and Yang homotopy perturbation method (YHPM) are employed in a very simple and straightforward manner to handle the current problems. The derivative of fractional order is displayed in a Caputo form operator. To illustrate the conclusion given from the findings, a few numerical cases are taken into account for their approximate analytical solutions. We looked at two cases and contrasted them with the actual result to validate the methodologies. These techniques create recurrence relations representing the proposed problem’s solution. It is possible to find the series solutions to the given problems, and these solutions have components that converge to precise solutions more quickly. Tables and graphs are used to describe the new results, which demonstrate the present methods’ adequate accuracy. The actual and estimated outcomes are demonstrated in graphs and tables to be quite similar, demonstrating the usefulness of the proposed approaches. The innovation of the current work resides in the application of effective methods that require less calculation and achieve a greater level of accuracy. Additionally, the suggested approaches can be applied in the future to resolve other nonlinear fractional problems, which will be a scientific contribution to the research community. Full article
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14 pages, 4055 KiB  
Article
Scaled Conjugate Gradient for the Numerical Simulations of the Mathematical Model-Based Monkeypox Transmission
by Suthep Suantai, Zulqurnain Sabir, Muhammad Umar and Watcharaporn Cholamjiak
Fractal Fract. 2023, 7(1), 63; https://doi.org/10.3390/fractalfract7010063 - 05 Jan 2023
Cited by 3 | Viewed by 1323
Abstract
The current study presents the numerical solutions of a fractional order monkeypox virus model. The fractional order derivatives in the sense of Caputo are applied to achieve more realistic results for the nonlinear model. The dynamics of the monkeypox virus model are categorized [...] Read more.
The current study presents the numerical solutions of a fractional order monkeypox virus model. The fractional order derivatives in the sense of Caputo are applied to achieve more realistic results for the nonlinear model. The dynamics of the monkeypox virus model are categorized into eight classes, namely susceptible human, exposed human, infectious human, clinically ill human, recovered human, susceptible rodent, exposed rodent and infected rodent. Three different fractional order cases have been presented for the numerical solutions of the mathematical monkeypox virus model by applying the stochastic computing performances through the artificial intelligence-based scaled conjugate gradient neural networks. The statics for the system were selected as 83%, 10% and 7% for training, testing and validation, respectively. The exactness of the stochastic procedure is presented through the performances of the obtained results and the reference Adams results. The rationality and constancy are presented through the stochastic solutions together with simulations based on the state transition measures, regression, error histogram performances and correlation. Full article
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31 pages, 2299 KiB  
Article
Sensitive Demonstration of the Twin-Core Couplers including Kerr Law Non-Linearity via Beta Derivative Evolution
by Adeel Asad, Muhammad Bilal Riaz and Yanfeng Geng
Fractal Fract. 2022, 6(12), 697; https://doi.org/10.3390/fractalfract6120697 - 24 Nov 2022
Viewed by 1094
Abstract
To obtain new solitary wave solutions for non-linear directional couplers using optical meta-materials, a new extended direct algebraic technique (EDAT) is used. This model investigates solitary wave propagation inside a fiber. As a result, twin couplers are the subject of this study. Kerr [...] Read more.
To obtain new solitary wave solutions for non-linear directional couplers using optical meta-materials, a new extended direct algebraic technique (EDAT) is used. This model investigates solitary wave propagation inside a fiber. As a result, twin couplers are the subject of this study. Kerr law is the sort of non-linearity addressed there. Because it offers solutions to problems with large tails or infinite fluctuations, the resulting solution set is more generalized than the current solution because it is turned into a fractional-order derivative. Furthermore, the found solutions are fractional solitons with spatial–temporal fractional beta derivative evolution. In intensity-dependent switches, these nonlinear directional couplers also serve as limiters. Non-linearity alters the transmission constants of a system’s modes. The significance of the beta derivative parameter and mathematical approach is demonstrated graphically, with a few of the extracted solutions. A parametric analysis revealed that the fractional beta derivative parameter has a significant impact on the soliton amplitudes. With the aid of the advanced software tools for numerical computations, the categories of semi-dark solitons, singular dark-pitch solitons, single solitons of Type-1 along with 2, intermingled hyperbolically, trigonometric, and rational solitons were established and evaluated. We also discussed sensitivity analysis, which is an inquiry that determines how sensitive our system is. A comparative investigation via different fractional derivatives was also studied in this paper so that one can easily understand the correlation with other fractional derivatives. The findings demonstrate that the approach is simple and efficient and that it yields generalized analytical results. The findings will be extremely beneficial in examining and comprehending physical issues in nonlinear optics, specifically in twin-core couplers with optical metamaterials. Full article
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15 pages, 5541 KiB  
Article
A Theoretical and Numerical Study on Fractional Order Biological Models with Caputo Fabrizio Derivative
by Mati ur Rahman, Ali Althobaiti, Muhammad Bilal Riaz and Fuad S. Al-Duais
Fractal Fract. 2022, 6(8), 446; https://doi.org/10.3390/fractalfract6080446 - 17 Aug 2022
Cited by 10 | Viewed by 1201
Abstract
This article studies a biological population model in the context of a fractional Caputo-Fabrizio operator using double Laplace transform combined with the Adomian method. The conditions for the existence and uniqueness of solution of the problem under consideration is established with the use [...] Read more.
This article studies a biological population model in the context of a fractional Caputo-Fabrizio operator using double Laplace transform combined with the Adomian method. The conditions for the existence and uniqueness of solution of the problem under consideration is established with the use of the Banach principle and some theorems from fixed point theory. Furthermore, the convergence analysis is presented. For the accuracy and validation of the technique, some applications are presented. The numerical simulations present the obtained approximate solutions with a variety of fractional orders. From the numerical simulations, it is observed that when the fractional order is large, then the population density is also large; on the other hand, population density decreases with the decrease in the fractional order. The obtained results reveal that the considered technique is suitable and highly accurate in terms of the cost of computing, and can be used to analyze a wide range of complex non-linear fractional differential equations. Full article
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