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Fractal Fract
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Advances in Fractional Modeling and Computation
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Advances in Fractional Modeling and Computation

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: 22 December 2024 | Viewed by 18899

Special Issue Editors

Dr. Yuri Dimitrov

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Guest Editor
Department of Mathematics, Physics and Informatics, University of Forestry, 1756 Sofia, Bulgaria
Interests: fractional calculus; numerical methods for fractional differential equations; Monte-Carlo methods
Dr. Venelin Todorov

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Guest Editor
Department of Information Modeling, Institute of Mathematics and Informatics, 1113 Sofia, Bulgaria
Interests: applied mathematics; mathematical modeling; fractional calculus; numerical methods; stochastic and Monte Carlo methods
Dr. Slavi Georgiev

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Guest Editor
1. Department of Information Modeling, Institute of Mathematics and Informatics, 1113 Sofia, Bulgaria
2. Department of Applied Mathematics and Statistics, University of Ruse, 7017 Ruse, Bulgaria
Interests: mathematical modeling; fractional calculus; scientific computing; financial mathematics
Prof. Dr. Jordan Hristov

E-Mail Website1 Website2
Guest Editor
Department of Chemical Engineering, University of Chemical Technology and Metallurgy, 1756 Sofia, Bulgaria
Interests: mathematical modeling; fractional calculus; non-linear diffusion; viscoelasticity
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Fractional calculus is a branch of mathematics that deals with the study of fractional order derivatives. Today, fractional calculus has many applications in various fields, including physics, engineering, finance, and biology. It can be used to model complex systems that exhibit non-local or long-range interactions, as well as to solve differential equations involving fractional derivatives. Many models of complex systems which use ordinary and partial differential equations do not have analytic solutions. There is an urgent need to develop effective computational methods for solution and analysis of fractional models.

The focus of the Special Issue is the development and advancement of models using fractional differential equations and processes. We welcome original and review papers on theory, computational and Monte Carlo methods, and practical applications of fractional models in physics, chemistry, biology, engineering,  economics, probability, and statistics. Topics that are invited for submission include (but are not limited to):

  • Fractional models in natural sciences
  • Fractional models in economics and engineering
  • Numerical algorithms and discretization
  • Fractional differential systems with control theory
  • Fractional dynamical systems
  • Analysis of fractional models
  • Stochastic methods for fractional models
  • Monte Carlo methods
  • Markov chains and processes
  • Stochastic modeling and simulation
  • Related fractional models

Dr. Yuri Dimitrov
Dr. Venelin Todorov
Dr. Slavi Georgiev
Prof. Dr. Jordan Hristov
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 models in natural sciences
  • fractional models in economics and engineering
  • numerical algorithms and discretization
  • fractional differential systems with control theory
  • fractional dynamical systems
  • analysis of fractional models
  • stochastic methods for fractional models
  • Monte Carlo methods
  • Markov chains and processes
  • stochastic modeling and simulation
  • related fractional models

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Published Papers (16 papers)

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21 pages, 1998 KiB  
Article
Analysis of Infection and Diffusion Coefficient in an SIR Model by Using Generalized Fractional Derivative
by Ibtehal Alazman, Manvendra Narayan Mishra, Badr Saad Alkahtani and Ravi Shanker Dubey
Fractal Fract. 2024, 8(9), 537; https://doi.org/10.3390/fractalfract8090537 - 15 Sep 2024
Abstract
In this article, a diffusion component in an SIR model is introduced, and its impact is analyzed using fractional calculus. We have included the diffusion component in the SIR model. in order to illustrate the variations. Here, we have applied the general fractional [...] Read more.
In this article, a diffusion component in an SIR model is introduced, and its impact is analyzed using fractional calculus. We have included the diffusion component in the SIR model. in order to illustrate the variations. Here, we have applied the general fractional derivative to analyze the impact. The Laplace decomposition technique is employed to obtain the numerical outcomes of the model. In order to observe the effect of the diffusion component in the SIR model, graphical solutions are also displayed. Full article
(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)
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13 pages, 4488 KiB  
Article
Application of Fractional Calculus in Predicting the Temperature-Dependent Creep Behavior of Concrete
by Jiecheng Chen, Lingwei Gong and Ruifan Meng
Fractal Fract. 2024, 8(8), 482; https://doi.org/10.3390/fractalfract8080482 - 18 Aug 2024
Viewed by 332
Abstract
Creep is an essential aspect of the durability and longevity of concrete structures. Based on fractional-order viscoelastic theory, this study investigated a creep model for predicting the temperature-dependent creep behavior of concrete. The order of the proposed fractional-order creep model can intuitively reflect [...] Read more.
Creep is an essential aspect of the durability and longevity of concrete structures. Based on fractional-order viscoelastic theory, this study investigated a creep model for predicting the temperature-dependent creep behavior of concrete. The order of the proposed fractional-order creep model can intuitively reflect the evolution of the material characteristics between solids and fluids, which provides a quantitative way to directly reveal the influence of loading conditions on the temperature-dependent mechanical properties of concrete during creep. The effectiveness of the model was verified using the experimental data of lightweight expansive shale concrete under various temperature and stress conditions, and the comparison of the results with those of the model in the literature showed that the proposed model has good accuracy while maintaining simplicity. Further analysis of the fractional order showed that temperature, not stress level, is the key factor affecting the creep process of concrete. At the same temperature, the fractional order is almost a fixed value and increases with the increase in temperature, reflecting the gradual softening of the mechanical properties of concrete at higher temperature. Finally, a novel prediction formula containing the average fractional-order value at each temperature was established, and the creep deformation of concrete can be predicted only by changing the applied stress, which provides a simple and practical method for predicting the temperature-dependent creep behavior of concrete. Full article
(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)
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14 pages, 1328 KiB  
Article
A Compartmental Approach to Modeling the Measles Disease: A Fractional Order Optimal Control Model
by Amar Nath Chatterjee, Santosh Kumar Sharma and Fahad Al Basir
Fractal Fract. 2024, 8(8), 446; https://doi.org/10.3390/fractalfract8080446 - 30 Jul 2024
Viewed by 527
Abstract
Measles is the most infectious disease with a high basic reproduction number (R0). For measles, it is reported that R0 lies between 12 and 18 in an endemic situation. In this paper, a fractional order mathematical model for measles [...] Read more.
Measles is the most infectious disease with a high basic reproduction number (R0). For measles, it is reported that R0 lies between 12 and 18 in an endemic situation. In this paper, a fractional order mathematical model for measles disease is proposed to identify the dynamics of disease transmission following a declining memory process. In the proposed model, a fractional order differential operator is used to justify the effect and success rate of vaccination. The total population of the model is subdivided into five sub-compartments: susceptible (S), exposed (E), infected (I), vaccinated (V), and recovered (R). Here, we consider the first dose of measles vaccination and convert the model to a controlled system. Finally, we transform the control-induced model to an optimal control model using control theory. Both models are analyzed to find the stability of the system, the basic reproduction number, the optimal control input, and the adjoint equations with the boundary conditions. Also, the numerical simulation of the model is presented along with using the analytical findings. We also verify the effective role of the fractional order parameter alpha on the model dynamics and changes in the dynamical behavior of the model with R0=1. Full article
(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)
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11 pages, 1113 KiB  
Article
Fractional-Order Tabu Learning Neuron Models and Their Dynamics
by Yajuan Yu, Zhenhua Gu, Min Shi and Feng Wang
Fractal Fract. 2024, 8(7), 428; https://doi.org/10.3390/fractalfract8070428 - 20 Jul 2024
Viewed by 503
Abstract
In this paper, by replacing the exponential memory kernel function of a tabu learning single-neuron model with the power-law memory kernel function, a novel Caputo’s fractional-order tabu learning single-neuron model and a network of two interacting fractional-order tabu learning neurons are constructed firstly. [...] Read more.
In this paper, by replacing the exponential memory kernel function of a tabu learning single-neuron model with the power-law memory kernel function, a novel Caputo’s fractional-order tabu learning single-neuron model and a network of two interacting fractional-order tabu learning neurons are constructed firstly. Different from the integer-order tabu learning model, the order of the fractional-order derivative is used to measure the neuron’s memory decay rate and then the stabilities of the models are evaluated by the eigenvalues of the Jacobian matrix at the equilibrium point of the fractional-order models. By choosing the memory decay rate (or the order of the fractional-order derivative) as the bifurcation parameter, it is proved that Hopf bifurcation occurs in the fractional-order tabu learning single-neuron model where the value of bifurcation point in the fractional-order model is smaller than the integer-order model’s. By numerical simulations, it is shown that the fractional-order network with a lower memory decay rate is capable of producing tangent bifurcation as the learning rate increases from 0 to 0.4. When the learning rate is fixed and the memory decay increases, the fractional-order network enters into frequency synchronization firstly and then enters into amplitude synchronization. During the synchronization process, the oscillation frequency of the fractional-order tabu learning two-neuron network increases with an increase in the memory decay rate. This implies that the higher the memory decay rate of neurons, the higher the learning frequency will be. Full article
(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)
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31 pages, 1496 KiB  
Article
Performance Analysis of Fully Intuitionistic Fuzzy Multi-Objective Multi-Item Solid Fractional Transportation Model
by Sultan Almotairi, Elsayed Badr, M. A. Elsisy, F. A. Farahat and M. A. El Sayed
Fractal Fract. 2024, 8(7), 404; https://doi.org/10.3390/fractalfract8070404 - 9 Jul 2024
Cited by 1 | Viewed by 644
Abstract
An investigation is conducted in this paper into a performance analysis of fully intuitionistic fuzzy multi-objective multi-item solid fractional transport model (FIF-MMSFTM). It is to be anticipated that the parameters of the conveyance model will be imprecise by virtue of numerous uncontrollable factors. [...] Read more.
An investigation is conducted in this paper into a performance analysis of fully intuitionistic fuzzy multi-objective multi-item solid fractional transport model (FIF-MMSFTM). It is to be anticipated that the parameters of the conveyance model will be imprecise by virtue of numerous uncontrollable factors. The model under consideration incorporates intuitionistic fuzzy (IF) quantities of shipments, costs and profit coefficients, supplies, demands, and transport. The FIF-MMSFTM that has been devised is transformed into a linear form through a series of operations. The accuracy function and ordering relations of IF sets are then used to reduce the linearized model to a concise multi-objective multi-item solid transportation model (MMSTM). Furthermore, an examination is conducted on several theorems that illustrate the correlation between the FIF-MMSFTM and its corresponding crisp model, which is founded upon linear, hyperbolic, and parabolic membership functions. A numerical example was furnished to showcase the efficacy and feasibility of the suggested methodology. The numerical data acquired indicates that the linear, hyperbolic, and parabolic models require fewer computational resources to achieve the optimal solution. The parabolic model has the greatest number of iterations, in contrast to the hyperbolic model which has the fewest. Additionally, the elapsed run time for the three models is a negligible amount of time: 0.2, 0.15, and 1.37 s, respectively. In conclusion, suggestions for future research are provided. Full article
(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)
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15 pages, 323 KiB  
Article
Nonlocal Changing-Sign Perturbation Tempered Fractional Sub-Diffusion Model with Weak Singularity
by Xinguang Zhang, Jingsong Chen, Peng Chen, Lishuang Li and Yonghong Wu
Fractal Fract. 2024, 8(6), 337; https://doi.org/10.3390/fractalfract8060337 - 5 Jun 2024
Viewed by 664
Abstract
In this paper, we study the existence of positive solutions for a changing-sign perturbation tempered fractional model with weak singularity which arises from the sub-diffusion study of anomalous diffusion in Brownian motion. By two-step substitution, we first transform the higher-order sub-diffusion model to [...] Read more.
In this paper, we study the existence of positive solutions for a changing-sign perturbation tempered fractional model with weak singularity which arises from the sub-diffusion study of anomalous diffusion in Brownian motion. By two-step substitution, we first transform the higher-order sub-diffusion model to a lower-order mixed integro-differential sub-diffusion model, and then introduce a power factor to the non-negative Green function such that the linear integral operator has a positive infimum. This innovative technique is introduced for the first time in the literature and it is critical for controlling the influence of changing-sign perturbation. Finally, an a priori estimate and Schauder’s fixed point theorem are applied to show that the sub-diffusion model has at least one positive solution whether the perturbation is positive, negative or changing-sign, and also the main nonlinear term is allowed to have singularity for some space variables. Full article
(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)
21 pages, 1491 KiB  
Article
Unveiling the Complexity of HIV Transmission: Integrating Multi-Level Infections via Fractal-Fractional Analysis
by Yasir Nadeem Anjam, Rubayyi Turki Alqahtani, Nadiyah Hussain Alharthi and Saira Tabassum
Fractal Fract. 2024, 8(5), 299; https://doi.org/10.3390/fractalfract8050299 - 20 May 2024
Viewed by 759
Abstract
This article presents a non-linear deterministic mathematical model that captures the evolving dynamics of HIV disease spread, considering three levels of infection in a population. The model integrates fractal-fractional order derivatives using the Caputo operator and undergoes qualitative analysis to establish the existence [...] Read more.
This article presents a non-linear deterministic mathematical model that captures the evolving dynamics of HIV disease spread, considering three levels of infection in a population. The model integrates fractal-fractional order derivatives using the Caputo operator and undergoes qualitative analysis to establish the existence and uniqueness of solutions via fixed-point theory. Ulam-Hyer stability is confirmed through nonlinear functional analysis, accounting for small perturbations. Numerical solutions are obtained using the fractional Adam-Bashforth iterative scheme and corroborated through MATLAB simulations. The results, plotted across various fractional orders and fractal dimensions, are compared with integer orders, revealing trends towards HIV disease-free equilibrium points for infective and recovered populations. Meanwhile, susceptible individuals decrease towards this equilibrium state, indicating stability in HIV exposure. The study emphasizes the critical role of controlling transmission rates to mitigate fatalities, curb HIV transmission, and enhance recovery rates. This proposed strategy offers a competitive advantage, enhancing comprehension of the model’s intricate dynamics. Full article
(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)
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66 pages, 4599 KiB  
Article
Conformal and Non-Minimal Couplings in Fractional Cosmology
by Kevin Marroquín, Genly Leon, Alfredo D. Millano, Claudio Michea and Andronikos Paliathanasis
Fractal Fract. 2024, 8(5), 253; https://doi.org/10.3390/fractalfract8050253 - 25 Apr 2024
Cited by 1 | Viewed by 789
Abstract
Fractional differential calculus is a mathematical tool that has found applications in the study of social and physical behaviors considered “anomalous”. It is often used when traditional integer derivatives models fail to represent cases where the power law is observed accurately. Fractional calculus [...] Read more.
Fractional differential calculus is a mathematical tool that has found applications in the study of social and physical behaviors considered “anomalous”. It is often used when traditional integer derivatives models fail to represent cases where the power law is observed accurately. Fractional calculus must reflect non-local, frequency- and history-dependent properties of power-law phenomena. This tool has various important applications, such as fractional mass conservation, electrochemical analysis, groundwater flow problems, and fractional spatiotemporal diffusion equations. It can also be used in cosmology to explain late-time cosmic acceleration without the need for dark energy. We review some models using fractional differential equations. We look at the Einstein–Hilbert action, which is based on a fractional derivative action, and add a scalar field, ϕ, to create a non-minimal interaction theory with the coupling, ξRϕ2, between gravity and the scalar field, where ξ is the interaction constant. By employing various mathematical approaches, we can offer precise schemes to find analytical and numerical approximations of the solutions. Moreover, we comprehensively study the modified cosmological equations and analyze the solution space using the theory of dynamical systems and asymptotic expansion methods. This enables us to provide a qualitative description of cosmologies with a scalar field based on fractional calculus formalism. Full article
(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)
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34 pages, 495 KiB  
Article
Fundamental Matrix, Integral Representation and Stability Analysis of the Solutions of Neutral Fractional Systems with Derivatives in the Riemann—Liouville Sense
by Hristo Kiskinov, Mariyan Milev, Slav Ivanov Cholakov and Andrey Zahariev
Fractal Fract. 2024, 8(4), 195; https://doi.org/10.3390/fractalfract8040195 - 28 Mar 2024
Cited by 1 | Viewed by 907
Abstract
The paper studies a class of nonlinear disturbed neutral linear fractional systems with derivatives in the the Riemann–Liouville sense and distributed delays. First, it is proved that the initial problem for these systems with discontinuous initial functions under some natural assumptions possesses a [...] Read more.
The paper studies a class of nonlinear disturbed neutral linear fractional systems with derivatives in the the Riemann–Liouville sense and distributed delays. First, it is proved that the initial problem for these systems with discontinuous initial functions under some natural assumptions possesses a unique solution. The assumptions used for the proof are similar to those used in the case of systems with first-order derivatives. Then, with the obtained result, we derive the existence and uniqueness of a fundamental matrix and a generalized fundamental matrix for the homogeneous system. In the linear case, via these fundamental matrices we obtain integral representations of the solutions of the homogeneous system and the corresponding inhomogeneous system. Furthermore, for the fractional systems with Riemann–Liouville derivatives we introduce a new concept for weighted stabilities in the Lyapunov, Ulam–Hyers, and Ulam–Hyers–Rassias senses, which coincides with the classical stability concepts for the cases of integer-order or Caputo-type derivatives. It is proved that the zero solution of the homogeneous system is weighted stable if and only if all its solutions are weighted bounded. In addition, for the homogeneous system it is established that the weighted stability in the Lyapunov and Ulam–Hyers senses are equivalent if and only if the inequality appearing in the Ulam–Hyers definition possess only bounded solutions. Finally, we derive natural sufficient conditions under which the property of weighted global asymptotic stability of the zero solution of the homogeneous system is preserved under nonlinear disturbances. Full article
(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)
11 pages, 368 KiB  
Article
Some Results on Fractional Boundary Value Problem for Caputo-Hadamard Fractional Impulsive Integro Differential Equations
by Ymnah Alruwaily, Kuppusamy Venkatachalam and El-sayed El-hady
Fractal Fract. 2023, 7(12), 884; https://doi.org/10.3390/fractalfract7120884 - 14 Dec 2023
Cited by 1 | Viewed by 1244
Abstract
The results for a new modeling integral boundary value problem (IBVP) using Caputo-Hadamard impulsive fractional integro-differential equations (C-HIFI-DE) with Banach space are investigated, along with the existence and uniqueness of solutions. The Krasnoselskii fixed-point theorem (KFPT) and the Banach contraction principle (BCP) serve [...] Read more.
The results for a new modeling integral boundary value problem (IBVP) using Caputo-Hadamard impulsive fractional integro-differential equations (C-HIFI-DE) with Banach space are investigated, along with the existence and uniqueness of solutions. The Krasnoselskii fixed-point theorem (KFPT) and the Banach contraction principle (BCP) serve as the basis of this unique strategy, and are used to achieve the desired results. We develop the illustrated examples at the end of the paper to support the validity of the theoretical statements. Full article
(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)
26 pages, 787 KiB  
Article
Approximation of Caputo Fractional Derivative and Numerical Solutions of Fractional Differential Equations
by Yuri Dimitrov, Slavi Georgiev and Venelin Todorov
Fractal Fract. 2023, 7(10), 750; https://doi.org/10.3390/fractalfract7100750 - 11 Oct 2023
Cited by 2 | Viewed by 2521
Abstract
In this paper, we consider an approximation of the Caputo fractional derivative and its asymptotic expansion formula, whose generating function is the polylogarithm function. We prove the convergence of the approximation and derive an estimate for the error and order. The approximation is [...] Read more.
In this paper, we consider an approximation of the Caputo fractional derivative and its asymptotic expansion formula, whose generating function is the polylogarithm function. We prove the convergence of the approximation and derive an estimate for the error and order. The approximation is applied for the construction of finite difference schemes for the two-term ordinary fractional differential equation and the time fractional Black–Scholes equation for option pricing. The properties of the approximation are used to prove the convergence and order of the finite difference schemes and to obtain bounds for the error of the numerical methods. The theoretical results for the order and error of the methods are illustrated by the results of the numerical experiments. Full article
(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)
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18 pages, 468 KiB  
Article
On Stability of a Fractional Discrete Reaction–Diffusion Epidemic Model
by Omar Alsayyed, Amel Hioual, Gharib M. Gharib, Mayada Abualhomos, Hassan Al-Tarawneh, Maha S. Alsauodi, Nabeela Abu-Alkishik, Abdallah Al-Husban and Adel Ouannas
Fractal Fract. 2023, 7(10), 729; https://doi.org/10.3390/fractalfract7100729 - 2 Oct 2023
Cited by 4 | Viewed by 1223
Abstract
This paper considers the dynamical properties of a space and time discrete fractional reaction–diffusion epidemic model, introducing a novel generalized incidence rate. The linear stability of the equilibrium solutions of the considered discrete fractional reaction–diffusion model has been carried out, and a global [...] Read more.
This paper considers the dynamical properties of a space and time discrete fractional reaction–diffusion epidemic model, introducing a novel generalized incidence rate. The linear stability of the equilibrium solutions of the considered discrete fractional reaction–diffusion model has been carried out, and a global asymptotic stability analysis has been undertaken. We conducted a global stability analysis using a specialized Lyapunov function that captures the system’s historical data, distinguishing it from the integer-order version. This approach significantly advanced our comprehension of the complex stability properties within discrete fractional reaction–diffusion epidemic models. To substantiate the theoretical underpinnings, this paper is accompanied by numerical examples. These examples serve a dual purpose: not only do they validate the theoretical findings, but they also provide illustrations of the practical implications of the proposed discrete fractional system. Full article
(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)
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14 pages, 325 KiB  
Article
A Study on Generalized Degenerate Form of 2D Appell Polynomials via Fractional Operators
by Mohra Zayed and Shahid Ahmad Wani
Fractal Fract. 2023, 7(10), 723; https://doi.org/10.3390/fractalfract7100723 - 30 Sep 2023
Cited by 6 | Viewed by 800
Abstract
This paper investigates the significance of generating expressions, operational principles, and defining characteristics in the study and development of special polynomials. The focus is on a novel generalized family of degenerate 2D Appell polynomials, which were defined using a fractional operator. Motivated by [...] Read more.
This paper investigates the significance of generating expressions, operational principles, and defining characteristics in the study and development of special polynomials. The focus is on a novel generalized family of degenerate 2D Appell polynomials, which were defined using a fractional operator. Motivated by inquiries into degenerate 2D bivariate Appell polynomials, this research reveals that well-known 2D Appell polynomials and simple Appell polynomials can be regarded as specific instances within this new family for certain values. This paper presents the operational rule, generating relation, determinant form, and recurrence relations for this generalized family. Furthermore, it explores the practical applications of these degenerate 2D Appell polynomials and establishes their connections with equivalent results for the generalized family of degenerate 2D Bernoulli, Euler, and Genocchi polynomials. Full article
(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)
20 pages, 4222 KiB  
Article
A Physical Phenomenon for the Fractional Nonlinear Mixed Integro-Differential Equation Using a Quadrature Nystrom Method
by A. R. Jan, M. A. Abdou and M. Basseem
Fractal Fract. 2023, 7(9), 656; https://doi.org/10.3390/fractalfract7090656 - 31 Aug 2023
Cited by 1 | Viewed by 960
Abstract
In this work, the existence and uniqueness solution of the fractional nonlinear mixed integro-differential equation (FrNMIoDE) is guaranteed with a general discontinuous kernel based on position and time-space [...] Read more.
In this work, the existence and uniqueness solution of the fractional nonlinear mixed integro-differential equation (FrNMIoDE) is guaranteed with a general discontinuous kernel based on position and time-space  L2Ω×C0,T, T<1. The FrNMIoDE conformed to the Volterra-Hammerstein integral equation (V-HIE) of the second kind, after applying the characteristics of a fractional integral, with a general discontinuous kernel in position for the Hammerstein integral term and a continuous kernel in time to the Volterra integral (VI) term. Then, using a separation technique methodology, we developed HIE, whose physical coefficients were time-variable. By examining the system’s convergence, the product Nystrom technique (PNT) and associated schemes were employed to create a nonlinear algebraic system (NAS). Full article
(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)
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16 pages, 2844 KiB  
Article
Dynamical Behaviour, Control, and Boundedness of a Fractional-Order Chaotic System
by Lei Ren, Sami Muhsen, Stanford Shateyi and Hassan Saberi-Nik
Fractal Fract. 2023, 7(7), 492; https://doi.org/10.3390/fractalfract7070492 - 21 Jun 2023
Cited by 4 | Viewed by 1349
Abstract
In this paper, the fractional-order chaotic system form of a four-dimensional system with cross-product nonlinearities is introduced. The stability of the equilibrium points of the system and then the feedback control design to achieve this goal have been analyzed. Furthermore, further dynamical behaviors [...] Read more.
In this paper, the fractional-order chaotic system form of a four-dimensional system with cross-product nonlinearities is introduced. The stability of the equilibrium points of the system and then the feedback control design to achieve this goal have been analyzed. Furthermore, further dynamical behaviors including, phase portraits, bifurcation diagrams, and the largest Lyapunov exponent are presented. Finally, the global Mittag–Leffler attractive sets (MLASs) and Mittag–Leffler positive invariant sets (MLPISs) of the considered fractional order system are presented. Numerical simulations are provided to show the effectiveness of the results. Full article
(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)
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32 pages, 11512 KiB  
Article
A Hybrid Local Radial Basis Function Method for the Numerical Modeling of Mixed Diffusion and Wave-Diffusion Equations of Fractional Order Using Caputo’s Derivatives
by Raheel Kamal, Kamran, Saleh M. Alzahrani and Talal Alzahrani
Fractal Fract. 2023, 7(5), 381; https://doi.org/10.3390/fractalfract7050381 - 1 May 2023
Cited by 1 | Viewed by 1754
Abstract
This article presents an efficient method for the numerical modeling of time fractional mixed diffusion and wave-diffusion equations with two Caputo derivatives of order 0<α<1, and 1<β<2. The numerical method is based on [...] Read more.
This article presents an efficient method for the numerical modeling of time fractional mixed diffusion and wave-diffusion equations with two Caputo derivatives of order 0<α<1, and 1<β<2. The numerical method is based on the Laplace transform technique combined with local radial basis functions. The method consists of three main steps: (i) first, the Laplace transform is used to transform the given time fractional model into an equivalent time-independent inhomogeneous problem in the frequency domain; (ii) in the second step, the local radial basis functions method is utilized to obtain an approximate solution for the reduced problem; (iii) finally, the Stehfest method is employed to convert the obtained solution from the frequency domain back to the time domain. The use of the Laplace transform eliminates the need for classical time-stepping techniques, which often require very small time steps to achieve accuracy. Additionally, the application of local radial basis functions helps overcome issues related to ill-conditioning and sensitivity to shape parameters typically encountered in global radial basis function methods. To validate the efficiency and accuracy of the proposed method, several test problems in regular and irregular domains with uniform and non-uniform nodes are considered. Full article
(This article belongs to the Special Issue Advances in Fractional Modeling and Computation)
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