Mathematics

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24 pages, 3340 KiB

Open AccessArticle

Conservative Finite-Difference Scheme for 1D Ginzburg–Landau Equation

by Vyacheslav Trofimov, Maria Loginova, Mikhail Fedotov, Daniil Tikhvinskii, Yongqiang Yang and Boyuan Zheng

Mathematics 2022, 10(11), 1912; https://doi.org/10.3390/math10111912 - 2 Jun 2022

Viewed by 2196

Abstract

In this study, our attention is focused on deriving integrals of motion (conservation laws; invariants) for the problem of an optical pulse propagation in an optical fiber containing an optical amplifier or attenuator because, to date, such invariants are absent in the literature. [...] Read more.

In this study, our attention is focused on deriving integrals of motion (conservation laws; invariants) for the problem of an optical pulse propagation in an optical fiber containing an optical amplifier or attenuator because, to date, such invariants are absent in the literature. The knowledge of a problem’s invariants allows us develop finite-difference schemes possessing the conservativeness property, which is crucial for solving nonlinear problems. Laser pulse propagation is governed by the nonlinear Ginzburg–Landau equation. Firstly, the problem’s conservation laws are developed for the various parameters’ relations: for a linear case, for a nonlinear case without considering the linear absorption, and for a nonlinear case accounting for the linear absorption and homogeneous shift of the pulse’s phase. Hereafter, the Crank–Nicolson-type scheme is constructed for the problem difference approximation. To demonstrate the conservativeness of the constructed implicit finite-difference scheme in the sense of preserving difference analogs of the problem’s invariants, the corresponding theorems are formulated and proved. The problem of the finite-difference scheme’s nonlinearity is solved by means of an iterative process. Finally, several numerical examples are presented to support the theoretical results. Full article

(This article belongs to the Special Issue Mathematical Models and Methods in Engineering and Social Sciences)

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11 pages, 294 KiB

Open AccessArticle

Maximal Regularity Estimates and the Solvability of Nonlinear Differential Equations

by Myrzagali Ospanov and Kordan Ospanov

Mathematics 2022, 10(10), 1717; https://doi.org/10.3390/math10101717 - 17 May 2022

Cited by 2 | Viewed by 1434

Abstract

We study a type of third-order linear differential equations with variable and unbounded coefficients, which are defined in an infinite interval. We also consider a non-linear generalization with coefficients that depends on an unknown function. We establish sufficient conditions for the correctness of [...] Read more.

We study a type of third-order linear differential equations with variable and unbounded coefficients, which are defined in an infinite interval. We also consider a non-linear generalization with coefficients that depends on an unknown function. We establish sufficient conditions for the correctness of this linear equation and the maximal regularity estimate for their solution. Using these results, we prove the solvability of a nonlinear differential equation and estimate the norms of its terms. Full article

(This article belongs to the Special Issue Mathematical Models and Methods in Engineering and Social Sciences)

15 pages, 3674 KiB

Open AccessArticle

Minimizing Dependency Ratio in Spain through Demographic Variables

by Joan C. Micó, David Soler, Maria T. Sanz, Antonio Caselles and Salvador Amigó

Mathematics 2022, 10(9), 1471; https://doi.org/10.3390/math10091471 - 27 Apr 2022

Cited by 1 | Viewed by 1952

Abstract

The population pyramids in Europe have changed in the last decades, particularly in Spain, where population aging is observed and implies an increase in the dependency ratio. The present study aims to modify a mathematical model of population defined by age and sex [...] Read more.

The population pyramids in Europe have changed in the last decades, particularly in Spain, where population aging is observed and implies an increase in the dependency ratio. The present study aims to modify a mathematical model of population defined by age and sex based on the von Foerster-Mckendrick equations in order to have birth and migration rates as control variables. To achieve this objective, the necessary mathematical changes are made in the model, and the new mathematical model is verified and validated for the case of Spain, in the period 2008–2019, being considered successful both in its deterministic and stochastic formulation. Finally, through the method of strategies and scenarios and a genetic algorithm, a modification of the population pyramid is obtained in the direction of a decrease in the dependency ratio, increasing the birth rate and immigration and decreasing the emigration rate. These changes lead to a modification of the population pyramid in the year 2040; in particular, an increase of the female population in the age range from 20 to 39 years old. Full article

(This article belongs to the Special Issue Mathematical Models and Methods in Engineering and Social Sciences)

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12 pages, 362 KiB

Open AccessArticle

Modeling Immigration in Spain: Causes, Size and Consequences

by Sheila Torres, Rafael Company and Lucas Jódar

Mathematics 2022, 10(9), 1371; https://doi.org/10.3390/math10091371 - 20 Apr 2022

Cited by 1 | Viewed by 2080

Abstract

This paper deals with the construction of a discrete dynamic population model addressed to estimate the expected size of the immigration population in a finite short period of time in Spain. By paying attention to a special subpopulation of interest, such as an [...] Read more.

This paper deals with the construction of a discrete dynamic population model addressed to estimate the expected size of the immigration population in a finite short period of time in Spain. By paying attention to a special subpopulation of interest, such as an irregular immigrant, unaccompanied minor immigrant and regular immigrant, a vector discrete population model is built after the discussion and introduction of proper hypotheses linked to economy, host and country of origin regulation policies, political interest and others. The model allows us to study the change of the results under variation of the parameters. Full article

(This article belongs to the Special Issue Mathematical Models and Methods in Engineering and Social Sciences)

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20 pages, 1490 KiB

Open AccessArticle

Optimization of the Cognitive Processes in a Virtual Classroom: A Multi-objective Integer Linear Programming Approach

by María Luisa Nolé, David Soler, Juan Luis Higuera-Trujillo and Carmen Llinares

Mathematics 2022, 10(7), 1184; https://doi.org/10.3390/math10071184 - 5 Apr 2022

Cited by 2 | Viewed by 2332

Abstract

A fundamental problem in the design of a classroom is to identify what characteristics it should have in order to optimize learning. This is a complex problem because learning is a construct related to several cognitive processes. The aim of this study is [...] Read more.

A fundamental problem in the design of a classroom is to identify what characteristics it should have in order to optimize learning. This is a complex problem because learning is a construct related to several cognitive processes. The aim of this study is to maximize learning, represented by the processes of attention, memory, and preference, depending on six classroom parameters: height, width, color hue, color saturation, color temperature, and illuminance. Multi-objective integer linear programming with three objective functions and 56 binary variables was used to solve this optimization problem. Virtual reality tools were used to gather the data; novel software was used to create variations of virtual classrooms for a sample of 112 students. Using an interactive method, more than 4700 integer linear programming problems were optimally solved to obtain 13 efficient solutions to the multi-objective problem, which allowed the decision maker to analyze all the information and make a final choice. The results showed that achieving the best cognitive processing performance involves using different classroom configurations. The use of a multi-objective interactive approach is interesting because in human behavioral studies, it is important to consider the judgement of an expert in order to make decisions. Full article

(This article belongs to the Special Issue Mathematical Models and Methods in Engineering and Social Sciences)

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12 pages, 2181 KiB

Open AccessArticle

Border Irrigation Modeling with the Barré de Saint-Venant and Green and Ampt Equations

by Sebastián Fuentes, Carlos Fuentes, Heber Saucedo and Carlos Chávez

Mathematics 2022, 10(7), 1039; https://doi.org/10.3390/math10071039 - 24 Mar 2022

Cited by 5 | Viewed by 3188

Abstract

In gravity irrigation, how water is distributed in the soil profile makes it necessary to study and develop methodologies to model the process of water infiltration and redistribution. In this work, a model is shown to simulate the advancing front in border irrigation [...] Read more.

In gravity irrigation, how water is distributed in the soil profile makes it necessary to study and develop methodologies to model the process of water infiltration and redistribution. In this work, a model is shown to simulate the advancing front in border irrigation based on the one dimensional equations of Barré de Saint-Venant for the surface flow and the equation of Green and Ampt for the flow in a porous medium. The solutions were obtained numerically using a finite difference Lagrangian scheme for the surface flow and the Raphson method for the subsurface flow. The model was validated with data obtained from the literature from an irrigation test and its predictive capacity was compared with another model and showed excellent results. The hydrodynamic parameters of the soil, necessary to obtain the optimal irrigation discharge, were obtained through the solution of the inverse problem using the Levenberg–Marquardt optimization algorithm. Finally, the results found here allow us to recommend that this model be used to design and model border irrigation, since the infiltration equation uses characteristic parameters of the physical soil. Full article

(This article belongs to the Special Issue Mathematical Models and Methods in Engineering and Social Sciences)

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18 pages, 311 KiB

Open AccessArticle

Mathematical Modeling of the Financial Impact of Air Crashes on Airlines and Involved Manufacturers

by Maria Ángeles Alcaide, Alberto Celani, Paula Cervera Chasan and Elena De La Poza

Mathematics 2022, 10(5), 715; https://doi.org/10.3390/math10050715 - 24 Feb 2022

Viewed by 2439

Abstract

Despite air transport being the safest way to travel, accidents still happen, which incur massive costs and many consequences for industry and society. The main objective of this research is to determine the financial impact of air crashes by distinguishing between fatal and [...] Read more.

Despite air transport being the safest way to travel, accidents still happen, which incur massive costs and many consequences for industry and society. The main objective of this research is to determine the financial impact of air crashes by distinguishing between fatal and non-fatal events and their effect on the market stock price of the involved companies of airlines and manufacturers. This study also aims to contribute to the literature about the Event Study Methodology by determining which model of the two most employed in this methodology (Market model or Fama-French model) provides more accurate results. The results reveal that the companies harmed when an air crash occurs include the involved airline, regardless of the causes of the crash if it was a fatal event. However, with non-fatal events, the impact on airlines differs depending on the event’s outcome. In any case, effects are immediate, especially on the same day the event occurred. Nevertheless, manufacturing firms show no negative abnormal returns after an air crash. Finally, the Market model is more accurate in this study. These results are important for investors since they show mistrust in air transport and losses only occur in the airline involved, especially if the accident is fatal. In turn, our results provide reassurance to investors in manufacturing companies in the event of such an occurrence. In any case, this study has shown that both airlines and manufacturers must continue to promote and improve safety. Full article

(This article belongs to the Special Issue Mathematical Models and Methods in Engineering and Social Sciences)

17 pages, 817 KiB

Open AccessArticle

Delays in Plant Virus Models and Their Stability

by Benito Chen-Charpentier

Mathematics 2022, 10(4), 603; https://doi.org/10.3390/math10040603 - 16 Feb 2022

Cited by 4 | Viewed by 2433

Abstract

Viruses infect humans and animals but also infect plants and cause great economic and ecological damage. In most cases, the virus is transmitted by a vector. After being bitten by an infected vector, the virus takes some time to replicate and spread in [...] Read more.

Viruses infect humans and animals but also infect plants and cause great economic and ecological damage. In most cases, the virus is transmitted by a vector. After being bitten by an infected vector, the virus takes some time to replicate and spread in the plant. We present two models of the spread of viruses in plants based on ordinary differential equations, and then add either a delay or an exposed plant population. We study two ways of adding the delay. In the first one, a plant infected by a vector changes from susceptible to infective after a time equal to the delay. In the second one, immediately after the contact between a susceptible plant and infective vector, the plant is no longer susceptible, but it takes time equal to the delay for it to turn infective. To better explain the two ways of incorporating the delays, we first introduce them in a simple SIRS model. We analyze the models and study their stability numerically. We conclude by studying the interactions and the conservation of the total plant population that the first way of introducing the delay is better justified. Full article

(This article belongs to the Special Issue Mathematical Models and Methods in Engineering and Social Sciences)

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20 pages, 1615 KiB

Open AccessArticle

Mathematical Modeling of Toxoplasmosis Considering a Time Delay in the Infectivity of Oocysts

by Gilberto González-Parra, Sharmin Sultana and Abraham J. Arenas

Mathematics 2022, 10(3), 354; https://doi.org/10.3390/math10030354 - 24 Jan 2022

Cited by 10 | Viewed by 2809

Abstract

In this paper, we study the effect of the introduction of a time delay on the dynamics of toxoplasmosis. This time delay is the elapsed time from when oocysts become present in the environment and when they become infectious. We construct a mathematical [...] Read more.

In this paper, we study the effect of the introduction of a time delay on the dynamics of toxoplasmosis. This time delay is the elapsed time from when oocysts become present in the environment and when they become infectious. We construct a mathematical model that includes cats and oocysts in the environment. We include the effect of oocysts, since they are crucial for the dynamics of toxoplasmosis. The likelihood of the acquisition of Toxoplasma gondii infection depends on the environmental load of the parasite. Furthermore, the model considers the possibility of vaccination of the feline host. In the mathematical model, we consider directly the infection of cats through the oocysts shed by other cats. We prove that the basic reproduction number

R_{0}

is a secondary parameter that determines the global dynamics of toxoplasmosis in cat populations. We study the effect of the time delay on the stability of the steady states. We find that the time delay cannot change the stability of the endemic state, which is an important result from the biological point of view. Numerical simulations are performed to support the theoretical results and obtain further insight into understanding toxoplasmosis dynamics in cat populations. Full article

(This article belongs to the Special Issue Mathematical Models and Methods in Engineering and Social Sciences)

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22 pages, 551 KiB

Open AccessArticle

Accurate Estimations of Any Eigenpairs of N-th Order Linear Boundary Value Problems

by Pedro Almenar and Lucas Jódar

Mathematics 2021, 9(21), 2663; https://doi.org/10.3390/math9212663 - 21 Oct 2021

Cited by 1 | Viewed by 1691

Abstract

This paper provides a method to bound and calculate any eigenvalues and eigenfunctions of n-th order boundary value problems with sign-regular kernels subject to two-point boundary conditions. The method is based on the selection of a particular type of cone for each [...] Read more.

This paper provides a method to bound and calculate any eigenvalues and eigenfunctions of n-th order boundary value problems with sign-regular kernels subject to two-point boundary conditions. The method is based on the selection of a particular type of cone for each eigenpair to be determined, the recursive application of the operator associated to the equivalent integral problem to functions belonging to such a cone, and the calculation of the Collatz–Wielandt numbers of the resulting functions. Full article

(This article belongs to the Special Issue Mathematical Models and Methods in Engineering and Social Sciences)

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18 pages, 414 KiB

Open AccessArticle

On the Approximated Solution of a Special Type of Nonlinear Third-Order Matrix Ordinary Differential Problem

by Emilio Defez, Javier Ibáñez, José M. Alonso, Michael M. Tung and Teresa Real-Herráiz

Mathematics 2021, 9(18), 2262; https://doi.org/10.3390/math9182262 - 15 Sep 2021

Cited by 1 | Viewed by 2099

Abstract

Matrix differential equations are at the heart of many science and engineering problems. In this paper, a procedure based on higher-order matrix splines is proposed to provide the approximated numerical solution of special nonlinear third-order matrix differential equations, having the form [...] Read more.

Matrix differential equations are at the heart of many science and engineering problems. In this paper, a procedure based on higher-order matrix splines is proposed to provide the approximated numerical solution of special nonlinear third-order matrix differential equations, having the form

Y^{(3)} (x) = f (x, Y (x))

. Some numerical test problems are also included, whose solutions are computed by our method. Full article

(This article belongs to the Special Issue Mathematical Models and Methods in Engineering and Social Sciences)

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11 pages, 582 KiB

Open AccessArticle

Metamaterial Acoustics on the (2 + 1)D Einstein Cylinder

by Michael M. Tung

Mathematics 2021, 9(17), 2079; https://doi.org/10.3390/math9172079 - 28 Aug 2021

Viewed by 1840

Abstract

The Einstein cylinder is the first cosmological model for our universe in modern history. Its geometry not only describes a static universe—a universe being invariant under time reversal—but it is also the prototype for a maximally symmetric spacetime with constant positive curvature. As [...] Read more.

The Einstein cylinder is the first cosmological model for our universe in modern history. Its geometry not only describes a static universe—a universe being invariant under time reversal—but it is also the prototype for a maximally symmetric spacetime with constant positive curvature. As such, it is still of crucial importance in numerous areas of physics and engineering, offering a fruitful playground for simulations and new theories. Here, we focus on the implementation and simulation of acoustic wave propagation on the Einstein cylinder. Engineering such an extraordinary device is the territory of metamaterial science, and we will propose an appropriate tuning of the relevant acoustic parameters in such a way as to mimic the geometric properties of this spacetime in acoustic space. Moreover, for probing such a space, we derive the corresponding wave equation from a variational principle for the underlying curved spacetime manifold and examine some of its solutions. In particular, fully analytical results are obtained for concentric wave propagation. We present predictions for this case and thereby investigate the most significant features of this spacetime. Finally, we produce simulation results for a more sophisticated test model which can only be tackled numerically. Full article

(This article belongs to the Special Issue Mathematical Models and Methods in Engineering and Social Sciences)

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19 pages, 635 KiB

Open AccessArticle

An Improved Taylor Algorithm for Computing the Matrix Logarithm

by Javier Ibáñez, Jorge Sastre, Pedro Ruiz, José M. Alonso and Emilio Defez

Mathematics 2021, 9(17), 2018; https://doi.org/10.3390/math9172018 - 24 Aug 2021

Cited by 4 | Viewed by 3115

Abstract

The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Padé approximation, sometimes accompanied by the Schur decomposition. In this work, we present a Taylor series algorithm, based on the [...] Read more.

The most popular method for computing the matrix logarithm is a combination of the inverse scaling and squaring method in conjunction with a Padé approximation, sometimes accompanied by the Schur decomposition. In this work, we present a Taylor series algorithm, based on the free-transformation approach of the inverse scaling and squaring technique, that uses recent matrix polynomial formulas for evaluating the Taylor approximation of the matrix logarithm more efficiently than the Paterson–Stockmeyer method. Two MATLAB implementations of this algorithm, related to relative forward or backward error analysis, were developed and compared with different state-of-the art MATLAB functions. Numerical tests showed that the new implementations are generally more accurate than the previously available codes, with an intermediate execution time among all the codes in comparison. Full article

(This article belongs to the Special Issue Mathematical Models and Methods in Engineering and Social Sciences)

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23 pages, 446 KiB

Open AccessArticle

Efficient Evaluation of Matrix Polynomials beyond the Paterson–Stockmeyer Method

by Jorge Sastre and Javier Ibáñez

Mathematics 2021, 9(14), 1600; https://doi.org/10.3390/math9141600 - 7 Jul 2021

Cited by 3 | Viewed by 3418

Abstract

Recently, two general methods for evaluating matrix polynomials requiring one matrix product less than the Paterson–Stockmeyer method were proposed, where the cost of evaluating a matrix polynomial is given asymptotically by the total number of matrix product evaluations. An analysis of the stability [...] Read more.

Recently, two general methods for evaluating matrix polynomials requiring one matrix product less than the Paterson–Stockmeyer method were proposed, where the cost of evaluating a matrix polynomial is given asymptotically by the total number of matrix product evaluations. An analysis of the stability of those methods was given and the methods have been applied to Taylor-based implementations for computing the exponential, the cosine and the hyperbolic tangent matrix functions. Moreover, a particular example for the evaluation of the matrix exponential Taylor approximation of degree 15 requiring four matrix products was given, whereas the maximum polynomial degree available using Paterson–Stockmeyer method with four matrix products is 9. Based on this example, a new family of methods for evaluating matrix polynomials more efficiently than the Paterson–Stockmeyer method was proposed, having the potential to achieve a much higher efficiency, i.e., requiring less matrix products for evaluating a matrix polynomial of certain degree, or increasing the available degree for the same cost. However, the difficulty of these family of methods lies in the calculation of the coefficients involved for the evaluation of general matrix polynomials and approximations. In this paper, we provide a general matrix polynomial evaluation method for evaluating matrix polynomials requiring two matrix products less than the Paterson-Stockmeyer method for degrees higher than 30. Moreover, we provide general methods for evaluating matrix polynomial approximations of degrees 15 and 21 with four and five matrix product evaluations, respectively, whereas the maximum available degrees for the same cost with the Paterson–Stockmeyer method are 9 and 12, respectively. Finally, practical examples for evaluating Taylor approximations of the matrix cosine and the matrix logarithm accurately and efficiently with these new methods are given. Full article

(This article belongs to the Special Issue Mathematical Models and Methods in Engineering and Social Sciences)

20 pages, 4156 KiB

Open AccessArticle

Energy and Personality: A Bridge between Physics and Psychology

by Antonio Caselles, Joan C. Micó and Salvador Amigó

Mathematics 2021, 9(12), 1339; https://doi.org/10.3390/math9121339 - 9 Jun 2021

Cited by 2 | Viewed by 2757

Abstract

The objective of this paper is to present a mathematical formalism that states a bridge between physics and psychology, concretely between analytical dynamics and personality theory, in order to open new insights in this theory. In this formalism, energy plays a central role. [...] Read more.

The objective of this paper is to present a mathematical formalism that states a bridge between physics and psychology, concretely between analytical dynamics and personality theory, in order to open new insights in this theory. In this formalism, energy plays a central role. First, the short-term personality dynamics can be measured by the General Factor of Personality (GFP) response to an arbitrary stimulus. This GFP dynamical response is modeled by a stimulus–response model: an integro-differential equation. The bridge between physics and psychology appears when the stimulus–response model can be formulated as a linear second order differential equation and, subsequently, reformulated as a Newtonian equation. This bridge is strengthened when the Newtonian equation is derived from a minimum action principle, obtaining the current Lagrangian and Hamiltonian functions. However, the Hamiltonian function is non-conserved energy. Then, some changes lead to a conserved Hamiltonian function: Ermakov–Lewis energy. This energy is presented, as well as the GFP dynamical response that can be derived from it. An application case is also presented: an experimental design in which 28 individuals consumed 26.51 g of alcohol. This experiment provides an ordinal scale for the Ermakov–Lewis energy that predicts the effect of a single dose of alcohol. Full article

(This article belongs to the Special Issue Mathematical Models and Methods in Engineering and Social Sciences)

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