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A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: closed (30 June 2021) | Viewed by 26019

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Prof. Dr. Jesús Martín Vaquero

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Department of Applied Mathematics, Institute of Fundamental Physics and Mathematics, Universidad de Salamanca, 37008 Salamanca, Spain
Interests: Runge–Kutta methods for nonlinear PDEs; approximation theory; mathematical education
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Prof. Dr. Deolinda M. L. Dias Rasteiro

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Department of Mathematics and Physics, Coimbra Polytechnic—ISEC, 3045-093 Coimbra, Portugal
Interests: network optimization; image processing; research on new methods to present mathematics to students
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Prof. Dr. Araceli Queiruga-Dios

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Department of Applied Mathematics, School of Industrial Engineering, University of Salamanca, 37008 Salamanca, Spain
Interests: cryptography; algorithms; applied mathematics; applied and computational mathematics
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Dr. Fatih Yilmaz

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Department of Mathematics, Ankara Hacı Bayram Veli University, 06900 Ankara, Turkey
Interests: applications of matrix theory; graph theory; number theory and mathematical education based on projects and competencies
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Special Issue Information

Dear Colleagues,

Mathematical modeling is an active area of applied mathematics. At its beginning, engineers were the main practitioners of this area of mathematics, developing mathematical models to solve engineering problems in natural sciences.

However, analysis methods and models in social sciences are similar to those of nature sciences, including engineering, with the only difference being that instead of using principles of the nature, one uses principles or theories from experts of such social sciences.

Models based on ordinary or partial differential equations describe a wide variety of phenomena such as sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, or quantum mechanics, for example. Further, stochastic models have recently received increasing attention. Obviously, some of these types of complex problems also require a deep analysis of the tools utilized to solve these situations.

In this Special Issue, we will attempt to integrate models, methods, and also applications, not only in the scope of traditional natural sciences, but also opening the scope to education and other social sciences. Theory and data-driven models, even in a synergy that gives rise to producing fertile, multidisciplinary, and hybrid models, can be considered. Potential topics include but are not limited to:

Numerical and modelling analysis
Optimization and evolutionary algorithms models and methods
Deterministic differential equations: methods and models
Random differential equations: methods and models
Numerical solution of large systems of linear and nonlinear equations
Educational models and methods
Social networks models and methods
Engineering models and simulation
Analysis modelling in economics and finance
Algebraic models and methods with applications
Intelligent data analysis models and methods

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Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at: https://susy.mdpi.com/ in accordance with the timetable.

Dr. Jesús Martín Vaquero
Prof. Dr. Deolinda M. L. Dias Rasteiro
Dr. Araceli Queiruga-Dios
Dr. Fatih Yılmaz
Guest Editors

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Published Papers (11 papers)

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18 pages, 1338 KiB

Open AccessArticle

Using Free Mathematical Software in Engineering Classes

by Víctor Gayoso Martínez, Luis Hernández Encinas, Agustín Martín Muñoz and Araceli Queiruga Dios

Axioms 2021, 10(4), 253; https://doi.org/10.3390/axioms10040253 - 12 Oct 2021

Cited by 4 | Viewed by 2954

Abstract

There are many computational applications and engines used in mathematics, with some of the best-known arguably being Maple, Mathematica, MATLAB, and Mathcad. However, although they are very complete and powerful, they demand the use of commercial licences, which can be a problem for some education institutions or in cases where students desire to use the software on an unlimited number of devices or to access it from several of them simultaneously. In this contribution, we show how GeoGebra, WolframAlpha, Python, and SageMath can be applied to the teaching of different mathematical courses in engineering studies, as they are some of the most interesting representatives of free (and mostly open source) mathematical software. As the best way to show a topic in mathematics is by providing examples, this article explains how to make calculations for some of the main topics associated with Calculus, Algebra, and Coding theories. In addition to this, we provide some results associated with the usage of Mathematica in different graded activities. Moreover, the comparison between the results from students that use Mathematica and students that participate in a “traditional” course, solving problems and attending to master classes, is shown. Full article

(This article belongs to the Special Issue Numerical Analysis and Computational Mathematics)

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14 pages, 407 KiB

Open AccessArticle

Numerical Algorithms for Computing an Arbitrary Singular Value of a Tensor Sum

by Asuka Ohashi and Tomohiro Sogabe

Axioms 2021, 10(3), 211; https://doi.org/10.3390/axioms10030211 - 31 Aug 2021

Cited by 1 | Viewed by 1801

Abstract

We consider computing an arbitrary singular value of a tensor sum: [...] Read more.

We consider computing an arbitrary singular value of a tensor sum:

T : = I_{n} \otimes I_{m} \otimes A + I_{n} \otimes B \otimes I_{ℓ} + C \otimes I_{m} \otimes I_{ℓ} \in R^{ℓ m n \times ℓ m n},

where

A \in R^{ℓ \times ℓ}

B \in R^{m \times m}

C \in R^{n \times n}

. We focus on the shift-and-invert Lanczos method, which solves a shift-and-invert eigenvalue problem of

{(T^{T} T - {\tilde{σ}}^{2} I_{ℓ m n})}^{- 1}

, where

\tilde{σ}

is set to a scalar value close to the desired singular value. The desired singular value is computed by the maximum eigenvalue of the eigenvalue problem. This shift-and-invert Lanczos method needs to solve large-scale linear systems with the coefficient matrix

T^{T} T - {\tilde{σ}}^{2} I_{ℓ m n}

. The preconditioned conjugate gradient (PCG) method is applied since the direct methods cannot be applied due to the nonzero structure of the coefficient matrix. However, it is difficult in terms of memory requirements to simply implement the shift-and-invert Lanczos and the PCG methods since the size of T grows rapidly by the sizes of A, B, and C. In this paper, we present the following two techniques: (1) efficient implementations of the shift-and-invert Lanczos method for the eigenvalue problem of

T^{T} T

and the PCG method for

T^{T} T - {\tilde{σ}}^{2} I_{ℓ m n}

using three-dimensional arrays (third-order tensors) and the n-mode products, and (2) preconditioning matrices of the PCG method based on the eigenvalue and the Schur decomposition of T. Finally, we show the effectiveness of the proposed methods through numerical experiments. Full article

(This article belongs to the Special Issue Numerical Analysis and Computational Mathematics)

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13 pages, 318 KiB

Open AccessArticle

Gauss–Newton–Secant Method for Solving Nonlinear Least Squares Problems under Generalized Lipschitz Conditions

by Ioannis K. Argyros, Stepan Shakhno, Roman Iakymchuk, Halyna Yarmola and Michael I. Argyros

Axioms 2021, 10(3), 158; https://doi.org/10.3390/axioms10030158 - 21 Jul 2021

Viewed by 2071

Abstract

We develop a local convergence of an iterative method for solving nonlinear least squares problems with operator decomposition under the classical and generalized Lipschitz conditions. We consider the case of both zero and nonzero residuals and determine their convergence orders. We use two types of Lipschitz conditions (center and restricted region conditions) to study the convergence of the method. Moreover, we obtain a larger radius of convergence and tighter error estimates than in previous works. Hence, we extend the applicability of this method under the same computational effort. Full article

(This article belongs to the Special Issue Numerical Analysis and Computational Mathematics)

14 pages, 390 KiB

Open AccessArticle

Unveiling the Dynamics of the European Entrepreneurial Framework Conditions over the Last Two Decades: A Cluster Analysis

by Eliana Costa e Silva, Aldina Correia and Ana Borges

Axioms 2021, 10(3), 149; https://doi.org/10.3390/axioms10030149 - 6 Jul 2021

Cited by 5 | Viewed by 2126

Abstract

Entrepreneurship is a theme of global interest, and it is the subject of investigations conducted by many researchers and projects. In particular, the Global Entrepreneurship Monitor project is a global project that involves several countries and years of surveys on entrepreneurship indicators. This study focuses on the 12 indicators of the entrepreneurial ecosystem defined by the Entrepreneurial Framework Conditions (EFCs). The EFCs are specifically related to the quality of the entrepreneurial ecosystem. Using clustering techniques, the present study analyzes how European experts’ perceptions on the EFCs of their home country have changed between 2000 and 2019. The main finding is the existence of significant differences between the clusters obtained over the years and between countries. Therefore, in theoretical terms, this dynamical behavior in relation to the entrepreneurial conditions of economies should be considered in future works, namely, those concerning the definition of the number of clusters, which, according to the internal validation measures computed in this work, should be two. Full article

(This article belongs to the Special Issue Numerical Analysis and Computational Mathematics)

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19 pages, 2021 KiB

Open AccessArticle

Numerical Solution of an Interval-Based Uncertain SIR (Susceptible–Infected–Recovered) Epidemic Model by Homotopy Analysis Method

by Emmanuel A. Bakare, Snehashish Chakraverty and Radovan Potucek

Axioms 2021, 10(2), 114; https://doi.org/10.3390/axioms10020114 - 6 Jun 2021

Viewed by 2228

Abstract

This work proposes an interval-based uncertain Susceptible–Infected–Recovered (SIR) epidemic model. The interval model has been numerically solved by the homotopy analysis method (HAM). The SIR epidemic model is proposed and solved under different uncertain intervals by the HAM to obtain the numerical solution of the model. Furthermore, the SIR ODE model was transformed into a stochastic differential equation (SDE) model and the results of the stochastic and deterministic models were compared using numerical simulations. The results obtained were compared with the numerical solution and found to be in good agreement. Finally, various simulations were done to discuss the solution. Full article

(This article belongs to the Special Issue Numerical Analysis and Computational Mathematics)

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10 pages, 650 KiB

Open AccessArticle

Mathematical Approach for System Repair Rate Analysis Used in Maintenance Decision Making

by Nataša Kontrec, Stefan Panić, Biljana Panić, Aleksandar Marković and Dejan Stošović

Axioms 2021, 10(2), 96; https://doi.org/10.3390/axioms10020096 - 20 May 2021

Cited by 2 | Viewed by 2110

Abstract

Reliability, the number of spare parts and repair time have a great impact on system availability. In this paper, we observed a repairable system comprised of several components. The aim was to determine the repair rate by emphasizing its stochastic nature. A model for the statistical analysis of the component repair rate in function of the desired level of availability is presented. Furthermore, based on the presented model, the approach for the calculation of probability density functions of maximal and minimal repair times for a system comprised of observed components was developed as an important measure that unambiguously defines the total annual repair time. The obtained generalized analytical expressions that can be used to predict the total repair time for an observed entity are the main contributions of the manuscript. The outputs of the model can be useful for making decisions in which time interval repair or replacement should be done to maintain the system and component availability. In addition to planning maintenance activities, the presented models could be used for service capacity planning and the dynamic forecasting of system characteristics. Full article

(This article belongs to the Special Issue Numerical Analysis and Computational Mathematics)

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20 pages, 503 KiB

Open AccessArticle

Approximations of an Equilibrium Problem without Prior Knowledge of Lipschitz Constants in Hilbert Spaces with Applications

by Chainarong Khanpanuk, Nuttapol Pakkaranang, Nopparat Wairojjana and Nattawut Pholasa

Axioms 2021, 10(2), 76; https://doi.org/10.3390/axioms10020076 - 27 Apr 2021

Cited by 1 | Viewed by 1817

Abstract

The objective of this paper is to introduce an iterative method with the addition of an inertial term to solve equilibrium problems in a real Hilbert space. The proposed iterative scheme is based on the Mann-type iterative scheme and the extragradient method. By imposing certain mild conditions on a bifunction, the corresponding theorem of strong convergence in real Hilbert space is well-established. The proposed method has the advantage of requiring no knowledge of Lipschitz-type constants. The applications of our results to solve particular classes of equilibrium problems is presented. Numerical results are established to validate the proposed method’s efficiency and to compare it to other methods in the literature. Full article

(This article belongs to the Special Issue Numerical Analysis and Computational Mathematics)

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21 pages, 569 KiB

Open AccessArticle

Towards the Dependence on Parameters for the Solution of the Thermostatted Kinetic Framework

by Bruno Carbonaro and Marco Menale

Axioms 2021, 10(2), 59; https://doi.org/10.3390/axioms10020059 - 12 Apr 2021

Cited by 7 | Viewed by 1764

Abstract

A complex system is a system involving particles whose pairwise interactions cannot be composed in the same way as in classical Mechanics, i.e., the result of interaction of each particle with all the remaining ones cannot be expressed as a sum of its interactions with each of them (we cannot even know the functional dependence of the total interaction on the single interactions). Moreover, in view of the wide range of its applications to biologic, social, and economic problems, the variables describing the state of the system (i.e., the states of all of its particles) are not always (only) the usual mechanical variables (position and velocity), but (also) many additional variables describing e.g., health, wealth, social condition, social rôle ⋯, and so on. Thus, in order to achieve a mathematical description of the problems of everyday’s life of any human society, either at a microscopic or at a macroscpoic scale, a new mathematical theory (or, more precisely, a scheme of mathematical models), called KTAP, has been devised, which provides an equation which is a generalized version of the Boltzmann equation, to describe in terms of probability distributions the evolution of a non-mechanical complex system. In connection with applications, the classical problems about existence, uniqueness, continuous dependence, and stability of its solutions turn out to be particularly relevant. As far as we are aware, however, the problem of continuous dependence and stability of solutions with respect to perturbations of the parameters expressing the interaction rates of particles and the transition probability densities (see Section The Basic Equations has not been tackled yet). Accordingly, the present paper aims to give some initial results concerning these two basic problems. In particular, Theorem 2 reveals to be stable with respect to small perturbations of parameters, and, as far as instability of solutions with respect to perturbations of parameters is concerned, Theorem 3 shows that solutions are unstable with respect to “large” perturbations of interaction rates; these hints are illustrated by numerical simulations that point out how much solutions corresponding to different values of parameters stay away from each other as

t \to + \infty

. Full article

(This article belongs to the Special Issue Numerical Analysis and Computational Mathematics)

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20 pages, 1222 KiB

Open AccessArticle

Approximation Results for Equilibrium Problems Involving Strongly Pseudomonotone Bifunction in Real Hilbert Spaces

by Wiyada Kumam and Kanikar Muangchoo

Axioms 2020, 9(4), 137; https://doi.org/10.3390/axioms9040137 - 26 Nov 2020

Viewed by 1914

Abstract

A plethora of applications in non-linear analysis, including minimax problems, mathematical programming, the fixed-point problems, saddle-point problems, penalization and complementary problems, may be framed as a problem of equilibrium. Most of the methods used to solve equilibrium problems involve iterative methods, which is why the aim of this article is to establish a new iterative method by incorporating an inertial term with a subgradient extragradient method to solve the problem of equilibrium, which includes a bifunction that is strongly pseudomonotone and meets the Lipschitz-type condition in a real Hilbert space. Under certain mild conditions, a strong convergence theorem is proved, and a required sequence is generated without the information of the Lipschitz-type cost bifunction constants. Thus, the method operates with the help of a slow-converging step size sequence. In numerical analysis, we consider various equilibrium test problems to validate our proposed results. Full article

(This article belongs to the Special Issue Numerical Analysis and Computational Mathematics)

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21 pages, 1771 KiB

Open AccessArticle

Inertial Iterative Self-Adaptive Step Size Extragradient-Like Method for Solving Equilibrium Problems in Real Hilbert Space with Applications

by Wiyada Kumam and Kanikar Muangchoo

Axioms 2020, 9(4), 127; https://doi.org/10.3390/axioms9040127 - 31 Oct 2020

Cited by 1 | Viewed by 1840

Abstract

A number of applications from mathematical programmings, such as minimization problems, variational inequality problems and fixed point problems, can be written as equilibrium problems. Most of the schemes being used to solve this problem involve iterative methods, and for that reason, in this paper, we introduce a modified iterative method to solve equilibrium problems in real Hilbert space. This method can be seen as a modification of the paper titled “A new two-step proximal algorithm of solving the problem of equilibrium programming” by Lyashko et al. (Optimization and its applications in control and data sciences, Springer book pp. 315–325, 2016). A weak convergence result has been proven by considering the mild conditions on the cost bifunction. We have given the application of our results to solve variational inequality problems. A detailed numerical study on the Nash–Cournot electricity equilibrium model and other test problems is considered to verify the convergence result and its performance. Full article

(This article belongs to the Special Issue Numerical Analysis and Computational Mathematics)

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18 pages, 530 KiB

Open AccessArticle

Application of Bernoulli Polynomials for Solving Variable-Order Fractional Optimal Control-Affine Problems

by Somayeh Nemati and Delfim F. M. Torres

Axioms 2020, 9(4), 114; https://doi.org/10.3390/axioms9040114 - 13 Oct 2020

Cited by 7 | Viewed by 3363

Abstract

We propose two efficient numerical approaches for solving variable-order fractional optimal control-affine problems. The variable-order fractional derivative is considered in the Caputo sense, which together with the Riemann–Liouville integral operator is used in our new techniques. An accurate operational matrix of variable-order fractional integration for Bernoulli polynomials is introduced. Our methods proceed as follows. First, a specific approximation of the differentiation order of the state function is considered, in terms of Bernoulli polynomials. Such approximation, together with the initial conditions, help us to obtain some approximations for the other existing functions in the dynamical control-affine system. Using these approximations, and the Gauss—Legendre integration formula, the problem is reduced to a system of nonlinear algebraic equations. Some error bounds are then given for the approximate optimal state and control functions, which allow us to obtain an error bound for the approximate value of the performance index. We end by solving some test problems, which demonstrate the high accuracy of our results. Full article

(This article belongs to the Special Issue Numerical Analysis and Computational Mathematics)

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