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Mathematical Methods and Models in Nature and Society
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Mathematical Methods and Models in Nature and Society

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 March 2024) | Viewed by 5740

Special Issue Editors

Prof. Dr. Lucas Jódar

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Guest Editor
Institute for Multidisciplinary Mathematics, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain
Interests: mathematical modeling of human behavior; analytic and numerical methods for partial differential equations
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Rafael Company

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Guest Editor
Institute for Multidisciplinary Mathematics, Universitat Politècnica de València, Camino de Vera s/n, 46022 Valencia, Spain
Interests: numerical methods for partial differential equations; numerical analysis; mathematical finance; random differential models
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Benito Chen-Charpentier

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Guest Editor
Department of Mathematics, University of Texas at Arlington, Arlington, TX 76019-0408, USA
Interests: mathematical biology; numerical analysis; problems of fluid flow; random problems
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This volume deals with the novel high-quality research results of a wide class of mathematical models, with applications in engineering, nature, and social sciences. Analytical and numeric, deterministic and uncertain dimensions are treated. Complex and multidisciplinary models are included. Innovation and challenges are welcome. Among the examples of treated problems, we include problems related to finance, engineering, social sciences, physics, biology and politics. Novelty arises with respect to both the mathematical treatment of the problem and, from within a given mathematical problem, the treatment of the problem.

Prof. Dr. Lucas Jódar
Prof. Dr. Rafael Company
Prof. Dr. Benito Chen-Charpentier
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.

Keywords

  • mathematical modelling
  • numerical methods
  • random differential equations
  • optimization problems
  • engineering applications

Published Papers (4 papers)

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Research

19 pages, 481 KiB  
Article
Qualitative Numerical Analysis of a Free-Boundary Diffusive Logistic Model
by María Consuelo Casabán, Rafael Company, Vera N. Egorova and Lucas Jódar
Mathematics 2023, 11(6), 1296; https://doi.org/10.3390/math11061296 - 08 Mar 2023
Cited by 1 | Viewed by 953
Abstract
A two-dimensional free-boundary diffusive logistic model with radial symmetry is considered. This model is used in various fields to describe the dynamics of spreading in different media: fire propagation, spreading of population or biological invasions. Due to the radial symmetry, the free boundary [...] Read more.
A two-dimensional free-boundary diffusive logistic model with radial symmetry is considered. This model is used in various fields to describe the dynamics of spreading in different media: fire propagation, spreading of population or biological invasions. Due to the radial symmetry, the free boundary can be treated by a front-fixing approach resulting in a fixed-domain non-linear problem, which is solved by an explicit finite difference method. Qualitative numerical analysis establishes the stability, positivity and monotonicity conditions. Special attention is paid to the spreading–vanishing dichotomy and a numerical algorithm for the spreading–vanishing boundary is proposed. Theoretical statements are illustrated by numerical tests. Full article
(This article belongs to the Special Issue Mathematical Methods and Models in Nature and Society)
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22 pages, 924 KiB  
Article
Accurate Approximation of the Matrix Hyperbolic Cosine Using Bernoulli Polynomials
by José M. Alonso, Javier Ibáñez, Emilio Defez and Fernando Alvarruiz
Mathematics 2023, 11(3), 520; https://doi.org/10.3390/math11030520 - 18 Jan 2023
Viewed by 1087
Abstract
This paper presents three different alternatives to evaluate the matrix hyperbolic cosine using Bernoulli matrix polynomials, comparing them from the point of view of accuracy and computational complexity. The first two alternatives are derived from two different Bernoulli series expansions of the matrix [...] Read more.
This paper presents three different alternatives to evaluate the matrix hyperbolic cosine using Bernoulli matrix polynomials, comparing them from the point of view of accuracy and computational complexity. The first two alternatives are derived from two different Bernoulli series expansions of the matrix hyperbolic cosine, while the third one is based on the approximation of the matrix exponential by means of Bernoulli matrix polynomials. We carry out an analysis of the absolute and relative forward errors incurred in the approximations, deriving corresponding suitable values for the matrix polynomial degree and the scaling factor to be used. Finally, we use a comprehensive matrix testbed to perform a thorough comparison of the alternative approximations, also taking into account other current state-of-the-art approaches. The most accurate and efficient options are identified as results. Full article
(This article belongs to the Special Issue Mathematical Methods and Models in Nature and Society)
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20 pages, 2324 KiB  
Article
A Population Pyramid Dynamics Model and Its Analytical Solution. Application Case for Spain
by Joan C. Micó
Mathematics 2022, 10(19), 3443; https://doi.org/10.3390/math10193443 - 22 Sep 2022
Cited by 2 | Viewed by 1949
Abstract
This paper presents the population pyramid dynamics model (PPDM) to study the evolution of the population pyramid of a determined country or society, deducing as a crucial objective its exact analytical solution. The PPDM is a first-order linear partial differential equation whose unknown [...] Read more.
This paper presents the population pyramid dynamics model (PPDM) to study the evolution of the population pyramid of a determined country or society, deducing as a crucial objective its exact analytical solution. The PPDM is a first-order linear partial differential equation whose unknown variable is the population density (population per age unit) depending on time and age, jointly an initial condition in the initial time and a boundary condition given by the births in the zero age. In addition, the dynamical patterns of the crude birth, death, immigration and emigration rates depending on time, jointly with the mathematical pattern of the initial population pyramid depending on ages, take part of the PPDM. These patterns can be obtained from the historical data. An application case of the PPDM analytical solution is presented: Spain, in the 2007–2021 period for its validation, and in the 2021–2026 period for its future forecasting. This application case also permits to obtain the forecasting limits of the PPDM by comparing the historical data with those provided by the PPDM. Other variables that can be obtained from the historical population pyramids data, such as the dependency ratio and the life expectancy at birth, are considered. Full article
(This article belongs to the Special Issue Mathematical Methods and Models in Nature and Society)
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18 pages, 560 KiB  
Article
A Matrix Spline Method for a Class of Fourth-Order Ordinary Differential Problems
by Michael M. Tung, Emilio Defez, Javier Ibáñez, José M. Alonso and Julia Real-Herráiz
Mathematics 2022, 10(16), 2826; https://doi.org/10.3390/math10162826 - 09 Aug 2022
Viewed by 1129
Abstract
Differential matrix models provide an elementary blueprint for the adequate and efficient treatment of many important applications in science and engineering. In the present work, we suggest a procedure, extending our previous research results, to represent the solutions of nonlinear matrix differential problems [...] Read more.
Differential matrix models provide an elementary blueprint for the adequate and efficient treatment of many important applications in science and engineering. In the present work, we suggest a procedure, extending our previous research results, to represent the solutions of nonlinear matrix differential problems of fourth order given in the form Y(4)(x)=f(x,Y(x)) in terms of higher-order matrix splines. The corresponding algorithm is explained, and some numerical examples for the illustration of the method are included. Full article
(This article belongs to the Special Issue Mathematical Methods and Models in Nature and Society)
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