Topic Editors

School of International Education, Neijiang Normal University, Neijiang 641004, China
Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 702 00 Ostrava, Czech Republic

Mathematical Modeling

Abstract submission deadline
closed (31 March 2024)
Manuscript submission deadline
31 May 2024
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38483

Topic Information

Dear Colleagues,

Mathematical modeling is the basis of technology, design, and control. Mathematical modeling can be a theory or a database. Theory and data involve each other to better understand physical or metaphysical worlds. We seek high-quality studies on new and innovative approaches, written to be simple and readable. We welcome submissions presenting new theoretical results, structural investigations, new models, algorithmic approaches, and new applications of mathematical physics problems. New scientific research works in the fields of epidemiology, biology, economy, ecology, medicine, physiology, sociology, chemistry, physics, and electronic and mechanical engineering, as well as civil engineering, are welcome.

Dr. Babak Shiri
Dr. Zahra Alijani
Topic Editors

Keywords

  • factional theory
  • neural network
  • fuzzy theory
  • dynamical systems
  • epidemiological model
  • differential equations
  • integral equations
  • partial differential equations
  • algebraic equations
  • boundary value problems and inverse problems

Participating Journals

Journal Name Impact Factor CiteScore Launched Year First Decision (median) APC
Axioms
axioms
2.0 - 2012 21.8 Days CHF 2400 Submit
Computation
computation
2.2 3.3 2013 18 Days CHF 1800 Submit
Mathematical and Computational Applications
mca
1.9 - 1996 22.5 Days CHF 1400 Submit
Mathematics
mathematics
2.4 3.5 2013 16.9 Days CHF 2600 Submit
Symmetry
symmetry
2.7 4.9 2009 16.2 Days CHF 2400 Submit

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Published Papers (29 papers)

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24 pages, 8167 KiB  
Article
A Robust FOPD Controller That Allows Faster Detection of Defects for Touch Panels
by Yuan-Jay Wang
Math. Comput. Appl. 2024, 29(2), 29; https://doi.org/10.3390/mca29020029 - 16 Apr 2024
Viewed by 245
Abstract
This study aims to synthesize and implement a robust fractional order PD (RFOPD) controller to increase the speed at which defects in automated touch panel inspection systems (ATPISs) are detected. A three-dimensional orthogonal stage (TDOS) driven by BLDC servo motors moves the inspection [...] Read more.
This study aims to synthesize and implement a robust fractional order PD (RFOPD) controller to increase the speed at which defects in automated touch panel inspection systems (ATPISs) are detected. A three-dimensional orthogonal stage (TDOS) driven by BLDC servo motors moves the inspection pen (IP) vertically and horizontally. The dynamic equation relating the BLDC servo motor input to the tip motion is established. A touch position identification (TPI) system is used to locate the touch point rapidly. An RFOPD controller is used to actuate the BLDC servo motors and move the TDOS rapidly and accurately in three dimensions. This method displaces the IP to any specified position and shows user-defined inspection trajectories on the touch screens. The gain-phase margin tester (GPMT) and stability equation methods are exploited to schedule the RFOPD controller gain settings and to maintain the specific safety margins for the controlled system. The simulation studies show that the proposed RFOPD controller exhibits better tracking and disturbance rejection responses than a conventional PID controller. The robustness of the RFOPD-controlled ATPIS, considering unmodeled uncertainties and friction-induced disturbances, is verified through simulation and experimental studies. Several user-defined inspection patterns are used to verify performance, and the experimental results show that the proposed RFOPD controller is effective. Full article
(This article belongs to the Topic Mathematical Modeling)
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16 pages, 268 KiB  
Article
Existence and Hyers–Ulam Stability for Random Impulsive Stochastic Pantograph Equations with the Caputo Fractional Derivative
by Dongdong Gao and Jianli Li
Mathematics 2024, 12(8), 1145; https://doi.org/10.3390/math12081145 - 10 Apr 2024
Viewed by 277
Abstract
In this paper, we study the existence, uniqueness and Hyers–Ulam stability of a class of fractional stochastic pantograph equations with random impulses. Firstly, we establish sufficient conditions to ensure the existence of solutions for the considered equations by applying Schaefer’s fixed point theorem [...] Read more.
In this paper, we study the existence, uniqueness and Hyers–Ulam stability of a class of fractional stochastic pantograph equations with random impulses. Firstly, we establish sufficient conditions to ensure the existence of solutions for the considered equations by applying Schaefer’s fixed point theorem under relaxed linear growth conditions. Secondly, we prove the solution for the considered equations is Hyers–Ulam stable via Gronwall’s inequality. Moreover, the previous literature will be significantly generalized in our paper. Finally, an example is given to explain the efficiency of the obtained results. Full article
(This article belongs to the Topic Mathematical Modeling)
25 pages, 5302 KiB  
Article
SSA-Deep Learning Forecasting Methodology with SMA and KF Filters and Residual Analysis
by Juan Frausto-Solís, José Christian de Jesús Galicia-González, Juan Javier González-Barbosa, Guadalupe Castilla-Valdez and Juan Paulo Sánchez-Hernández
Math. Comput. Appl. 2024, 29(2), 19; https://doi.org/10.3390/mca29020019 - 05 Mar 2024
Viewed by 1268
Abstract
Accurate forecasting remains a challenge, even with advanced techniques like deep learning (DL), ARIMA, and Holt–Winters (H&W), particularly for chaotic phenomena such as those observed in several areas, such as COVID-19, energy, and financial time series. Addressing this, we introduce a Forecasting Method [...] Read more.
Accurate forecasting remains a challenge, even with advanced techniques like deep learning (DL), ARIMA, and Holt–Winters (H&W), particularly for chaotic phenomena such as those observed in several areas, such as COVID-19, energy, and financial time series. Addressing this, we introduce a Forecasting Method with Filters and Residual Analysis (FMFRA), a hybrid methodology specifically applied to datasets of COVID-19 time series, which we selected for their complexity and exemplification of current forecasting challenges. FMFFRA consists of the following two approaches: FMFRA-DL, employing deep learning, and FMFRA-SSA, using singular spectrum analysis. This proposed method applies the following three phases: filtering, forecasting, and residual analysis. Initially, each time series is split into filtered and residual components. The second phase involves a simple fine-tuning for the filtered time series, while the third phase refines the forecasts and mitigates noise. FMFRA-DL is adept at forecasting complex series by distinguishing primary trends from insufficient relevant information. FMFRA-SSA is effective in data-scarce scenarios, enhancing forecasts through automated parameter search and residual analysis. Chosen for their geographical and substantial populations and chaotic dynamics, time series for Mexico, the United States, Colombia, and Brazil permitted a comparative perspective. FMFRA demonstrates its efficacy by improving the common forecasting performance measures of MAPE by 22.91%, DA by 13.19%, and RMSE by 25.24% compared to the second-best method, showcasing its potential for providing essential insights into various rapidly evolving domains. Full article
(This article belongs to the Topic Mathematical Modeling)
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28 pages, 407 KiB  
Article
Interior Multi-Peak Solutions for a Slightly Subcritical Nonlinear Neumann Equation
by Mohamed Ben Ayed, Khalil El Mehdi and Fatimetou Mohamed Salem
Symmetry 2024, 16(3), 291; https://doi.org/10.3390/sym16030291 - 02 Mar 2024
Viewed by 571
Abstract
In this paper, we consider the nonlinear Neumann problem (Pε): Δu+V(x)u=K(x)u(n+2)/(n2)ε, [...] Read more.
In this paper, we consider the nonlinear Neumann problem (Pε): Δu+V(x)u=K(x)u(n+2)/(n2)ε, u>0 in Ω, u/ν=0 on Ω, where Ω is a smooth bounded domain in Rn, n4, ε is a small positive real, and V and K are non-constant smooth positive functions on Ω¯. First, we study the asymptotic behavior of solutions for (Pε) which blow up at interior points as ε moves towards zero. In particular, we give the precise location of blow-up points and blow-up rates. This description of the interior blow-up picture of solutions shows that, in contrast to a case where K1, problem (Pε) has no interior bubbling solutions with clustered bubbles. Second, we construct simple interior multi-peak solutions for (Pε) which allow us to provide multiplicity results for (Pε). The strategy of our proofs consists of testing the equation with vector fields which make it possible to obtain balancing conditions which are satisfied by the concentration parameters. Thanks to a careful analysis of these balancing conditions, we were able to obtain our results. Our results are proved without any assumptions of the symmetry or periodicity of the function K. Furthermore, no assumption of the symmetry of the domain is needed. Full article
(This article belongs to the Topic Mathematical Modeling)
11 pages, 2567 KiB  
Article
Breaking a Combinatorial Symmetry Resolves the Paradox of Einstein-Podolsky-Rosen and Bell
by Jürgen Jakumeit and Karl Hess
Symmetry 2024, 16(3), 255; https://doi.org/10.3390/sym16030255 - 20 Feb 2024
Viewed by 594
Abstract
We present a Monte Carlo model of Einstein–Podolsky–Rosen experiments that may be implemented on two independent computers and resembles the measurements of the Clauser–Aspect–Zeilinger-type which are performed in two distant stations SA and SB. Our computer model is local deterministic [...] Read more.
We present a Monte Carlo model of Einstein–Podolsky–Rosen experiments that may be implemented on two independent computers and resembles the measurements of the Clauser–Aspect–Zeilinger-type which are performed in two distant stations SA and SB. Our computer model is local deterministic because we show that a theorist in station SB is able to conceive the products of the measurement outcomes of both stations, conditional to any possible equipment configuration in station SA. We show that the Monte Carlo model violates Bell-type inequalities and approaches the results of quantum theory for certain relationships between the number of measurements performed and the number of possible different physical properties of the entangled photon pairs. These relationships are clearly linked to Vorob’ev cyclicities, which always enforce Bell-type inequalities. The realization of this cyclicity depends, however, on combinatorial symmetry considerations that, in turn, depend on the mathematical properties of Einstein’s elements of physical reality. Because these mathematical properties have never been investigated and, therefore, may be free to be chosen in the models for all published experiments, Einstein’s physics does not contradict the experimental findings, instantaneous influences at a distance are put into question and the paradox of Einstein–Podolsky–Rosen and Bell is, thus, resolved. Full article
(This article belongs to the Topic Mathematical Modeling)
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22 pages, 847 KiB  
Article
On the Parallelization of Square-Root Vélu’s Formulas
by Jorge Chávez-Saab, Odalis Ortega and Amalia Pizarro-Madariaga
Math. Comput. Appl. 2024, 29(1), 14; https://doi.org/10.3390/mca29010014 - 16 Feb 2024
Viewed by 813
Abstract
A primary challenge in isogeny-based cryptography lies in the substantial computational cost associated to computing and evaluating prime-degree isogenies. This computation traditionally relied on Vélu’s formulas, an approach with time complexity linear in the degree but which was further enhanced by Bernstein, De [...] Read more.
A primary challenge in isogeny-based cryptography lies in the substantial computational cost associated to computing and evaluating prime-degree isogenies. This computation traditionally relied on Vélu’s formulas, an approach with time complexity linear in the degree but which was further enhanced by Bernstein, De Feo, Leroux, and Smith to a square-root complexity. The improved square-root Vélu’s formulas exhibit a degree of parallelizability that has not been exploited in major implementations. In this study, we introduce a theoretical framework for parallelizing isogeny computations and provide a proof-of-concept implementation in C with OpenMP. While the parallelization effectiveness exhibits diminishing returns with the number of cores, we still obtain strong results when using a small number of cores. Concretely, our implementation shows that for large degrees it is easy to achieve speedup factors of up to 1.74, 2.54, and 3.44 for two, four, and eight cores, respectively. Full article
(This article belongs to the Topic Mathematical Modeling)
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15 pages, 269 KiB  
Article
A SIR Epidemic Model Allowing Recovery
by Anthony G. Pakes
Axioms 2024, 13(2), 115; https://doi.org/10.3390/axioms13020115 - 08 Feb 2024
Viewed by 752
Abstract
The deterministic SIR model for disease spread in a closed population is extended to allow infected individuals to recover to the susceptible state. This extension preserves the second constant of motion, i.e., a functional relationship of susceptible and removed numbers, [...] Read more.
The deterministic SIR model for disease spread in a closed population is extended to allow infected individuals to recover to the susceptible state. This extension preserves the second constant of motion, i.e., a functional relationship of susceptible and removed numbers, S(t) and R(t), respectively. This feature allows a substantially complete elucidation of qualitative properties. The model exhibits three modes of behaviour classified in terms of the sign of S(0), the initial value of the epidemic curve. Model behaviour is similar to that of the SIS model if S(0)>0 and to the SIR model if S(0)<0. The separating case is completely soluble and S(t) is constant-valued. Long-term outcomes are determined for all cases, together with determination of the rate of convergence. Determining the shape of the epidemic curve motivates an investigation of curvature properties of all three state functions and quite complete results are obtained that are new, even for the SIR model. Finally, the second threshold theorem for the SIR model is extended in refined and generalised forms. Full article
(This article belongs to the Topic Mathematical Modeling)
17 pages, 1215 KiB  
Article
Investigating the Surrogate Modeling Capabilities of Continuous Time Echo State Networks
by Saakaar Bhatnagar
Math. Comput. Appl. 2024, 29(1), 9; https://doi.org/10.3390/mca29010009 - 24 Jan 2024
Viewed by 1234
Abstract
Continuous Time Echo State Networks (CTESNs) are a promising yet under-explored surrogate modeling technique for dynamical systems, particularly those governed by stiff Ordinary Differential Equations (ODEs). A key determinant of the generalization accuracy of a CTESN surrogate is the method of projecting the [...] Read more.
Continuous Time Echo State Networks (CTESNs) are a promising yet under-explored surrogate modeling technique for dynamical systems, particularly those governed by stiff Ordinary Differential Equations (ODEs). A key determinant of the generalization accuracy of a CTESN surrogate is the method of projecting the reservoir state to the output. This paper shows that of the two common projection methods (linear and nonlinear), the surrogates developed via the nonlinear projection consistently outperform those developed via the linear method. CTESN surrogates are developed for several challenging benchmark cases governed by stiff ODEs, and for each case, the performance of the linear and nonlinear projections is compared. The results of this paper demonstrate the applicability of CTESNs to a variety of problems while serving as a reference for important algorithmic and hyper-parameter choices for CTESNs. Full article
(This article belongs to the Topic Mathematical Modeling)
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17 pages, 466 KiB  
Article
Separation of the Linear and Nonlinear Covariates in the Sparse Semi-Parametric Regression Model in the Presence of Outliers
by Morteza Amini, Mahdi Roozbeh and Nur Anisah Mohamed
Mathematics 2024, 12(2), 172; https://doi.org/10.3390/math12020172 - 05 Jan 2024
Viewed by 522
Abstract
Determining the predictor variables that have a non-linear effect as well as those that have a linear effect on the response variable is crucial in additive semi-parametric models. This issue has been extensively investigated by many researchers in the area of semi-parametric linear [...] Read more.
Determining the predictor variables that have a non-linear effect as well as those that have a linear effect on the response variable is crucial in additive semi-parametric models. This issue has been extensively investigated by many researchers in the area of semi-parametric linear additive models, and various separation methods are proposed by the authors. A popular issue that might affect both estimation and separation results is the existence of outliers among the observations. In order to address this lack of sensitivity towards extreme observations, robust estimating approaches are frequently applied. We propose a robust method for simultaneously identifying the linear and nonlinear components of a semi-parametric linear additive model, even in the presence of outliers in the observations. Additionally, this model is sparse in that it may be used to determine which explanatory variables are ineffective by giving accurate zero estimates for their coefficients. To assess the effectiveness of the proposed method, a comprehensive Monte Carlo simulation study is conducted along with an application to investigate the dataset, which includes Boston property prices dataset. Full article
(This article belongs to the Topic Mathematical Modeling)
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28 pages, 1776 KiB  
Article
On the Method of Transformations: Obtaining Solutions of Nonlinear Differential Equations by Means of the Solutions of Simpler Linear or Nonlinear Differential Equations
by Nikolay K. Vitanov
Axioms 2023, 12(12), 1106; https://doi.org/10.3390/axioms12121106 - 08 Dec 2023
Viewed by 3837
Abstract
Transformations are much used to connect complicated nonlinear differential equations to simple equations with known exact solutions. Two examples of this are the Hopf–Cole transformation and the simple equations method. In this article, we follow an idea that is opposite to the idea [...] Read more.
Transformations are much used to connect complicated nonlinear differential equations to simple equations with known exact solutions. Two examples of this are the Hopf–Cole transformation and the simple equations method. In this article, we follow an idea that is opposite to the idea of Hopf and Cole: we use transformations in order to transform simpler linear or nonlinear differential equations (with known solutions) to more complicated nonlinear differential equations. In such a way, we can obtain numerous exact solutions of nonlinear differential equations. We apply this methodology to the classical parabolic differential equation (the wave equation), to the classical hyperbolic differential equation (the heat equation), and to the classical elliptic differential equation (Laplace equation). In addition, we use the methodology to obtain exact solutions of nonlinear ordinary differential equations by means of the solutions of linear differential equations and by means of the solutions of the nonlinear differential equations of Bernoulli and Riccati. Finally, we demonstrate the capacity of the methodology to lead to exact solutions of nonlinear partial differential equations on the basis of known solutions of other nonlinear partial differential equations. As an example of this, we use the Korteweg–de Vries equation and its solutions. Traveling wave solutions of nonlinear differential equations are of special interest in this article. We demonstrate the existence of the following phenomena described by some of the obtained solutions: (i) occurrence of the solitary wave–solitary antiwave from the solution, which is zero at the initial moment (analogy of an occurrence of particle and antiparticle from the vacuum); (ii) splitting of a nonlinear solitary wave into two solitary waves (analogy of splitting of a particle into two particles); (iii) soliton behavior of some of the obtained waves; (iv) existence of solitons which move with the same velocity despite the different shape and amplitude of the solitons. Full article
(This article belongs to the Topic Mathematical Modeling)
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25 pages, 8602 KiB  
Article
Numerical Methodology to Reduce the Drag and Control Flow around a Cam-Shaped Cylinder Integrated with Backward Splitter Plate
by Sunil Chamoli, Amit Joshi, Sumit Rana, Suvanjan Bhattacharaya, Ashutosh Gupta, Siddharth Ghansela, Chinaruk Thianpong and Smith Eiamsa-ard
Computation 2023, 11(10), 196; https://doi.org/10.3390/computation11100196 - 03 Oct 2023
Cited by 2 | Viewed by 1114
Abstract
After publishing a research article in the year 2019, a cam-shaped cylinder was introduced, and the results expressed its ability to prevent the vortex from shedding. This makes the cam-shaped cylinder a better performer than the circular cylinder. This work is an extension [...] Read more.
After publishing a research article in the year 2019, a cam-shaped cylinder was introduced, and the results expressed its ability to prevent the vortex from shedding. This makes the cam-shaped cylinder a better performer than the circular cylinder. This work is an extension of past work with the aim of further reducing drag by attaching a backward splitter plate to a cam-shaped cylinder. In an attempt to decrease drag and regulate the wake regime more efficiently than the traditional splitter plate control devices, a splitter plate flow departure control device is presented in this paper for a low Reynolds number flow range (Re = 50–200). It has been noted that when plate length increases, integral parameters like drag, lift, and Strouhal number do not change monotonically. The Strouhal number (St) increases with a drop in D2/Deq, but the average drag reduces with a rise in Re and a decrease in D2/Deq, respectively. In terms of decreased drag, the current cam-shaped cylinders attached to a rearward splitter plate have shown their superiority to other bluff bodies. Full article
(This article belongs to the Topic Mathematical Modeling)
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14 pages, 2648 KiB  
Article
Efficient Numerical Methods of Inverse Coefficient Problem Solution for One Inhomogeneous Body
by Alexandr Vatulyan, Pavel Uglich, Vladimir Dudarev and Roman Mnukhin
Axioms 2023, 12(10), 912; https://doi.org/10.3390/axioms12100912 - 25 Sep 2023
Viewed by 631
Abstract
In the present paper, the problems of longitudinal and flexural vibrations of an inhomogeneous rod are considered. The Young’s modulus and density are variable in longitudinal coordinate. Vibrations are caused by a load applied at the right end. The proposed method allows us [...] Read more.
In the present paper, the problems of longitudinal and flexural vibrations of an inhomogeneous rod are considered. The Young’s modulus and density are variable in longitudinal coordinate. Vibrations are caused by a load applied at the right end. The proposed method allows us to consider a wider class of inhomogeneity laws in comparison with other numerical solutions. Sensitivity analysis is carried out. A new inverse problem related to the simultaneous identification of the variation laws of Young’s modulus and density from amplitude–frequency data, which are measured in given frequency ranges, is considered. Its solution is based on an iterative process: at each step, a system of two Fredholm integral equations of the first kind with smooth kernels is solved numerically. The analysis of the kernels is carried out for different frequency values. To find the initial approximation, several approaches are proposed: a genetic algorithm, minimization of the residual functional on a compact set, and additional information about the values of the sought-for functions at the ends of the rod. The Tikhonov regularization and the LSQR method are proposed. Examples of reconstruction of monotonic and non-monotonic functions are presented. Full article
(This article belongs to the Topic Mathematical Modeling)
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30 pages, 3022 KiB  
Article
Catastrophe Bond Diversification Strategy Using Probabilistic–Possibilistic Bijective Transformation and Credibility Measures in Fuzzy Environment
by Wulan Anggraeni, Sudradjat Supian, Sukono and Nurfadhlina Abdul Halim
Mathematics 2023, 11(16), 3513; https://doi.org/10.3390/math11163513 - 14 Aug 2023
Viewed by 812
Abstract
The variety of catastrophe bond issuances can be used for portfolio diversification. However, the structure of catastrophe bonds differs from traditional bonds in that the face value and coupons depend on triggering events. This study aims to build a diversification strategy model framework [...] Read more.
The variety of catastrophe bond issuances can be used for portfolio diversification. However, the structure of catastrophe bonds differs from traditional bonds in that the face value and coupons depend on triggering events. This study aims to build a diversification strategy model framework using probabilistic–possibilistic bijective transformation (PPBT) and credibility measures in fuzzy environments based on the payoff function. The stages of modeling include identifying the trigger distribution; determining the membership degrees for the face value and coupons using PPBT; calculating the average face value and coupons using the fuzzy quantification theory; formulating the fuzzy variables for the yield; defining the function of triangular fuzzy membership for the yield; defining the credibility distribution for the triangular fuzzy variables for the yield; determining the expectation and total variance for the yield; developing a model of the catastrophe bond diversification strategy; the numerical simulation of the catastrophe bond strategy model; and formulating a solution to the simulation model of the diversification strategy using the sequential method, quadratic programming, transformation, and linearization techniques. The simulation results show that the proposed model can overcome the self-duality characteristic not possessed by the possibilistic measures in the fuzzy variables. The results obtained are expected to contribute to describing the yield uncertainty of investing in catastrophe bond assets so that investors can make wise decisions. Full article
(This article belongs to the Topic Mathematical Modeling)
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17 pages, 1051 KiB  
Article
A Bimodal Extension of the Log-Normal Distribution on the Real Line with an Application to DNA Microarray Data
by Mai F. Alfahad, Mohamed E. Ghitany, Ahmad N. Alothman and Saralees Nadarajah
Mathematics 2023, 11(15), 3360; https://doi.org/10.3390/math11153360 - 31 Jul 2023
Viewed by 698
Abstract
A bimodal double log-normal distribution on the real line is proposed using the random sign mixture transform. Its associated statistical inferences are derived. Its parameters are estimated by the maximum likelihood method. The performance of the estimators and the corresponding confidence intervals is [...] Read more.
A bimodal double log-normal distribution on the real line is proposed using the random sign mixture transform. Its associated statistical inferences are derived. Its parameters are estimated by the maximum likelihood method. The performance of the estimators and the corresponding confidence intervals is checked by simulation studies. Application of the proposed distribution to a real data set from a DNA microarray is presented. Full article
(This article belongs to the Topic Mathematical Modeling)
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13 pages, 321 KiB  
Article
Moments of the Negative Multinomial Distribution
by Frédéric Ouimet
Math. Comput. Appl. 2023, 28(4), 85; https://doi.org/10.3390/mca28040085 - 24 Jul 2023
Cited by 1 | Viewed by 863
Abstract
The negative multinomial distribution appears in many areas of applications such as polarimetric image processing and the analysis of longitudinal count data. In previous studies, general formulas for the falling factorial moments and cumulants of the negative multinomial distribution were obtained. However, despite [...] Read more.
The negative multinomial distribution appears in many areas of applications such as polarimetric image processing and the analysis of longitudinal count data. In previous studies, general formulas for the falling factorial moments and cumulants of the negative multinomial distribution were obtained. However, despite the availability of the moment generating function, no comprehensive formulas for the moments have been calculated thus far. This paper addresses this gap by presenting general formulas for both central and non-central moments of the negative multinomial distribution. These formulas are expressed in terms of binomial coefficients and Stirling numbers of the second kind. Utilizing these formulas, we provide explicit expressions for all central moments up to the fourth order and all non-central moments up to the eighth order. Full article
(This article belongs to the Topic Mathematical Modeling)
11 pages, 1171 KiB  
Article
Head Waves in Modified Weiskopf Sandy Medium
by Anastasia Karakozova and Sergey Kuznetsov
Axioms 2023, 12(7), 679; https://doi.org/10.3390/axioms12070679 - 10 Jul 2023
Viewed by 720
Abstract
In this study, a modified anisotropic elastic Weiskopf model for a sandy medium was introduced, satisfying the highest rotational symmetry compatible with a Weiskopf sandiness parameter greater than the unity. The developed approach was applied for computing and comparing head SP waves propagating [...] Read more.
In this study, a modified anisotropic elastic Weiskopf model for a sandy medium was introduced, satisfying the highest rotational symmetry compatible with a Weiskopf sandiness parameter greater than the unity. The developed approach was applied for computing and comparing head SP waves propagating along the free surface of the halfspace. The comparison revealed a substantial discrepancy in the SP wave pulses detected at the points of observation. The developed models for the generation and detection of SP waves are based on the FEA (finite element analysis) of the inner Lamb problem. Full article
(This article belongs to the Topic Mathematical Modeling)
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13 pages, 760 KiB  
Article
Developing a Robust Bioventing Model
by Mohammad Khodabakhshi Soureshjani and Richard G. Zytner
Math. Comput. Appl. 2023, 28(3), 76; https://doi.org/10.3390/mca28030076 - 16 Jun 2023
Cited by 2 | Viewed by 1186
Abstract
Bioventing is a widely recognized technique for the remediation of petroleum hydrocarbon-contaminated soil. In this study, the objective was to identify an optimal mathematical model that balances accuracy and ease of implementation. A comprehensive review of various models developed for bioventing was conducted [...] Read more.
Bioventing is a widely recognized technique for the remediation of petroleum hydrocarbon-contaminated soil. In this study, the objective was to identify an optimal mathematical model that balances accuracy and ease of implementation. A comprehensive review of various models developed for bioventing was conducted wherein the advantages and disadvantages of each model were evaluated and compared regarding the different numerical methods used to solve relevant bioventing equations. After investigating the various assumptions and methods from the literature, an improved foundational bioventing model was developed that characterizes gas flow in unsaturated zones where water and non-aqueous phase liquid (NAPL) are present and immobile, accounting for interphase mass transfer and biodegradation, incorporating soil properties through a rate constant correlation. The proposed model was solved using the finite volume method in OpenFOAM, an independent dimensional open-source coding toolbox. The preliminary simulation results of a simple case indicate good agreement with the exact analytical solution of the same equations. This improved bioventing model has the potential to enhance predictions of the remediation process and support the development of efficient remediation strategies for petroleum hydrocarbon-contaminated soil. Full article
(This article belongs to the Topic Mathematical Modeling)
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31 pages, 1620 KiB  
Article
Routing, Modulation Level, and Spectrum Assignment in Elastic Optical Networks—A Serial Stage Approach with Multiple Sub-Sets of Requests Based on Integer Linear Programming
by Luis Víctor Maidana Benítez, Melisa María Rosa Villamayor Paredes, José Colbes, César F. Bogado-Martínez, Benjamin Barán and Diego P. Pinto-Roa
Math. Comput. Appl. 2023, 28(3), 67; https://doi.org/10.3390/mca28030067 - 10 May 2023
Viewed by 1807
Abstract
This paper addresses serialized approaches of the routing, modulation level, and spectrum assignment (RMLSA) problem in elastic optical networks, using multiple sequential sub-sets of requests, under Integer Linear Programming (ILP). The literature has reported two-stage serial optimization methods referred to as RML+SA, which [...] Read more.
This paper addresses serialized approaches of the routing, modulation level, and spectrum assignment (RMLSA) problem in elastic optical networks, using multiple sequential sub-sets of requests, under Integer Linear Programming (ILP). The literature has reported two-stage serial optimization methods referred to as RML+SA, which retain computational efficiency when the problem grows, compared to the classical one-stage RMLSA optimization approach. However, there still remain numerous issues in terms of the spectrum used that can be improved when compared to the RMLSA solution. Consequently, this paper proposes RML+SA solutions considering multiple sequential sub-sets of requests, split traffic flow, as well as path-oriented and link-oriented routing models. Simulation results on different test scenarios determine that: (a) the multiple sequential sub-sets of request-based models improve computation time without worsening the spectrum usage when compared to just one set of requests optimization approaches, (b) divisible traffic flow approaches show promise in cases where the number of request sub-sets is low compared to the non-divisible counterpart, and (c) path-oriented routing succeeds in improving the used spectrum by increasing the number of candidate routes compared to link-oriented routing. Full article
(This article belongs to the Topic Mathematical Modeling)
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19 pages, 8019 KiB  
Article
Study on the Anisotropy of Strength Properties of Columnar Jointed Rock Masses Using a Geometric Model Reconstruction Method Based on a Single-Random Movement Voronoi Diagram of Uniform Seed Points
by Zhende Zhu, Luxiang Wang, Shu Zhu and Junyu Wu
Symmetry 2023, 15(4), 944; https://doi.org/10.3390/sym15040944 - 20 Apr 2023
Cited by 6 | Viewed by 1147
Abstract
The unique structural characteristics and special symmetry of columnar jointed rock mass result in its complex mechanical properties and strong anisotropy, which seriously affects the safety of engineering construction. To better simulate natural columnar jointed rock mass, a geometric model reconstruction method based [...] Read more.
The unique structural characteristics and special symmetry of columnar jointed rock mass result in its complex mechanical properties and strong anisotropy, which seriously affects the safety of engineering construction. To better simulate natural columnar jointed rock mass, a geometric model reconstruction method based on a single-random movement Voronoi diagram of uniform seed points using the feasible geological parameters of horizontal polygon density, irregular factor, dip angle, strike angle, transverse joint spacing, and transverse joint penetration rate is proposed in this paper. Based on this method, numerical simulation of CJRM models with varying strike angles, dip angles, and irregular factors under uniaxial compression were conducted. The results show that the uniaxial compression strengths versus strike angle and dip angle both decrease with the increase in the irregular factor, showing a U-shape and a gentle W-shape, respectively. The strength anisotropy of the strike angle decreases from 1.1057 to 1.0395 with the increase in the irregular factor, indicating relatively isotropy. With the increase int the irregular factor, the strength anisotropy of the dip angle increases from 4.3381 to 6.7953, indicating an increasing strong anisotropy at a high degree, and the effect of the irregular factor on strength behavior has the strongest and weakest impact at the dip angles of 60° and 90°, respectively. Full article
(This article belongs to the Topic Mathematical Modeling)
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13 pages, 4485 KiB  
Article
About Oscillations in Nonlinear Systems with Elastic Bonds
by Valery Ochkov, Inna Vasileva, Yulia Chudova and Anton Tikhonov
Mathematics 2023, 11(8), 1765; https://doi.org/10.3390/math11081765 - 07 Apr 2023
Viewed by 1106
Abstract
This article deals with oscillations that occur with a certain combination of parameters in a nonlinear system with elastic bonds, formed by three cylinders pulled together by an elastic thread. The aim of this work was to develop a methodology for studying nonlinear [...] Read more.
This article deals with oscillations that occur with a certain combination of parameters in a nonlinear system with elastic bonds, formed by three cylinders pulled together by an elastic thread. The aim of this work was to develop a methodology for studying nonlinear oscillatory systems based on the analysis of the potential energy of a system and the balance of the forces acting on it. The novelty of the work lies in the proposed methods for determining the conditions for the occurrence of vibrations and in those for calculating the threshold value of the elasticity coefficient of an elastic thread, at which an oscillatory process is guaranteed to occur. The differential equations of oscillations were compiled and numerically solved both with and without allowance for friction forces. The critical value of the elasticity coefficient of the thread at which periodic oscillations occurred was determined. A study of the motion of the system was carried out. Full article
(This article belongs to the Topic Mathematical Modeling)
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12 pages, 572 KiB  
Article
A Nonparametric Dual Control Algorithm of Multidimensional Objects with Interval-Valued Observations
by Manuel Arana-Jiménez, Alexander V. Medvedev and Ekaterina Chzhan
Axioms 2023, 12(2), 193; https://doi.org/10.3390/axioms12020193 - 11 Feb 2023
Viewed by 1094
Abstract
We focus on the dual interval control problem of multidimensional objects with delay. We propose a new nonparametric algorithm. In such a case, it is not necessary to determine a parametric structure of the investigated object. Another difficulty lies in the complex nature [...] Read more.
We focus on the dual interval control problem of multidimensional objects with delay. We propose a new nonparametric algorithm. In such a case, it is not necessary to determine a parametric structure of the investigated object. Another difficulty lies in the complex nature of the decision-making field as it might not be flexible or convenient for decision-makers to exactly quantify their opinions with crisp numbers. Due to this fact, we introduce the interval-valued observations into the algorithm by means of the single-level constraint interval arithmetic. The results of computational experiments illustrate the effectiveness of the algorithm in the case of using intervals instead of crisp values. Full article
(This article belongs to the Topic Mathematical Modeling)
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27 pages, 2349 KiB  
Article
Deterioration Control Decision Support System for the Retailer during Availability of Trade Credit and Shortages
by Mrudul Y. Jani, Heta A. Patel, Amrita Bhadoriya, Urmila Chaudhari, Mohamed Abbas and Malak S. Alqahtani
Mathematics 2023, 11(3), 580; https://doi.org/10.3390/math11030580 - 21 Jan 2023
Cited by 4 | Viewed by 1454
Abstract
The deterioration rate is a significant aspect of perishable goods. Since perishable items will always deteriorate, there are effective methods for reducing the rate of deterioration. Furthermore, in the existing inventory control literature, the deterioration rate is often viewed as an exogenous component. [...] Read more.
The deterioration rate is a significant aspect of perishable goods. Since perishable items will always deteriorate, there are effective methods for reducing the rate of deterioration. Furthermore, in the existing inventory control literature, the deterioration rate is often viewed as an exogenous component. Keeping this problem in mind, this article develops the perishable inventory control system from the retailer’s perspective in which: (i) the deterioration rate is a controllable factor and suggests a new fresh quality technology (FQT) indicator, (ii) demand is determined by the perishable product’s quality, that is controlled by its rate of deterioration, (iii) the credit duration is predefined, and (iv) shortages are expected. The key goal is to demonstrate that there is an ideal quantity of the order that minimizes the retailer’s overall cost in terms of cycle time and deterioration rate. Finally, theoretical results are validated by solving two numerical illustrations and conducting a sensitivity analysis of the main factors resulting from the following managerial implications: (i) if the range of deterioration is between zero and one then the retailer should invest in the preservation factor to preserve the perishable product and if greater than one the retailer should not invest in the preservation factor, (ii) credit period significantly reduces the total cost. Hence, this trade credit strategy is more beneficial to the model. Full article
(This article belongs to the Topic Mathematical Modeling)
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11 pages, 295 KiB  
Article
Application of the ARA Method in Solving Integro-Differential Equations in Two Dimensions
by Rania Saadeh
Computation 2023, 11(1), 4; https://doi.org/10.3390/computation11010004 - 29 Dec 2022
Cited by 8 | Viewed by 1282
Abstract
The main purpose of this study is to investigate solutions of some integral equations of different classes using a new scheme. This research introduces and implements the new double ARA transform to solve integral and partial integro-differential equations. We introduce basic theorems and [...] Read more.
The main purpose of this study is to investigate solutions of some integral equations of different classes using a new scheme. This research introduces and implements the new double ARA transform to solve integral and partial integro-differential equations. We introduce basic theorems and properties of the double ARA transform in two dimensions, and some results related to the double convolution theorem and partial derivatives are presented. In addition, to show the validity of the proposed technique, we introduce and solve some examples using the new approach. Full article
(This article belongs to the Topic Mathematical Modeling)
20 pages, 4278 KiB  
Article
A Microscopic Heterogeneous Traffic Flow Model Considering Distance Headway
by Faryal Ali, Zawar Hussain Khan, Khurram Shehzad Khattak, Thomas Aaron Gulliver and Akhtar Nawaz Khan
Mathematics 2023, 11(1), 184; https://doi.org/10.3390/math11010184 - 29 Dec 2022
Cited by 6 | Viewed by 1337
Abstract
The intelligent driver (ID) model characterizes traffic behavior with a constant acceleration exponent and does not follow traffic physics. This results in unrealistic traffic behavior. In this paper, a new microscopic heterogeneous traffic flow model is proposed which improves the performance of the [...] Read more.
The intelligent driver (ID) model characterizes traffic behavior with a constant acceleration exponent and does not follow traffic physics. This results in unrealistic traffic behavior. In this paper, a new microscopic heterogeneous traffic flow model is proposed which improves the performance of the ID model. The forward and lateral distance headways are used to characterize traffic behavior. The stability of the ID and proposed models is examined over a 1000 m circular road with a traffic disturbance after 30 s. The results obtained show that the proposed model is more stable than the ID model. The performance of the proposed and ID models is evaluated over an 1800 m circular road for 150 s with a platoon of 51 vehicles. Results are presented which indicate that traffic evolves realistically with the proposed model. This is because it is based on the lateral distance headway. Full article
(This article belongs to the Topic Mathematical Modeling)
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23 pages, 1092 KiB  
Article
Dynamical Behaviour of a Modified Tuberculosis Model with Impact of Public Health Education and Hospital Treatment
by Fatima Sulayman and Farah Aini Abdullah
Axioms 2022, 11(12), 723; https://doi.org/10.3390/axioms11120723 - 13 Dec 2022
Cited by 4 | Viewed by 1730
Abstract
Tuberculosis (TB), caused by Mycobacterium tuberculosis is one of the treacherous infectious diseases of global concern. In this paper, we consider a deterministic model of TB infection with the public health education and hospital treatment impact. The effective reproductive number, [...] Read more.
Tuberculosis (TB), caused by Mycobacterium tuberculosis is one of the treacherous infectious diseases of global concern. In this paper, we consider a deterministic model of TB infection with the public health education and hospital treatment impact. The effective reproductive number, Rph, that measures the potential spread of TB is presented by employing the next generation matrix approach. We investigate local and global stability of the TB-free equilibrium point, endemic equilibrium point, and sensitivity analysis. The analyses of the proposed model show that the model undergoes the phenomenon of backward bifurcation when the effective reproduction number (Rph) is less than one, where two stable equilibria, namely, the DFE and an EEP coexist. Further, we compute the sensitivity of the impact of each parameter on the effective reproductive number of the model by employing a normalized sensitivity index formula. Numerical simulation of the proposed model was conducted using Maple 2016 and MatLab R2020b software and compared with the theoretical results for illustration purposes. The investigation results can be useful in providing information to policy makers and public health authorities in mitigating the spread of TB infection by public health education and hospital treatment. Full article
(This article belongs to the Topic Mathematical Modeling)
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14 pages, 306 KiB  
Article
Applications of Double ARA Integral Transform
by Rania Saadeh
Computation 2022, 10(12), 216; https://doi.org/10.3390/computation10120216 - 08 Dec 2022
Cited by 12 | Viewed by 1383
Abstract
This paper describes our construction of a new double transform, which we call the double ARA transform (DARAT). Our novel double-integral transform can be used to solve partial differential equations and other problems. We discuss some fundamental characteristics of our approach, including existence, [...] Read more.
This paper describes our construction of a new double transform, which we call the double ARA transform (DARAT). Our novel double-integral transform can be used to solve partial differential equations and other problems. We discuss some fundamental characteristics of our approach, including existence, linearity, and several findings relating to partial derivatives and the double convolution theorem. DARAT can be used to precisely solve a variety of partial differential equations, including the heat equation, wave equation, telegraph equation, Klein–Gordon equation, and others, all of which are crucial for physical applications. Herein, we use DARAT to solve model integral equations to obtain exact solutions. We conclude that our novel method is easier to use than comparable transforms. Full article
(This article belongs to the Topic Mathematical Modeling)
11 pages, 354 KiB  
Article
Calculation of the Free Energy of the Ising Model on a Cayley Tree via the Self-Similarity Method
by Hasan Akın
Axioms 2022, 11(12), 703; https://doi.org/10.3390/axioms11120703 - 07 Dec 2022
Cited by 4 | Viewed by 1490
Abstract
In this study, an interactive Ising model having the nearest and prolonged next-nearest neighbors defined on a Cayley tree is considered. Inspired by the results obtained for the one-dimensional Ising model, we will construct the partition function and then calculate the free energy [...] Read more.
In this study, an interactive Ising model having the nearest and prolonged next-nearest neighbors defined on a Cayley tree is considered. Inspired by the results obtained for the one-dimensional Ising model, we will construct the partition function and then calculate the free energy of the Ising model having the prolonged next nearest and nearest neighbor interactions and external field on a two-order Cayley tree using the self-similarity of the semi-infinite Cayley tree. The phase transition problem for the Ising system is investigated under the given conditions. Full article
(This article belongs to the Topic Mathematical Modeling)
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15 pages, 300 KiB  
Article
Well-Posedness Results of Certain Variational Inequalities
by Savin Treanţă
Mathematics 2022, 10(20), 3809; https://doi.org/10.3390/math10203809 - 15 Oct 2022
Viewed by 1217
Abstract
Well-posedness and generalized well-posedness results are examined for a class of commanded variational inequality problems. In this regard, by using the concepts of hemicontinuity, monotonicity, and pseudomonotonicity of the considered functional, and by introducing the set of approximating solutions of the considered commanded [...] Read more.
Well-posedness and generalized well-posedness results are examined for a class of commanded variational inequality problems. In this regard, by using the concepts of hemicontinuity, monotonicity, and pseudomonotonicity of the considered functional, and by introducing the set of approximating solutions of the considered commanded variational inequality problems, we establish several well-posedness and generalized well-posedness results. Moreover, some illustrative examples are provided to highlight the effectiveness of the results obtained in the paper. Full article
(This article belongs to the Topic Mathematical Modeling)
25 pages, 2032 KiB  
Article
Mathematical Assessment of the Impact of the Imperfect Vaccination on Diphtheria Transmission Dynamics
by Siwaphorn Kanchanarat, Settapat Chinviriyasit and Wirawan Chinviriyasit
Symmetry 2022, 14(10), 2000; https://doi.org/10.3390/sym14102000 - 24 Sep 2022
Cited by 3 | Viewed by 2098
Abstract
Diphtheria is a vaccine-preventable disease in which the outbreaks will not occur if a high enough proportion of individuals in a population are immune. Recent reports reveal that vaccinated individuals with low coverage levels of immunity may be at risk of subclinical diphtheria [...] Read more.
Diphtheria is a vaccine-preventable disease in which the outbreaks will not occur if a high enough proportion of individuals in a population are immune. Recent reports reveal that vaccinated individuals with low coverage levels of immunity may be at risk of subclinical diphtheria infection. Therefore, the development of an epidemiology model that will predict the optimal vaccine coverage level needed to prevent the spread of these diseases is crucial. In this paper, a mathematical model for diphtheria transmission with asymptomatic infection, logistic growth, and vaccination is formulated and rigorously analyzed to gain insights into its global dynamical features. The study results show that the disease is eradicated whenever the vaccination coverage is greater than the optimal vaccination coverage level needed for diphtheria eradication. The reported cases of diphtheria in Thailand are applied to estimate the appropriate parameters of the model. Sensitivity analysis reveals the rate of vaccination and the asymptomatic infection are influential factors in controlling and preventing diphtheria. Numerical simulations are illustrated in the theoretical results and show that the incubation period of asymptomatic individuals has an impact on the optimal vaccination coverage level needed for diphtheria eradication. Full article
(This article belongs to the Topic Mathematical Modeling)
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