Mathematical Models and Simulations

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Physics".

Deadline for manuscript submissions: closed (22 January 2024) | Viewed by 16490

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Department of Mathematics and Computer Science, University of Catania, Viale Andrea Doria 6, 95125 Catania, Italy
Interests: semiconductor modeling and simulations; kinetic models; numerical solutions of PDEs; Monte Carlo methods; optimization
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Dear Colleagues,

Mathematical models constitute a fundamental tool for the understanding of physical phenomena, biological systems, and finance and engineering. In addition to theoretical aspects, simulations play a primary role in applications, because they allow for the prediction of the behavior of quantities of interest.

The scope of this Special Issue is to collect papers in the field of mathematical physics, where different categories of mathematical models are presented both deterministic, i.e., based on ordinary or partial differential equations, and stochastic, i.e., defined by stochastic processes or based on stochastic differential equations. The study of mathematical aspects of the presented models is desirable. To provide realistic applications, the presence of numerical simulations is encouraged. Several numerical methods suited to the specific problem can be adopted, i.e., finite differences and finite volume schemes, finite elements, and discontinuous Galerkin and Monte Carlo methods. Usually, the simulations are performed by adopting real data for the parameters, and the models can also be optimized on datasets if available.

Dr. Giovanni Nastasi
Guest Editor

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Keywords

  • mathematical models
  • ordinary differential equations
  • partial differential equations
  • stochastic processes
  • stochastic differential equations
  • finite difference schemes
  • finite volume schemes
  • finite element method
  • discontinuous Galerkin method
  • Monte Carlo method

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Published Papers (14 papers)

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Editorial

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3 pages, 376 KiB  
Editorial
Mathematical Models and Simulations
by Giovanni Nastasi
Axioms 2024, 13(3), 149; https://doi.org/10.3390/axioms13030149 - 25 Feb 2024
Viewed by 853
Abstract
In this editorial, we present the Special Issue of the scientific journal Axioms entitled “Mathematical Models and Simulations” [...] Full article
(This article belongs to the Special Issue Mathematical Models and Simulations)

Research

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13 pages, 2748 KiB  
Article
Oscillatory Behavior of the Solutions for a Parkinson’s Disease Model with Discrete and Distributed Delays
by Chunhua Feng
Axioms 2024, 13(2), 75; https://doi.org/10.3390/axioms13020075 - 23 Jan 2024
Viewed by 833
Abstract
In this paper, the oscillatory behavior of the solutions for a Parkinson’s disease model with discrete and distributed delays is discussed. The distributed delay terms can be changed to new functions such that the original model is equivalent to a system in which [...] Read more.
In this paper, the oscillatory behavior of the solutions for a Parkinson’s disease model with discrete and distributed delays is discussed. The distributed delay terms can be changed to new functions such that the original model is equivalent to a system in which it only has discrete delays. Using Taylor’s expansion, the system can be linearized at the equilibrium to obtain both the linearized part and the nonlinearized part. One can see that the nonlinearized part is a disturbed term of the system. Therefore, the instability of the linearized system implies the instability of the whole system. If a system is unstable for a small delay, then the instability of this system will be maintained as the delay increased. By analyzing the linearized system at the smallest delay, some sufficient conditions to guarantee the existence of oscillatory solutions for a delayed Parkinson’s disease system can be obtained. It is found that under suitable conditions on the parameters, time delay affects the stability of the system. The present method does not need to consider a bifurcating equation. Some numerical simulations are provided to illustrate the theoretical result. Full article
(This article belongs to the Special Issue Mathematical Models and Simulations)
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13 pages, 4072 KiB  
Article
Analysis of the Burgers–Huxley Equation Using the Nondimensionalisation Technique: Universal Solution for Dirichlet and Symmetry Boundary Conditions
by Juan Francisco Sánchez-Pérez, Joaquín Solano-Ramírez, Enrique Castro, Manuel Conesa, Fulgencio Marín-García and Gonzalo García-Ros
Axioms 2023, 12(12), 1113; https://doi.org/10.3390/axioms12121113 - 11 Dec 2023
Viewed by 888
Abstract
The Burgers–Huxley equation is important because it involves the phenomena of accumulation, drag, diffusion, and the generation or decay of species, which are common in various problems in science and engineering, such as heat transmission, the diffusion of atmospheric contaminants, etc. On the [...] Read more.
The Burgers–Huxley equation is important because it involves the phenomena of accumulation, drag, diffusion, and the generation or decay of species, which are common in various problems in science and engineering, such as heat transmission, the diffusion of atmospheric contaminants, etc. On the other hand, the mathematical technique of nondimensionalisation has proven to be very useful in the appropriate grouping of the variables involved in a physical–chemical phenomenon and in obtaining universal solutions to different complex engineering problems. Therefore, a deep analysis using this technique of the Burgers–Huxley equation and its possible boundary conditions can facilitate a common understanding of these problems through the appropriate grouping of variables and propose common universal solutions. Thus, in this case, the technique is applied to obtain a universal solution for Dirichlet and symmetric boundary conditions. The validation of the methodology is carried out by comparing different cases, where the coefficients or the value of the boundary condition are varied, with the results obtained through a numerical simulation. Furthermore, one of the cases presented presents a boundary condition that changes at a certain time. Finally, after applying the technique, it is studied which phenomenon is predominant, concluding that from a certain value diffusion predominates, with the rest being practically negligible. Full article
(This article belongs to the Special Issue Mathematical Models and Simulations)
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34 pages, 3811 KiB  
Article
Dynamics Analysis of a Discrete-Time Commensalism Model with Additive Allee for the Host Species
by Yanbo Chong, Ankur Jyoti Kashyap, Shangming Chen and Fengde Chen
Axioms 2023, 12(11), 1031; https://doi.org/10.3390/axioms12111031 - 02 Nov 2023
Viewed by 891
Abstract
We propose and study a class of discrete-time commensalism systems with additive Allee effects on the host species. First, the single species with additive Allee effects is analyzed for existence and stability, then the existence of fixed points of discrete systems is given, [...] Read more.
We propose and study a class of discrete-time commensalism systems with additive Allee effects on the host species. First, the single species with additive Allee effects is analyzed for existence and stability, then the existence of fixed points of discrete systems is given, and the local stability of fixed points is given by characteristic root analysis. Second, we used the center manifold theorem and bifurcation theory to study the bifurcation of a codimension of one of the system at non-hyperbolic fixed points, including flip, transcritical, pitchfork, and fold bifurcations. Furthermore, this paper used the hybrid chaos method to control the chaos that occurs in the flip bifurcation of the system. Finally, the analysis conclusions were verified by numerical simulations. Compared with the continuous system, the similarities are that both species’ densities decrease with increasing Allee values under the weak Allee effect and that the host species hastens extinction under the strong Allee effect. Further, when the birth rate of the benefited species is low and the time is large enough, the benefited species will be locally asymptotically stabilized. Thus, our new finding is that both strong and weak Allee effects contribute to the stability of the benefited species under certain conditions. Full article
(This article belongs to the Special Issue Mathematical Models and Simulations)
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20 pages, 727 KiB  
Article
Modeling Environmental Pollution Using Varying-Coefficients Quantile Regression Models under Log-Symmetric Distributions
by Luis Sánchez, Germán Ibacache-Pulgar, Carolina Marchant and Marco Riquelme
Axioms 2023, 12(10), 976; https://doi.org/10.3390/axioms12100976 - 17 Oct 2023
Viewed by 842
Abstract
Many phenomena can be described by random variables that follow asymmetrical distributions. In the context of regression, when the response variable Y follows such a distribution, it is preferable to estimate the response variable for predictor values using the conditional median. Quantile regression [...] Read more.
Many phenomena can be described by random variables that follow asymmetrical distributions. In the context of regression, when the response variable Y follows such a distribution, it is preferable to estimate the response variable for predictor values using the conditional median. Quantile regression models can be employed for this purpose. However, traditional models do not incorporate a distributional assumption for the response variable. To introduce a distributional assumption while preserving model flexibility, we propose new varying-coefficients quantile regression models based on the family of log-symmetric distributions. We achieve this by reparametrizing the distribution of the response variable using quantiles. Parameter estimation is performed using a maximum likelihood penalized method, and a back-fitting algorithm is developed. Additionally, we propose diagnostic techniques to identify potentially influential local observations and leverage points. Finally, we apply and illustrate the methodology using real pollution data from Padre Las Casas city, one of the most polluted cities in Latin America and the Caribbean according to the World Air Quality Index Ranking. Full article
(This article belongs to the Special Issue Mathematical Models and Simulations)
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15 pages, 890 KiB  
Article
Mathematical Model of Cyber Risks Management Based on the Expansion of Piecewise Continuous Analytical Approximation Functions of Cyber Attacks in the Fourier Series
by Valentyn Sobchuk, Oleg Barabash, Andrii Musienko, Iryna Tsyganivska and Oleksandr Kurylko
Axioms 2023, 12(10), 924; https://doi.org/10.3390/axioms12100924 - 28 Sep 2023
Viewed by 670
Abstract
The comprehensive system of information security of an enterprise includes both tactical aspects of information and strategic priorities, reflecting the information policy and information strategy of the enterprise. Ensuring a given level of cybersecurity requires the identification of threat actors, their purpose, intentions [...] Read more.
The comprehensive system of information security of an enterprise includes both tactical aspects of information and strategic priorities, reflecting the information policy and information strategy of the enterprise. Ensuring a given level of cybersecurity requires the identification of threat actors, their purpose, intentions of attacks on the IT infrastructure, and weak points of the enterprise’s information security. To achieve these goals, enterprises need new information security solutions. In this work, a mathematical model of the process of cyber risk management in the enterprise, which is based on the distribution of piecewise continuous analytical approximating functions of cyber attacks in the Fourier series, is obtained. A constant continuous monitoring and conduction of cyber regulatory control of the enterprise on time makes it possible to effectively ensure the cybersecurity of the enterprise in real time—predicting the emergence of cyber threats to some extent—which, in turn, determines the management of cyber risks arising in the field of information security of the enterprise. Such a Fourier series expansion of the piecewise continuous analytical approximating function of the intensity of cyber attacks on damage to standard software, obtained by approximating empirical–statistical slices of the intensity of cyber attacks on damage to standard software for each time period by analytical functions, opens up new mathematical possibilities of transition to systems of regulatory control of cyber threats of the enterprise from discrete to continuous automated process for such types of control. Full article
(This article belongs to the Special Issue Mathematical Models and Simulations)
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23 pages, 362 KiB  
Article
Some Relations on the rRs(P,Q,z) Matrix Function
by Ayman Shehata, Ghazi S. Khammash and Carlo Cattani
Axioms 2023, 12(9), 817; https://doi.org/10.3390/axioms12090817 - 25 Aug 2023
Viewed by 634
Abstract
In this paper, we derive some classical and fractional properties of the rRs matrix function by using the Hilfer fractional operator. The theory of special matrix functions is the theory of those matrices that correspond to special matrix functions such as [...] Read more.
In this paper, we derive some classical and fractional properties of the rRs matrix function by using the Hilfer fractional operator. The theory of special matrix functions is the theory of those matrices that correspond to special matrix functions such as the gamma, beta, and Gauss hypergeometric matrix functions. We will also show the relationship with other generalized special matrix functions in the context of the Konhauser and Laguerre matrix polynomials. Full article
(This article belongs to the Special Issue Mathematical Models and Simulations)
11 pages, 2677 KiB  
Article
Effects of the Wiener Process and Beta Derivative on the Exact Solutions of the Kadomtsev–Petviashvili Equation
by Farah M. Al-Askar, Clemente Cesarano and Wael W. Mohammed
Axioms 2023, 12(8), 748; https://doi.org/10.3390/axioms12080748 - 29 Jul 2023
Cited by 4 | Viewed by 560
Abstract
We take into account the (2 + 1)-dimensional stochastic Kadomtsev–Petviashvili equation with beta-derivative (SKPE-BD) in this paper. To develop new hyperbolic, trigonometric, elliptic, and rational solutions, the Riccati equation and Jacobi elliptic function methods are employed. Because the KP equation is required for [...] Read more.
We take into account the (2 + 1)-dimensional stochastic Kadomtsev–Petviashvili equation with beta-derivative (SKPE-BD) in this paper. To develop new hyperbolic, trigonometric, elliptic, and rational solutions, the Riccati equation and Jacobi elliptic function methods are employed. Because the KP equation is required for explaining the development of quasi-one-dimensional shallow-water waves, the solutions obtained can be used to interpret various attractive physical phenomena. To display how the multiplicative white noise and beta-derivative impact the exact solutions of the SKPE-BD, we plot a few graphs in MATLAB and display different 3D and 2D figures. We deduce how multiplicative noise stabilizes the solutions of SKPE-BD at zero. Full article
(This article belongs to the Special Issue Mathematical Models and Simulations)
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20 pages, 1002 KiB  
Article
Study of Time-Fractional Nonlinear Model Governing Unsteady Flow of Polytropic Gas
by Brajesh K. Singh, Haci Mehmet Baskonus, Neetu Singh, Mukesh Gupta and D. G. Prakasha
Axioms 2023, 12(3), 285; https://doi.org/10.3390/axioms12030285 - 08 Mar 2023
Viewed by 867
Abstract
The present study is concerned with studying the dynamical behavior of two space-dimensional nonlinear time-fractional models governing the unsteady-flow of polytropic-gas (in brief, pGas) that occurred in cosmology and astronomy. For this purpose, two efficient hybrid methods so-called optimal homotopy analysis J-transform [...] Read more.
The present study is concerned with studying the dynamical behavior of two space-dimensional nonlinear time-fractional models governing the unsteady-flow of polytropic-gas (in brief, pGas) that occurred in cosmology and astronomy. For this purpose, two efficient hybrid methods so-called optimal homotopy analysis J-transform method (OHAJTM) and J-variational iteration transform method (J-VITM) have been adopted. The OHAJTM is the hybrid method, where optimal-homotopy analysis method (OHAM) is utilized after implementing the properties of J-transform (JT), and in J-VITM is the J-transform-based variational iteration method. Banach’s fixed point approach is adopted to analyze the convergence of these methods. It is demonstrated that J-VITM is T-stable, and the evaluated dynamics of pGas are described in terms of Mittag–Leffler functions. The proposed evaluation confirms that the implemented methods perform better for the referred model equation of pGas. In addition, for a given iteration, the proposed behavior via OHAJTM performs better in producing more accurate behavior in comparison to J-VITM and the methods introduced recently. Full article
(This article belongs to the Special Issue Mathematical Models and Simulations)
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11 pages, 581 KiB  
Article
Electrothermal Monte Carlo Simulation of a GaAs Resonant Tunneling Diode
by Orazio Muscato
Axioms 2023, 12(2), 216; https://doi.org/10.3390/axioms12020216 - 19 Feb 2023
Viewed by 1003
Abstract
This paper deals with the electron transport and heat generation in a Resonant Tunneling Diode semiconductor device. A new electrothermal Monte Carlo method is introduced. The method couples a Monte Carlo solver of the Boltzmann–Wigner transport equation with a steady-state solution of the [...] Read more.
This paper deals with the electron transport and heat generation in a Resonant Tunneling Diode semiconductor device. A new electrothermal Monte Carlo method is introduced. The method couples a Monte Carlo solver of the Boltzmann–Wigner transport equation with a steady-state solution of the heat diffusion equation. This methodology provides an accurate microscopic description of the spatial distribution of self-heating and its effect on the detailed nonequilibrium carrier dynamics. Full article
(This article belongs to the Special Issue Mathematical Models and Simulations)
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40 pages, 1240 KiB  
Article
Dynamic Behaviors of a COVID-19 and Influenza Co-Infection Model with Time Delays and Humoral Immunity
by Ahmed M. Elaiw, Raghad S. Alsulami and Aatef D. Hobiny
Axioms 2023, 12(2), 151; https://doi.org/10.3390/axioms12020151 - 01 Feb 2023
Cited by 1 | Viewed by 3447
Abstract
Co-infections with respiratory viruses were reported in hospitalized patients in several cases. Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) and influenza A virus (IAV) are two respiratory viruses and are similar in terms of their seasonal occurrence, clinical manifestations, transmission routes, and related [...] Read more.
Co-infections with respiratory viruses were reported in hospitalized patients in several cases. Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) and influenza A virus (IAV) are two respiratory viruses and are similar in terms of their seasonal occurrence, clinical manifestations, transmission routes, and related immune responses. SARS-CoV-2 is the cause of coronavirus disease 2019 (COVID-19). In this paper, we study the dynamic behaviors of an influenza and COVID-19 co-infection model in vivo. The role of humoral (antibody) immunity in controlling the co-infection is modeled. The model considers the interactions among uninfected epithelial cells (ECs), SARS-CoV-2-infected ECs, IAV-infected ECs, SARS-CoV-2 particles, IAV particles, SARS-CoV-2 antibodies, and IAV antibodies. The model is given by a system of delayed ordinary differential equations (DODEs), which include four time delays: (i) a delay in the SARS-CoV-2 infection of ECs, (ii) a delay in the IAV infection of ECs, (iii) a maturation delay of newly released SARS-CoV-2 virions, and (iv) a maturation delay of newly released IAV virions. We establish the non-negativity and boundedness of the solutions. We examine the existence and stability of all equilibria. The Lyapunov method is used to prove the global stability of all equilibria. The theoretical results are supported by performing numerical simulations. We discuss the effects of antiviral drugs and time delays on the dynamics of influenza and COVID-19 co-infection. It is noted that increasing the delay length has a similar influence to that of antiviral therapies in eradicating co-infection from the body. Full article
(This article belongs to the Special Issue Mathematical Models and Simulations)
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20 pages, 13296 KiB  
Article
Analysis of Finite Solution Spaces of Second-Order ODE with Dirac Delta Periodic Forcing
by Susmit Bagchi
Axioms 2023, 12(1), 85; https://doi.org/10.3390/axioms12010085 - 13 Jan 2023
Viewed by 1226
Abstract
Second-order Ordinary Differential Equations (ODEs) with discontinuous forcing have numerous applications in engineering and computational sciences. The analysis of the solution spaces of non-homogeneous ODEs is difficult due to the complexities in multidimensional systems, with multiple discontinuous variables present in forcing functions. Numerical [...] Read more.
Second-order Ordinary Differential Equations (ODEs) with discontinuous forcing have numerous applications in engineering and computational sciences. The analysis of the solution spaces of non-homogeneous ODEs is difficult due to the complexities in multidimensional systems, with multiple discontinuous variables present in forcing functions. Numerical solutions are often prone to failures in the presence of discontinuities. Algebraic decompositions are employed for analysis in such cases, assuming that regularities exist, operators are present in Banach (solution) spaces, and there is finite measurability. This paper proposes a generalized, finite-dimensional algebraic analysis of the solution spaces of second-order ODEs equipped with periodic Dirac delta forcing. The proposed algebraic analysis establishes the conditions for the convergence of responses within the solution spaces without requiring relative smoothness of the forcing functions. The Lipschitz regularizations and Lebesgue measurability are not considered as preconditions maintaining generality. The analysis shows that smooth and locally finite responses can be admitted in an exponentially stable solution space. The numerical analysis of the solution spaces is computed based on combinatorial changes in coefficients. It exhibits a set of locally uniform responses in the solution spaces. In contrast, the global response profiles show localized as well as oriented instabilities at specific neighborhoods in the solution spaces. Furthermore, the bands of the expansions–contractions of the stable response profiles are observable within the solution spaces depending upon the values of the coefficients and time intervals. The application aspects and distinguishing properties of the proposed approaches are outlined in brief. Full article
(This article belongs to the Special Issue Mathematical Models and Simulations)
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23 pages, 363 KiB  
Article
Boundary-Value Problem for Nonlinear Fractional Differential Equations of Variable Order with Finite Delay via Kuratowski Measure of Noncompactness
by Benoumran Telli, Mohammed Said Souid and Ivanka Stamova
Axioms 2023, 12(1), 80; https://doi.org/10.3390/axioms12010080 - 12 Jan 2023
Cited by 8 | Viewed by 1151
Abstract
This paper is devoted to boundary-value problems for Riemann–Liouville-type fractional differential equations of variable order involving finite delays. The existence of solutions is first studied using a Darbo’s fixed-point theorem and the Kuratowski measure of noncompactness. Secondly, the Ulam–Hyers stability criteria are examined. [...] Read more.
This paper is devoted to boundary-value problems for Riemann–Liouville-type fractional differential equations of variable order involving finite delays. The existence of solutions is first studied using a Darbo’s fixed-point theorem and the Kuratowski measure of noncompactness. Secondly, the Ulam–Hyers stability criteria are examined. All of the results in this study are established with the help of generalized intervals and piecewise constant functions. We convert the Riemann–Liouville fractional variable-order problem to equivalent standard Riemann–Liouville problems of fractional-constant orders. Finally, two examples are constructed to illustrate the validity of the observed results. Full article
(This article belongs to the Special Issue Mathematical Models and Simulations)
11 pages, 944 KiB  
Article
Analytical and Numerical Simulations of a Delay Model: The Pantograph Delay Equation
by Essam Roshdy El-Zahar and Abdelhalim Ebaid
Axioms 2022, 11(12), 741; https://doi.org/10.3390/axioms11120741 - 17 Dec 2022
Cited by 5 | Viewed by 1104
Abstract
In this paper, the pantograph delay differential equation y(t)=ay(t)+byct subject to the condition y(0)=λ is reanalyzed for the real constants a, b [...] Read more.
In this paper, the pantograph delay differential equation y(t)=ay(t)+byct subject to the condition y(0)=λ is reanalyzed for the real constants a, b, and c. In the literature, it has been shown that the pantograph delay differential equation, for λ=1, is well-posed if c<1, but not if c>1. In addition, the solution is available in the form of a standard power series when λ=1. In the present research, we are able to determine the solution of the pantograph delay differential equation in a closed series form in terms of exponential functions. The convergence of such a series is analysed. It is found that the solution converges for c(1,1) such that ba<1 and it also converges for c>1 when a<0. For c=1, the exact solution is obtained in terms of trigonometric functions, i.e., a periodic solution with periodicity 2πb2a2 when b>a. The current results are introduced for the first time and have not been reported in the relevant literature. Full article
(This article belongs to the Special Issue Mathematical Models and Simulations)
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