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Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications
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Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Engineering Mathematics".

Deadline for manuscript submissions: closed (31 December 2023) | Viewed by 34262

Special Issue Editors

Prof. Dr. Juan Francisco Sánchez-Pérez

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Guest Editor
Department of Applied Physics and Naval Technology, Technical University of Cartagena, 30203 Cartagena, Spain
Interests: mathematical modeling; numerical simulation; network simulation method; nondimensionalization; dimensionless groups; engineering; applied physics
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Gonzalo García Ros

E-Mail Website
Guest Editor
Department of Civil Engineering Department, Technical University of Cartagena, Paseo Alfonso XIII 52, 30203 Cartagena, Spain
Interests: partial differential equations; mathematical modeling; applied mathematics in civil engineering; statistical treatment of acoustic emissions; nondimensionalization of engineering problems; ground engineeringstatistical treatment of acoustic emissions; nondimensionalization of engineering problems; ground engineering
Special Issues, Collections and Topics in MDPI journals
Dr. Manuel Conesa

E-Mail Website
Guest Editor
Department of Applied Physics and Naval Technology, Technical University of Cartagena, 30203 Cartagena, Spain
Interests: dimensional analysis; free convection; ordinary differential equations; network method; electrical simulation; multidisciplinary tools; education in science and engineeringon, multidisciplinary tools, education in science and engineering
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue invites you to present papers that solve problems of interest in the fields of science and engineering, through the use of different mathematical methodologies.

The main objective is to solve the proposed mathematical model composed of complex systems of coupled or not differential equations that represent the study problem through novel methodologies, such as the use of dimensional analysis, nondimensionalization, numerical simulation techniques or optimization techniques, among others.

In addition, this Special Issue integrates those teaching methodologies that are used to facilitate the learning of mathematical methods for undergraduate and graduate students in the different fields of science and engineering, along with their evaluation.

Finally, this Special Issue includes the following potential topics (but is not limited to):

  • Mathematical modeling for science and engineering applications;
  • Numerical analysis;
  • Mathematical and computational engineering;
  • Numerical methods for science and engineering applications;
  • Optimization and control in engineering applications;
  • Dynamical systems;
  • Mathematical physics;
  • Analysis of PDEs;
  • Classical analysis and ODEs;
  • Mathematics in engineering and sciences studies;
  • Teaching and assessment methodologies in science and engineering;
  • STEM and mathematics education.

Prof. Dr. Juan Francisco Sánchez-Pérez
Prof. Dr. Gonzalo García Ros
Dr. Manuel Conesa
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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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.

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Published Papers (27 papers)

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Research

14 pages, 535 KiB  
Article
Understanding Complex Traffic Dynamics with the Nondimensionalisation Technique
by Juan Francisco Sánchez-Pérez, Santiago Oviedo-Casado, Gonzalo García-Ros, Manuel Conesa and Enrique Castro
Mathematics 2024, 12(4), 532; https://0-doi-org.brum.beds.ac.uk/10.3390/math12040532 - 08 Feb 2024
Viewed by 467
Abstract
Hydrodynamic traffic models are crucial to optimizing transportation efficiency and urban planning. They usually comprise a set of coupled partial differential equations featuring an arbitrary number of terms that aim to describe the different nuances of traffic flow. Consequently, traffic models quickly become [...] Read more.
Hydrodynamic traffic models are crucial to optimizing transportation efficiency and urban planning. They usually comprise a set of coupled partial differential equations featuring an arbitrary number of terms that aim to describe the different nuances of traffic flow. Consequently, traffic models quickly become complicated to solve and difficult to interpret. In this article, we present a general traffic model that includes a relaxation term and an inflow of vehicles term and utilize the mathematical technique of nondimensionalisation to obtain universal solutions to the model. Thus, we are able to show extreme sensitivity to initial conditions and parameter changes, a classical signature of deterministic chaos. Moreover, we obtain simple relations among the different variables governing traffic, thus managing to efficiently describe the onset of traffic jams. We validate our model by comparing different scenarios and highlighting the model’s applicability regimes in traffic equations. We show that extreme speed values, or heavy traffic inflow, lead to divergences in the model, showing its limitations but also demonstrating how the problem of traffic jams can be alleviated. Our results pave the way to simulating and predicting traffic accurately on a real-time basis. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
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19 pages, 2703 KiB  
Article
Normalization Method as a Potent Tool for Grasping Linear and Nonlinear Systems in Physics and Soil Mechanics
by Manuel Conesa, Juan Francisco Sánchez-Pérez, Gonzalo García-Ros, Enrique Castro and Julio Valenzuela
Mathematics 2023, 11(20), 4321; https://0-doi-org.brum.beds.ac.uk/10.3390/math11204321 - 17 Oct 2023
Cited by 2 | Viewed by 757
Abstract
To address physical problems that require solving differential equations, both linear and nonlinear analytical methods are preferred when possible, but numerical methods are utilized when necessary. In this study, the normalization technique is established, which is a simple mathematical approach that requires only [...] Read more.
To address physical problems that require solving differential equations, both linear and nonlinear analytical methods are preferred when possible, but numerical methods are utilized when necessary. In this study, the normalization technique is established, which is a simple mathematical approach that requires only basic manipulation of the governing equations to obtain valuable information about the solution. The methodology of this technique involves adopting appropriate references to obtain the dimensionless form of the governing equation, after which the terms of the equation are balanced, obtaining the dimensionless monomials governing the solution. Thorough knowledge of the physical processes involved is necessary to find the best references. The main advantages of this technique are the simplicity of the methodology, the acquisition of valuable information about the solution without the need for complex mathematical calculations, and its applicability to nonlinear problems. However, it is important to consider the difficulty in selecting appropriate references in more complex scenarios. This study applies this normalization methodology to different scenarios, showing how choosing appropriate references lead to the independent dimensionless monomials. Once obtained, it was possible to identify different situations concerning the value of monomials. It will be when they are close to unity, and therefore normalized, when they fundamentally affect the solution of the problem. Finally, we present two cases, one linear and one complex, about the application of normalization to the challenging problem of soil consolidation in ground engineering, illustrating how the technique was used to obtain the solution and its many advantages. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
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24 pages, 7123 KiB  
Article
Multivariate Statistical and Correlation Analysis between Acoustic and Geotechnical Variables in Soil Compression Tests Monitored by the Acoustic Emission Technique
by Gonzalo García-Ros, Danny Xavier Villalva-León, Enrique Castro, Juan Francisco Sánchez-Pérez, Julio Valenzuela and Manuel Conesa
Mathematics 2023, 11(19), 4085; https://0-doi-org.brum.beds.ac.uk/10.3390/math11194085 - 26 Sep 2023
Viewed by 605
Abstract
In this research, a series of compression tests were carried out, under oedometric conditions, on sand samples prepared with different moisture contents. In these tests, in addition to the usual measurements of the stress and deformation of the sample, a series of acoustic [...] Read more.
In this research, a series of compression tests were carried out, under oedometric conditions, on sand samples prepared with different moisture contents. In these tests, in addition to the usual measurements of the stress and deformation of the sample, a series of acoustic emission sensors were used to monitor the parameters of the acoustic signals coming from inside the sample. This is a rather novel technique with great potential, but sometimes difficult to approach due to the large amount of acoustic emission data generated. In this paper, a correlation and regression analysis has been performed to quantify the correlations between the geotechnical variables and the parameters characterizing the acoustic emissions. The results presented open an interesting horizon of possibilities since, as it has been shown, it is possible to determine the values of the geotechnical properties from the acoustic variables, by means of the regression functions obtained for each type of soil or for each practical case. At the very least, this is a complementary tool in the determination of the mechanical properties of soils subjected to compression, although it could also be useful in those situations in which the monitoring of geotechnical variables describing the tenso-deformational behavior of the soil may be difficult or impossible. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
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13 pages, 302 KiB  
Article
On Ricci Curvature of a Homogeneous Generalized Matsumoto Finsler Space
by Yanlin Li, Manish Kumar Gupta, Suman Sharma and Sudhakar Kumar Chaubey
Mathematics 2023, 11(15), 3365; https://0-doi-org.brum.beds.ac.uk/10.3390/math11153365 - 01 Aug 2023
Cited by 17 | Viewed by 722
Abstract
The characterization of Finsler spaces with Ricci curvature is an ancient and cumbersome one. In this paper, we have derived an expression of Ricci curvature for the homogeneous generalized Matsumoto change. Moreover, we have deduced the expression of Ricci curvature for the aforementioned [...] Read more.
The characterization of Finsler spaces with Ricci curvature is an ancient and cumbersome one. In this paper, we have derived an expression of Ricci curvature for the homogeneous generalized Matsumoto change. Moreover, we have deduced the expression of Ricci curvature for the aforementioned space with vanishing the S-curvature. These findings contribute significantly to understanding the complex nature of Finsler spaces and their curvature properties. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
26 pages, 6077 KiB  
Article
Violation of Neumann Problem’s Solvability Condition for Poisson Equation, Involved in the Non-Linear PDEs, Containing the Reaction-Diffusion-Convection-Type Equation, at Numerical Solution by Direct Method
by Vyacheslav Trofimov, Maria Loginova, Vladimir Egorenkov, Yongqiang Yang and Zhongwei Yan
Mathematics 2023, 11(11), 2567; https://0-doi-org.brum.beds.ac.uk/10.3390/math11112567 - 03 Jun 2023
Viewed by 960
Abstract
In this paper, we consider the 3D problem of laser-induced semiconductor plasma generation under the action of the optical pulse, which is governed by the set of coupled time-dependent non-linear PDEs involving the Poisson equation with Neumann boundary conditions. The main feature of [...] Read more.
In this paper, we consider the 3D problem of laser-induced semiconductor plasma generation under the action of the optical pulse, which is governed by the set of coupled time-dependent non-linear PDEs involving the Poisson equation with Neumann boundary conditions. The main feature of this problem is the non-linear feedback between the Poisson equation with respect to induced electric field potential and the reaction-diffusion-convection-type equation with respect to free electron concentration and accounting for electron mobility (convection’s term). Herein, we focus on the choice of the numerical method for the Poisson equation solution with inhomogeneous Neumann boundary conditions. Despite the ubiquitous application of such a direct method as the Fast Fourier Transform for solving an elliptic problem in simple spatial domains, we demonstrate that applying a direct method for solving the problem under consideration results in a solution distortion. The reason for the Neumann problem’s solvability condition violation is the computational error’s accumulation. In contrast, applying an iterative method allows us to provide finite-difference scheme conservativeness, asymptotic stability, and high computation accuracy. For the iteration technique, we apply both an implicit alternating direction method and a new three-stage iteration process. The presented computer simulation results confirm the advantages of using iterative methods. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
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14 pages, 319 KiB  
Article
Harnack Estimation for Nonlinear, Weighted, Heat-Type Equation along Geometric Flow and Applications
by Yanlin Li, Sujit Bhattacharyya, Shahroud Azami, Apurba Saha and Shyamal Kumar Hui
Mathematics 2023, 11(11), 2516; https://0-doi-org.brum.beds.ac.uk/10.3390/math11112516 - 30 May 2023
Cited by 23 | Viewed by 1151
Abstract
The method of gradient estimation for the heat-type equation using the Harnack quantity is a classical approach used for understanding the nature of the solution of these heat-type equations. Most of the studies in this field involve the Laplace–Beltrami operator, but in our [...] Read more.
The method of gradient estimation for the heat-type equation using the Harnack quantity is a classical approach used for understanding the nature of the solution of these heat-type equations. Most of the studies in this field involve the Laplace–Beltrami operator, but in our case, we studied the weighted heat equation that involves weighted Laplacian. This produces a number of terms involving the weight function. Thus, in this article, we derive the Harnack estimate for a positive solution of a weighted nonlinear parabolic heat equation on a weighted Riemannian manifold evolving under a geometric flow. Applying this estimation, we derive the Li–Yau-type gradient estimation and Harnack-type inequality for the positive solution. A monotonicity formula for the entropy functional regarding the estimation is derived. We specify our results for various different flows. Our results generalize some works. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
8 pages, 311 KiB  
Article
Bias Due to Averaging the Logistic and SI Models
by Dorothy I. Wallace
Mathematics 2023, 11(10), 2321; https://0-doi-org.brum.beds.ac.uk/10.3390/math11102321 - 16 May 2023
Viewed by 734
Abstract
Modelers have choices in how they approach a problem, with different approaches potentially leading to different outcomes. Sometimes one approach gives a consistently lower (or higher) result than another. The theorem and corollaries in this study show that if the logistic equation or, [...] Read more.
Modelers have choices in how they approach a problem, with different approaches potentially leading to different outcomes. Sometimes one approach gives a consistently lower (or higher) result than another. The theorem and corollaries in this study show that if the logistic equation or, equivalently, the SI model, are perturbed at time zero by a range of values with mean zero, the resulting trajectories must average to a value below (for logistic and I) or above (for S) the solution with average initial condition. The proof of the theorem shows that this phenomenon is the result of algebraic properties of the nonlinear quadratic term, although we note it can be extended to a larger class of systems. More importantly it shows that the only necessary criterion is that the perturbations average to zero. The source of them and the properties of their distribution does not matter to the result of the theorem but does affect the magnitude of the proven difference. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
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28 pages, 9435 KiB  
Article
Methodology to Obtain Universal Solutions for Systems of Coupled Ordinary Differential Equations: Examples of a Continuous Flow Chemical Reactor and a Coupled Oscillator
by Juan Francisco Sánchez-Pérez, Gonzalo García-Ros, Manuel Conesa, Enrique Castro and Manuel Cánovas
Mathematics 2023, 11(10), 2303; https://0-doi-org.brum.beds.ac.uk/10.3390/math11102303 - 15 May 2023
Cited by 4 | Viewed by 918
Abstract
This paper presents a concise and orderly methodology to obtain universal solutions to different problems in science and engineering using the nondimensionalization of the governing equations of the physical–chemical problem posed. For its application, a deep knowledge of the problem is necessary since [...] Read more.
This paper presents a concise and orderly methodology to obtain universal solutions to different problems in science and engineering using the nondimensionalization of the governing equations of the physical–chemical problem posed. For its application, a deep knowledge of the problem is necessary since it will facilitate the adequate choice of the references necessary for its resolution. In addition, the application of the methodology to examples of coupled ordinary differential equations is shown, resulting in an interesting tool to teach postgraduate students in the branches of physics, mathematics, and engineering. The first example used for a system of coupled ordinary differential equations is a model of a continuous flow chemical reactor, where it is worth noting; on the one hand, the methodology used to choose the reference (characteristic) time and, on the other, the equivalence between the characteristic times obtained for each one of the species. The following universal curves are obtained, which are validated by comparing them with the results obtained by numerical simulation, where it stands out that the universal solution includes an unknown that must be previously obtained. The resolution of this unknown implies having a deep knowledge of the problem, a common characteristic when using the methodology proposed in this work for different engineering or physicochemical problems. Finally, the second example is a coupled oscillator, where it is worth noting that the appearance of characteristic periods that implicitly or explicitly affect the particles’ movement is striking. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
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14 pages, 3973 KiB  
Article
Neural Networks Simulation of Distributed SEIR System
by Tibor Kmet, Maria Kmetova and Ladislav Végh
Mathematics 2023, 11(9), 2113; https://0-doi-org.brum.beds.ac.uk/10.3390/math11092113 - 29 Apr 2023
Cited by 1 | Viewed by 968
Abstract
In this paper, a neural network-based optimal control synthesis is presented for distributed optimal control problems. We deal with solutions of systems controlled by parabolic differential equations with control and state constraints and discrete time delays. The given optimal control problem is transformed [...] Read more.
In this paper, a neural network-based optimal control synthesis is presented for distributed optimal control problems. We deal with solutions of systems controlled by parabolic differential equations with control and state constraints and discrete time delays. The given optimal control problem is transformed into a discrete nonlinear problem and then implemented into a feed-forward adaptive critic neural network. We propose a new algorithm to reach optimal control and an optimal trajectory using a feed-forward neural network. We present a concrete application of this simulation method on the SEIR (Susceptible—Exposed—Infectious—Recovered) optimal control problem of a distributed system for disease control. The results show that the adaptive-critic-based neural network approach is suitable for the solution of optimal distributed control problems with delay in state and control variables subject to control-state constraints and simulates the spread of disease in the SEIR system. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
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16 pages, 2533 KiB  
Article
Experimental Design for Progressive Type I Interval Censoring on the Lifetime Performance Index of Chen Lifetime Distribution
by Shu-Fei Wu and Meng-Zong Song
Mathematics 2023, 11(6), 1554; https://0-doi-org.brum.beds.ac.uk/10.3390/math11061554 - 22 Mar 2023
Cited by 3 | Viewed by 961
Abstract
The lifetime performance index is commonly utilized to assess the lifetime performance of products. Based on the testing procedure for the lifetime of products following Chen distribution, an experimental design for progressive type I interval censoring is determined to achieve the desired power [...] Read more.
The lifetime performance index is commonly utilized to assess the lifetime performance of products. Based on the testing procedure for the lifetime of products following Chen distribution, an experimental design for progressive type I interval censoring is determined to achieve the desired power level while minimizing total experimental cost. For fixed inspection interval lengths and an unfixed number of inspection intervals, the required number of inspection intervals and sample sizes to achieve the minimum experimental costs are computed and presented in a table format. For unfixed termination times, the required number of inspection intervals, minimum sample sizes, and equal interval lengths are obtained and presented in a table format, while the minimum experimental costs are achieved. Finally, a practical example is presented to demonstrate the utilization of this experimental design for collecting samples and conducting a testing procedure to evaluate the lifetime performance of products. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
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26 pages, 1126 KiB  
Article
Effect of Macrophages and Latent Reservoirs on the Dynamics of HTLV-I and HIV-1 Coinfection
by A. M. Elaiw, N. H. AlShamrani, E. Dahy, A. A. Abdellatif and Aeshah A. Raezah
Mathematics 2023, 11(3), 592; https://0-doi-org.brum.beds.ac.uk/10.3390/math11030592 - 22 Jan 2023
Cited by 1 | Viewed by 1386
Abstract
Human immunodeficiency virus type 1 (HIV-1) and human T-lymphotropic virus type I (HTLV-I) are two retroviruses that have a similar fashion of transmission via sharp objects contaminated by viruses, transplant surgery, transfusion, and sexual relations. Simultaneous infections with HTLV-I and HIV-1 usually occur [...] Read more.
Human immunodeficiency virus type 1 (HIV-1) and human T-lymphotropic virus type I (HTLV-I) are two retroviruses that have a similar fashion of transmission via sharp objects contaminated by viruses, transplant surgery, transfusion, and sexual relations. Simultaneous infections with HTLV-I and HIV-1 usually occur in areas where both viruses have become endemic. CD4+T cells are the main targets of HTLV-I, while HIV-1 can infect CD4+T cells and macrophages. It is the aim of this study to develop a model of HTLV-I and HIV-1 coinfection that describes the interactions of nine compartments: susceptible cells of both CD4+T cells and macrophages, HIV-1-infected cells that are latent/active in both CD4+T cells and macrophages, HTLV-I-infected CD4+T cells that are latent/active, and free HIV-1 particles. The well-posedness, existence of equilibria, and global stability analysis of our model are investigated. The Lyapunov function and LaSalle’s invariance principle were used to study the global asymptotic stability of all equilibria. The theoretically predicted outcomes were verified by utilizing numerical simulations. The effect of including the macrophages and latent reservoirs in the HTLV-I and HIV-1 coinfection model is discussed. We show that the presence of macrophages makes a coinfection model more realistic when the case of the coexistence of HIV-1 and HTLV-I is established. Moreover, we have shown that neglecting the latent reservoirs in HTLV-I and HIV-1 coinfection modeling will lead to the design of an overflow of anti-HIV-1 drugs. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
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22 pages, 6173 KiB  
Article
Characterization of Flow under Impervious Dams: Dimensionless Groups and Universal Solutions
by Encarnación Martínez-Moreno, Gonzalo García-Ros, Iván Alhama and Francisco Alhama
Mathematics 2023, 11(3), 540; https://0-doi-org.brum.beds.ac.uk/10.3390/math11030540 - 19 Jan 2023
Cited by 2 | Viewed by 941
Abstract
As far as we know, no dimensionless solutions for infiltrated flow under dams in anisotropic media exist since those that can be found in manuals refer to isotropic soils. The novelty of this work is the presentation of universal solutions in the form [...] Read more.
As far as we know, no dimensionless solutions for infiltrated flow under dams in anisotropic media exist since those that can be found in manuals refer to isotropic soils. The novelty of this work is the presentation of universal solutions in the form of abaci for water flow, average exit gradient, uplift force, and its application point for this type of soil. These solutions are obtained by the application of the discriminated nondimensionalization technique to the governing equations in order to find accurate dimensionless groups that control the results of the problem. In particular, the ratio of permeabilities corrected by a geometrical aspect relationship appears as a governing group, so anisotropy can be considered as input information. In this way, the sought solutions are a function of the emerging groups. Numerical solutions are used to successfully verify the results obtained, which in turn are compared to those of other authors for isotropic scenarios. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
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11 pages, 290 KiB  
Article
Sample Size Determination for Two-Stage Multiple Comparisons for Exponential Location Parameters with the Average
by Shu-Fei Wu
Mathematics 2023, 11(2), 441; https://0-doi-org.brum.beds.ac.uk/10.3390/math11020441 - 13 Jan 2023
Viewed by 701
Abstract
In this paper, the design constant is determined based on a criterion of probability of correct detection for a given deviation of k location parameters from the average being at least γ for the two-stage multiple comparisons for location parameters of exponential distributions [...] Read more.
In this paper, the design constant is determined based on a criterion of probability of correct detection for a given deviation of k location parameters from the average being at least γ for the two-stage multiple comparisons for location parameters of exponential distributions when scale parameters are unequal. All required values for determining the design constant and then the total sample size for a two-stage procedure are listed in tables for practical use. For illustrative purposes, two examples are given to demonstrate the sample size determination for this two-stage procedure. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
32 pages, 2910 KiB  
Article
Global Properties of a Diffusive SARS-CoV-2 Infection Model with Antibody and Cytotoxic T-Lymphocyte Immune Responses
by Ahmed. M. Elaiw, Abdullah J. Alsaedi, Aatef. D. Hobiny and Shaban. A. Aly
Mathematics 2023, 11(1), 190; https://0-doi-org.brum.beds.ac.uk/10.3390/math11010190 - 29 Dec 2022
Cited by 2 | Viewed by 963
Abstract
A severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection can lead to morbidity and mortality. SARS-CoV-2 infects the epithelial cells of the respiratory tract and causes coronavirus disease 2019 (COVID-19). The immune system’s response plays a significant role in viral progression. This article [...] Read more.
A severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection can lead to morbidity and mortality. SARS-CoV-2 infects the epithelial cells of the respiratory tract and causes coronavirus disease 2019 (COVID-19). The immune system’s response plays a significant role in viral progression. This article develops and analyzes a system of partial differential equations (PDEs), which describe the in-host dynamics of SARS-CoV-2 under the effect of cytotoxic T-lymphocyte (CTL) and antibody immune responses. The model characterizes the interplay between six compartments, healthy epithelial cells (ECs), latent infected ECs, active infected ECs, free SARS-CoV-2 particles, CTLs, and antibodies. We consider the logistic growth of healthy ECs. We first investigate the properties of the model’s solutions, then, we calculate all steady states and determine the conditions of their existence and global stability. The global asymptotic stability is examined by constructing Lyapunov functions. The analytical findings are supported via numerical simulations. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
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22 pages, 4740 KiB  
Article
Nonlinear Hammerstein System Identification: A Novel Application of Marine Predator Optimization Using the Key Term Separation Technique
by Khizer Mehmood, Naveed Ishtiaq Chaudhary, Zeshan Aslam Khan, Khalid Mehmood Cheema, Muhammad Asif Zahoor Raja, Ahmad H. Milyani and Abdullah Ahmed Azhari
Mathematics 2022, 10(22), 4217; https://0-doi-org.brum.beds.ac.uk/10.3390/math10224217 - 11 Nov 2022
Cited by 15 | Viewed by 1111
Abstract
The mathematical modelling and optimization of nonlinear problems arising in diversified engineering applications is an area of great interest. The Hammerstein structure is widely used in the modelling of various nonlinear processes found in a range of applications. This study investigates the parameter [...] Read more.
The mathematical modelling and optimization of nonlinear problems arising in diversified engineering applications is an area of great interest. The Hammerstein structure is widely used in the modelling of various nonlinear processes found in a range of applications. This study investigates the parameter optimization of the nonlinear Hammerstein model using the abilities of the marine predator algorithm (MPA) and the key term separation technique. MPA is a population-based metaheuristic inspired by the behavior of predators for catching prey, and utilizes Brownian/Levy movement for predicting the optimal interaction between predator and prey. A detailed analysis of MPA is conducted to verify the accurate and robust behavior of the optimization scheme for nonlinear Hammerstein model identification. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
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20 pages, 7094 KiB  
Article
A Numerical Intuition of Activation Energy in Transient Micropolar Nanofluid Flow Configured by an Exponentially Extended Plat Surface with Thermal Radiation Effects
by Zeeshan, N. Ameer Ahammad, Haroon Ur Rasheed, Ahmed A. El-Deeb, Barakah Almarri and Nehad Ali Shah
Mathematics 2022, 10(21), 4046; https://0-doi-org.brum.beds.ac.uk/10.3390/math10214046 - 31 Oct 2022
Cited by 16 | Viewed by 1373
Abstract
In recent times, heat and mass transportation have had some of the most recognized and attractive research areas in computational fluid dynamics. It is useful in the modeling of the flow of nuclear reactors, bioinformatics, the medical discipline, etc. Driven by the execution [...] Read more.
In recent times, heat and mass transportation have had some of the most recognized and attractive research areas in computational fluid dynamics. It is useful in the modeling of the flow of nuclear reactors, bioinformatics, the medical discipline, etc. Driven by the execution of the flow in the manufacturing application, the goal of the present analysis is to explore the novel effect of micropolar fluid configured by an exponentially elongated sheet positioned horizontally in a porous channel. The impact of activation energy, internal heating, and heat and mass transfer features are integrated into the revised flow model. A mathematical framework for different flow fields is developed in order to highlight the significant aspects of the thermal and concentration slip effects evaluated on the extended plat surface, with the aid of appropriate transformation factors to diminish the nonlinear fundamental flow equations (PDEs) to a system of (ODEs). Precise numerical treatment for a wide range of pertinent parameters is adopted to solve the nonlinear system through a built-in algorithm in the MATHEMATICA platform. The features of prominent emerging parameters against various flow fields are viewed and addressed through plotted visuals. The influence of the factors on skin friction, heat, and mass coefficients offered through 3D animation is evaluated. The temperature profile improves with ascending values of Brownian parameter and thermophoretic diffusion force but diminishes with subject expansions in Prandtl number and thermal slip parameter. It has been noticed that the concentration outlines increase for reaction rate and activation energy parameters but dwindle for expending values of porosity parameter, Lewis number, and concentration slip parameter. Skin fraction values increase due to the growing nature of the micropolar and second-grade fluid parameters. Nusselt numbers upsurge for increasing thermophoretic diffusion parameters while exhibiting a declining trend for Brownian motion parameters. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
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25 pages, 2232 KiB  
Article
Mathematical Modelling of Diffusion Flows in Two-Phase Stratified Bodies with Randomly Disposed Layers of Stochastically Set Thickness
by Olha Chernukha, Anastasiia Chuchvara, Yurii Bilushchak, Petro Pukach and Natalia Kryvinska
Mathematics 2022, 10(19), 3650; https://0-doi-org.brum.beds.ac.uk/10.3390/math10193650 - 05 Oct 2022
Cited by 3 | Viewed by 1159
Abstract
The work is dedicated to mathematical modelling of random diffusion flows of admixture particles in a two-phase stratified strip with stochastic disposition of phases and random thickness of inclusion-layers. The study of such models are especially important during the creation of composite layered [...] Read more.
The work is dedicated to mathematical modelling of random diffusion flows of admixture particles in a two-phase stratified strip with stochastic disposition of phases and random thickness of inclusion-layers. The study of such models are especially important during the creation of composite layered materials, in the research of the transmission properties of filters, and in the prediction of the spread of pollutants in the environment. Within the model we consider one case of uniform distribution of coordinates of upper boundaries of the layers of which the body is made up and two more cases, i.e., of uniform and triangular distributions of the inclusion thickness. The initial-boundary value problems of diffusion are formulated for flux functions; the boundary conditions at one of the body’s surfaces are set for flux and, at the other boundary, the conditions are given for admixture concentration; the initial condition being concerned with zero and non-zero constant initial concentrations. An equivalent integro-differential equation is constructed. Its solution is found in terms of Neumann series. For the first time it was obtained calculation formulae for diffusion flux averaged over the ensemble of phase configurations and over the inclusion thickness. It allowed to investigate the dependence of averaged diffusion fluxes on the medium’s characteristics on the basis of the developed software. The simulation of averaged fluxes of admixture in multilayered Fe−Cu and αFe−Ni materials is made. Comparative analysis of solutions, depending on the stage of averaging procedure over thickness, is carried out. It is shown that for some values of parameters the stage of averaging procedure over thickness has almost no effect on the diffusion flow value. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
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16 pages, 6386 KiB  
Article
Numerical Analysis of Building Cooling Using New Passive Downdraught Evaporative Tower Configuration in an Arid Climate
by Mohammad Abdullah Alshenaifi, Abdelhakim Mesloub, Walid Hassen, Mohammed Awad Abuhussain and Lioua Kolsi
Mathematics 2022, 10(19), 3616; https://0-doi-org.brum.beds.ac.uk/10.3390/math10193616 - 02 Oct 2022
Cited by 4 | Viewed by 1295
Abstract
Building energy consumption in hot arid climates is dominated by air conditioning use. Therefore, using passive cooling methods could reduce this demand, improve resource efficiency, and decrease carbon emissions. In this study, an innovative configuration of a passive downdraught evaporative cooling (PDEC) tower [...] Read more.
Building energy consumption in hot arid climates is dominated by air conditioning use. Therefore, using passive cooling methods could reduce this demand, improve resource efficiency, and decrease carbon emissions. In this study, an innovative configuration of a passive downdraught evaporative cooling (PDEC) tower is investigated numerically. The governing equations are solved using the finite element method (FEM), and the effects of inlet velocity (0.5 m·s−1 ≤ uin ≤ 3 m·s−1) and temperature (35 °C ≤ Tin ≤ 45 °C) on the fluid structure, temperature field, and relative humidity are studied for three cases related to the position of the air outlet. The flow is considered as turbulent, and the building walls and the tower are assumed to be thermally well insulated. The PDEC tower is equipped with two vertical isotropic saturated porous layers. The results revealed that the inlet velocity and temperature play an essential role in the quality of the indoor temperature. In fact, the temperature can be reduced by about 7 degrees, and the relative humidity can be enhanced by 9% for lower inlet velocities. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
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20 pages, 8116 KiB  
Article
Effects of Seat Belts and Shock Absorbers on the Safety of Racing Car Drivers
by Calin Itu, Ana Toderita, Lucia-Violeta Melnic and Sorin Vlase
Mathematics 2022, 10(19), 3593; https://0-doi-org.brum.beds.ac.uk/10.3390/math10193593 - 01 Oct 2022
Cited by 5 | Viewed by 2159
Abstract
This paper aimed to study the behavior of a body (dummy) that was in a race car in the event of a frontal collision with a wall in order to see what loads were acting on the dummy. Based on a complex car [...] Read more.
This paper aimed to study the behavior of a body (dummy) that was in a race car in the event of a frontal collision with a wall in order to see what loads were acting on the dummy. Based on a complex car model, equipped with two safety system seat belts and a shock absorption system, the behavior of the dummy was obtained following frontal collision of the car–dummy assembly. The accelerations were obtained at different points of the dummy’s body and the force that appeared on the seat belts were determined. The Gibbs–Appell method was used to assess the response of the system based on the equations of motion in a problem involving shocks. This paper demonstrates that the revisited old principle of mechanics can offer an interesting and convenient means to obtain results in a short time. FEM and Altair Hyperworks software II was used to model the system. It can be used to determine whether a seat belt is able to work if it has defects during use, such as scratches, cigarette burns or animal bites. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
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21 pages, 24947 KiB  
Article
Visualization in Mathematical Packages When Teaching with Information Technologies
by Valery Ochkov, Inna Vasileva, Konstantin Orlov, Julia Chudova and Anton Tikhonov
Mathematics 2022, 10(19), 3413; https://0-doi-org.brum.beds.ac.uk/10.3390/math10193413 - 20 Sep 2022
Cited by 3 | Viewed by 1771
Abstract
A method has been obtained for the use of visualization in computer mathematical packages, which is an effective means of overcoming difficult situations that arise for students when mastering such packages and solving computational problems. Depending on the complexity of the problem being [...] Read more.
A method has been obtained for the use of visualization in computer mathematical packages, which is an effective means of overcoming difficult situations that arise for students when mastering such packages and solving computational problems. Depending on the complexity of the problem being solved, either the teacher or the students themselves can create special visual graphic (animation) objects. Such objects allow, initially without going into the intricacies of the functioning of the package and the mathematical apparatus used, to competently describe a complete picture of a difficult situation for students and indicate ways to resolve it. The method is considered through the example of the process of solving systems of equations using the mathematical package Mathcad and the WolframAlpha online resource. Graphical and animated objects are presented that clearly demonstrate the areas of the location of initial approximations, allowing you to numerically obtain all the real roots of systems of trigonometric and nonlinear equations. Similar objects are built to find the critical points of the Himmelblau’s special test function. Visualization materials are confirmed by the presented computational calculations. The proposed method is implemented in the form of plans for lectures and practical classes on mathematical modeling using computer technologies. The method was tested with university students at the National Research University Moscow Power Engineering Institute. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
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18 pages, 5624 KiB  
Article
Morphology of Hybrid MHD Nanofluid Flow through Orthogonal Coaxial Porous Disks
by Qadeer Raza, M. Zubair Akbar Qureshi, Bagh Ali, Ahmed Kadhim Hussein, Behzad Ali Khan, Nehad Ali Shah and Wajaree Weera
Mathematics 2022, 10(18), 3280; https://0-doi-org.brum.beds.ac.uk/10.3390/math10183280 - 09 Sep 2022
Cited by 8 | Viewed by 1354
Abstract
In this article, we study the novel features of morphological effects for hybrid nanofluid flow subject to expanding/contracting geometry. The nanoparticles are incorporated due to their extraordinary thermal conductivity and innovative work for hybrid nanofluids, which are assembled of aluminum oxides, Al2 [...] Read more.
In this article, we study the novel features of morphological effects for hybrid nanofluid flow subject to expanding/contracting geometry. The nanoparticles are incorporated due to their extraordinary thermal conductivity and innovative work for hybrid nanofluids, which are assembled of aluminum oxides, Al2O3 metallic oxides, and metallic copper Cu. Cu nanoparticles demonstrate very strong catalytic activity, while Al2O3 nanoparticles perform well as an electrical insulator. The governing partial differential equations of the elaborated model are transformed into a system of nonlinear ordinary differential equations with the use of similarity variables, and these equations are numerically solved through a shooting technique based on the Runge–Kutta method. We develop a hybrid correlation for thermophysical properties based on a single-phase approach. A favorable comparison between shape and size factors for metallic and metallic-oxide nanoparticles is discussed via tables and figures. Moreover, the effect of embedding flow factors on concentration, velocity, and temperature is shaped in line with parametric studies, such as the permeable Reynolds number, nanoparticle volume fractions, and expansion/contraction parameters. The fluid velocity, temperature, and concentration are demonstrated in the presence of hybrid nanoparticles and are discussed in detail, while physical parameters such as the shear stress, flow of heat, and mass transfer at the lower and upper disks are demonstrated in a table. The hybrid nanoparticles show significant results as compared to the nanofluids. If we increase the nanoparticle volume fraction, this increases the thermal performance for an injection/suction case as well. The above collaborative research provides a strong foundation in the field of biomedical equipment and for the development of nanotechnology-oriented computers. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
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15 pages, 4608 KiB  
Article
Modeling and Optimization of Triple Diode Model of Dye-Sensitized Solar Panel Using Heterogeneous Marine Predators Algorithm
by Alaa A. Zaky, Hesham Alhumade, Dalia Yousri, Ahmed Fathy, Hegazy Rezk, Lida Givalou and Polycarpos Falaras
Mathematics 2022, 10(17), 3143; https://0-doi-org.brum.beds.ac.uk/10.3390/math10173143 - 01 Sep 2022
Cited by 4 | Viewed by 1492
Abstract
The reliable mathematical model construction of dye-sensitized solar cells (DSSCs) using the triple-diode model (TDM) is proposed but it is a challenge due to its complexity. This work implements a novel method incorporating a recent meta-heuristic optimizer called the heterogeneous marine predators algorithm [...] Read more.
The reliable mathematical model construction of dye-sensitized solar cells (DSSCs) using the triple-diode model (TDM) is proposed but it is a challenge due to its complexity. This work implements a novel method incorporating a recent meta-heuristic optimizer called the heterogeneous marine predators algorithm (H-MPA) to identify the nine parameters of the triple-diode equivalent circuit of DSSCs. In the optimization procedure, the nine unknown parameters of TDM are employed as decision variables, but the objective function to be minimized is the root mean square error (RMSE) between the experimental data and the estimated data. To prove the superiority of the H-MPA, the obtained results are compared with the slime mold algorithm (SMA), Transient search optimizer (TSO), Manta-Ray Foraging Optimization algorithm (MRFO), Forensic-Based Investigation (FBI), Equilibrium optimizer (EO), and Artificial ecosystem-based optimizer. The primary findings demonstrated the superiority of the proposed strategy in building a consistent model of the triple-diode model of DSSCs. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
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20 pages, 304 KiB  
Article
Linear Quadratic Optimal Control Problem for Linear Stochastic Generalized System in Hilbert Spaces
by Zhaoqiang Ge
Mathematics 2022, 10(17), 3118; https://0-doi-org.brum.beds.ac.uk/10.3390/math10173118 - 30 Aug 2022
Viewed by 913
Abstract
A finite-horizon linear stochastic quadratic optimal control problem is investigated by the GE-evolution operator in the sense of the mild solution in Hilbert spaces. We assume that the coefficient operator of the differential term is a bounded linear operator and that the state [...] Read more.
A finite-horizon linear stochastic quadratic optimal control problem is investigated by the GE-evolution operator in the sense of the mild solution in Hilbert spaces. We assume that the coefficient operator of the differential term is a bounded linear operator and that the state and input operators are time-varying in the dynamic equation of the problem. Optimal state feedback along with the well-posedness of the generalized Riccati equation is obtained for the finite-horizon case. The results are also applicable to the linear quadratic optimal control problem of ordinary time-varying linear stochastic systems. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
19 pages, 7040 KiB  
Article
Performance Comparison of Numerical Methods in a Predictive Controller for an AC–DC Power Converter
by Jazmin Ramirez-Hernandez, Oswaldo Ulises Juarez-Sandoval, Leobardo Hernandez-Gonzalez, Domingo Cortes, Juan C. Sanchez-Garcia and Pedro Guevara-Lopez
Mathematics 2022, 10(15), 2818; https://0-doi-org.brum.beds.ac.uk/10.3390/math10152818 - 08 Aug 2022
Cited by 1 | Viewed by 1726
Abstract
The use of model-based predictive control in power converters has substantially increased in recent years. This control technique always needs a discrete system model to be implemented. There are several methods to obtain a discrete model; in this paper, all common methods are [...] Read more.
The use of model-based predictive control in power converters has substantially increased in recent years. This control technique always needs a discrete system model to be implemented. There are several methods to obtain a discrete model; in this paper, all common methods are examined from a practical point of view. Their precision, simplicity, and implementation requirements are analyzed to establish their advantages and disadvantages. From this analysis, it is shown that different discretization methods result in different closed-loop converter performance. A model-based predictive control AC–DC converter is used to show that different discretization procedures result in different total harmonic distortion. For this evaluation, a simulation of a 1 kW three-phase active rectifier was performed in Matlab-Simulink. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
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19 pages, 7984 KiB  
Article
Dimensionless Characterization to Estimate Horizontal Groundwater Velocity from Temperature–Depth Profiles in Aquifers
by José Antonio Jiménez-Valera and Francisco Alhama
Mathematics 2022, 10(15), 2717; https://0-doi-org.brum.beds.ac.uk/10.3390/math10152717 - 01 Aug 2022
Cited by 7 | Viewed by 1010
Abstract
The outcome of a dimensionless characterization study in a two-dimensional porous media domain in which groundwater flows at a constant horizontal velocity is presented in this report. Using spatial discrimination, the dimensionless groups that govern the solution patterns are determined from dimensionless governing [...] Read more.
The outcome of a dimensionless characterization study in a two-dimensional porous media domain in which groundwater flows at a constant horizontal velocity is presented in this report. Using spatial discrimination, the dimensionless groups that govern the solution patterns are determined from dimensionless governing equations. As a boundary condition on the surface, the case of constant temperature is studied. From the mathematical deduction of the groups, a characteristic horizontal length emerges. This length determines the region in which temperature–depth profiles are affected by flow. Existing analytical solutions have been shown to be invalid due to the severe assumption that the horizontal thermal gradient has a constant value. Therefore, universal solutions based on pi theorem have been obtained for the characteristic horizontal length, temperature field, temperature–depth profiles and horizontal temperature profiles. Dependencies between dimensionless groups have been depicted by universal curves, abacuses and surfaces. These graphical solutions are used in an easy way to estimate groundwater velocity from experimental temperature measurements in the form of an inverse problem. In addition, an easy and fast protocol for estimating fluid flow velocity and groundwater inlet temperature from temperature profile measurements is proposed. This protocol is applied in a scenario of groundwater discharge from a quaternary aquifer to a salty lagoon located in the southeast of Spain. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
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13 pages, 1045 KiB  
Article
A Game for Learning How to Model in Graph Theory
by Alicia Cordero, Cristina Jordan, Marina Murillo-Arcila and Esther Sanabria-Codesal
Mathematics 2022, 10(12), 1969; https://0-doi-org.brum.beds.ac.uk/10.3390/math10121969 - 07 Jun 2022
Viewed by 1719
Abstract
In this article, we show how to introduce students to modeling while exposing the power of graph theory as a modeling tool. For that purpose, we propose a problem aimed at university students based on a game where the objective is to strengthen [...] Read more.
In this article, we show how to introduce students to modeling while exposing the power of graph theory as a modeling tool. For that purpose, we propose a problem aimed at university students based on a game where the objective is to strengthen the learning of reachability and the shortest path algorithms. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
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18 pages, 29860 KiB  
Article
Design of a Thermal Measurement System with Vandal Protection Used for the Characterization of New Asphalt Pavements through Discriminated Dimensionless Analysis
by Juan Francisco Sánchez-Pérez, Gloria Motos-Cascales, Manuel Conesa, Francisco Moral-Moreno, Enrique Castro and Gonzalo García-Ros
Mathematics 2022, 10(11), 1924; https://0-doi-org.brum.beds.ac.uk/10.3390/math10111924 - 03 Jun 2022
Cited by 2 | Viewed by 1541
Abstract
This work focuses on the protection of measurement sensors against accidents, vandalism, or theft and on the improvement of the data collected due to the interference produced by these protections. These sensors are part of a larger study, within the framework of a [...] Read more.
This work focuses on the protection of measurement sensors against accidents, vandalism, or theft and on the improvement of the data collected due to the interference produced by these protections. These sensors are part of a larger study, within the framework of a LIFE Heatland project, carried out in a Spanish city, Murcia, with the fundamental objective of minimizing the urban heat island effect using pavements with lower solar energy storage than traditional ones. The study presented here has been carried out through the implementation of aluminum tubes that protect the sensors installed in the street. Once the problem of sensor protection had been solved, the problem of thermal interference in the measurements due to overheating inside the tubes had to be overcome by means of discriminated dimensionless analysis techniques, focusing on heat transfer by convection of the air flow in the inner part of the tube, by finding the most suitable size and materials to complement the outer aluminum coating. In particular, the search for the critical radius of the tubes was essential since it allowed the insulator size to be optimized. Derived from the study carried out to avoid the overheating of the tube, a small part was covered with a dark material and holes were made to improve air circulation inside the tube, allowing adequate measurement results to be obtained. Finally, the results showed that the designed device was suitable for temperature measurement, since small variations were observed with respect to the control device. Full article
(This article belongs to the Special Issue Mathematical Modelling, Nonlinear Optimization, Nondimensionalization and Engineering Applications)
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