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Math. Comput. Appl., Volume 24, Issue 1 (March 2019)

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Open AccessArticle An Elastodiffusive Orthotropic Euler–Bernoulli Beam Considering Diffusion Flux Relaxation
Math. Comput. Appl. 2019, 24(1), 23; https://doi.org/10.3390/mca24010023
Received: 29 December 2018 / Revised: 30 January 2019 / Accepted: 30 January 2019 / Published: 6 February 2019
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Abstract
This article considers an unsteady elastic diffusion model of Euler–Bernoulli beam oscillations in the presence of diffusion flux relaxation. We used the model of coupled elastic diffusion for a homogeneous orthotropic multicomponent continuum to formulate the problem. A model of unsteady bending for [...] Read more.
This article considers an unsteady elastic diffusion model of Euler–Bernoulli beam oscillations in the presence of diffusion flux relaxation. We used the model of coupled elastic diffusion for a homogeneous orthotropic multicomponent continuum to formulate the problem. A model of unsteady bending for the elastic diffusive Euler–Bernoulli beam was obtained using Hamilton’s variational principle. The Laplace transform on time and the Fourier series expansion by the spatial coordinate were used to solve the obtained problem. Full article
(This article belongs to the Special Issue Mathematical Modeling in Physical Sciences)
Open AccessArticle Mathematical Analysis of a Prey–Predator System: An Adaptive Back-Stepping Control and Stochastic Approach
Math. Comput. Appl. 2019, 24(1), 22; https://doi.org/10.3390/mca24010022
Received: 4 December 2018 / Revised: 16 January 2019 / Accepted: 30 January 2019 / Published: 5 February 2019
Viewed by 135 | PDF Full-text (4198 KB)
Abstract
In this paper, stochastic analysis of a diseased prey–predator system involving adaptive back-stepping control is studied. The system was investigated for its dynamical behaviours, such as boundedness and local stability analysis. The global stability of the system was derived using the Lyapunov function. [...] Read more.
In this paper, stochastic analysis of a diseased prey–predator system involving adaptive back-stepping control is studied. The system was investigated for its dynamical behaviours, such as boundedness and local stability analysis. The global stability of the system was derived using the Lyapunov function. The uniform persistence condition for the system is obtained. The proposed system was studied with adaptive back-stepping control, and it is proved that the system stabilizes to its steady state in nonlinear feedback control. The value of the system is described mostly by the environmental stochasticity in the form of Gaussian white noise. We also established some conditions for oscillations of all positive solutions of the delayed system. Numerical simulations are illustrated, and sustained our analytical findings. We concluded that controlled harvesting on the susceptible and infected prey is able to control prey infection. Full article
(This article belongs to the Section Natural Sciences)
Open AccessArticle Minimizing an Insurer’s Ultimate Ruin Probability by Reinsurance and Investments
Math. Comput. Appl. 2019, 24(1), 21; https://doi.org/10.3390/mca24010021
Received: 9 January 2019 / Revised: 25 January 2019 / Accepted: 28 January 2019 / Published: 2 February 2019
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Abstract
In this paper, we work with a diffusion-perturbed risk model comprising a surplus generating process and an investment return process. The investment return process is of standard a Black–Scholes type, that is, it comprises a single risk-free asset that earns interest at a [...] Read more.
In this paper, we work with a diffusion-perturbed risk model comprising a surplus generating process and an investment return process. The investment return process is of standard a Black–Scholes type, that is, it comprises a single risk-free asset that earns interest at a constant rate and a single risky asset whose price process is modelled by a geometric Brownian motion. Additionally, the company is allowed to purchase noncheap proportional reinsurance priced via the expected value principle. Using the Hamilton–Jacobi–Bellman (HJB) approach, we derive a second-order Volterra integrodifferential equation which we transform into a linear Volterra integral equation of the second kind. We proceed to solve this integral equation numerically using the block-by-block method for the optimal reinsurance retention level that minimizes the ultimate ruin probability. The numerical results based on light- and heavy-tailed individual claim amount distributions show that proportional reinsurance and investments play a vital role in enhancing the survival of insurance companies. But the ruin probability exhibits sensitivity to the volatility of the stock price. Full article
(This article belongs to the Section Social Sciences)
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Open AccessFeature PaperArticle Reduced-Order Modelling and Homogenisation in Magneto-Mechanics: A Numerical Comparison of Established Hyper-Reduction Methods
Math. Comput. Appl. 2019, 24(1), 20; https://doi.org/10.3390/mca24010020
Received: 10 December 2018 / Revised: 29 January 2019 / Accepted: 29 January 2019 / Published: 1 February 2019
Viewed by 155 | PDF Full-text (2673 KB)
Abstract
The simulation of complex engineering structures built from magneto-rheological elastomers is a computationally challenging task. Using the FE2 method, which is based on computational homogenisation, leads to the repetitive solution of micro-scale FE problems, causing excessive computational effort. In this paper, the [...] Read more.
The simulation of complex engineering structures built from magneto-rheological elastomers is a computationally challenging task. Using the FE 2 method, which is based on computational homogenisation, leads to the repetitive solution of micro-scale FE problems, causing excessive computational effort. In this paper, the micro-scale FE problems are replaced by POD reduced models of comparable accuracy. As these models do not deliver the required reductions in computational effort, they are combined with hyper-reduction methods like the Discrete Empirical Interpolation Method (DEIM), Gappy POD, Gauss–Newton Approximated Tensors (GNAT), Empirical Cubature (EC) and Reduced Integration Domain (RID). The goal of this work is the comparison of the aforementioned hyper-reduction techniques focusing on accuracy and robustness. For the application in the FE 2 framework, EC and RID are favourable due to their robustness, whereas Gappy POD rendered both the most accurate and efficient reduced models. The well-known DEIM is discarded for this application as it suffers from serious robustness deficiencies. Full article
Open AccessArticle High-Order Methods Applied to Nonlinear Magnetostatic Problems
Math. Comput. Appl. 2019, 24(1), 19; https://doi.org/10.3390/mca24010019
Received: 18 January 2019 / Accepted: 26 January 2019 / Published: 29 January 2019
Viewed by 187 | PDF Full-text (854 KB)
Abstract
This paper presents a comparison between two high-order modeling methods for solving magnetostatic problems under magnetic saturation, focused on the extraction of machine parameters. Two formulations are compared, the first is based on the Newton-Raphson approach, and the second successively iterates the local [...] Read more.
This paper presents a comparison between two high-order modeling methods for solving magnetostatic problems under magnetic saturation, focused on the extraction of machine parameters. Two formulations are compared, the first is based on the Newton-Raphson approach, and the second successively iterates the local remanent magnetization and the incremental reluctivity of the nonlinear soft-magnetic material. The latter approach is more robust than the Newton-Raphson method, and uncovers useful properties for the fast and accurate calculation of incremental inductance. A novel estimate for the incremental inductance relying on a single additional computation is proposed to avoid multiple nonlinear simulations which are traditionally operated with finite difference linearization or spline interpolation techniques. Fast convergence and high accuracy of the presented methods are demonstrated for the force calculation, which demonstrates their applicability for the design and analysis of electromagnetic devices. Full article
Open AccessFeature PaperArticle Data Pruning of Tomographic Data for the Calibration of Strain Localization Models
Math. Comput. Appl. 2019, 24(1), 18; https://doi.org/10.3390/mca24010018
Received: 12 November 2018 / Revised: 9 January 2019 / Accepted: 22 January 2019 / Published: 28 January 2019
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Abstract
The development and generalization of Digital Volume Correlation (DVC) on X-ray computed tomography data highlight the issue of long-term storage. The present paper proposes a new model-free method for pruning experimental data related to DVC, while preserving the ability to identify constitutive equations [...] Read more.
The development and generalization of Digital Volume Correlation (DVC) on X-ray computed tomography data highlight the issue of long-term storage. The present paper proposes a new model-free method for pruning experimental data related to DVC, while preserving the ability to identify constitutive equations (i.e., closure equations in solid mechanics) reflecting strain localizations. The size of the remaining sampled data can be user-defined, depending on the needs concerning storage space. The proposed data pruning procedure is deeply linked to hyper-reduction techniques. The DVC data of a resin-bonded sand tested in uniaxial compression is used as an illustrating example. The relevance of the pruned data was tested afterwards for model calibration. A Finite Element Model Updating (FEMU) technique coupled with an hybrid hyper-reduction method aws used to successfully calibrate a constitutive model of the resin bonded sand with the pruned data only. Full article
Open AccessFeature PaperArticle Multiple Tensor Train Approximation of Parametric Constitutive Equations in Elasto-Viscoplasticity
Math. Comput. Appl. 2019, 24(1), 17; https://doi.org/10.3390/mca24010017
Received: 9 November 2018 / Revised: 11 January 2019 / Accepted: 23 January 2019 / Published: 28 January 2019
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Abstract
This work presents a novel approach to construct surrogate models of parametric differential algebraic equations based on a tensor representation of the solutions. The procedure consists of building simultaneously an approximation given in tensor-train format, for every output of the reference model. A [...] Read more.
This work presents a novel approach to construct surrogate models of parametric differential algebraic equations based on a tensor representation of the solutions. The procedure consists of building simultaneously an approximation given in tensor-train format, for every output of the reference model. A parsimonious exploration of the parameter space coupled with a compact data representation allows alleviating the curse of dimensionality. The approach is thus appropriate when many parameters with large domains of variation are involved. The numerical results obtained for a nonlinear elasto-viscoplastic constitutive law show that the constructed surrogate model is sufficiently accurate to enable parametric studies such as the calibration of material coefficients. Full article
Open AccessArticle On a Problem Arising in Application of the Re-Quantization Method to Construct Asymptotics of Solutions to Linear Differential Equations with Holomorphic Coefficients at Infinity
Math. Comput. Appl. 2019, 24(1), 16; https://doi.org/10.3390/mca24010016
Received: 26 December 2018 / Revised: 24 January 2019 / Accepted: 26 January 2019 / Published: 28 January 2019
Viewed by 186 | PDF Full-text (338 KB)
Abstract
The re-quantization method—one of the resurgent analysis methods of current importance—is developed in this study. It is widely used in the analytical theory of linear differential equations. With the help of the re-quantization method, the problem of constructing the asymptotics of the inverse [...] Read more.
The re-quantization method—one of the resurgent analysis methods of current importance—is developed in this study. It is widely used in the analytical theory of linear differential equations. With the help of the re-quantization method, the problem of constructing the asymptotics of the inverse Laplace–Borel transform is solved for a particular type of functions with holomorphic coefficients that exponentially grow at zero. Two examples of constructing the uniform asymptotics at infinity for the second- and forth-order differential equations with the help of the re-quantization method and the result obtained in this study are considered. Full article
(This article belongs to the Special Issue Mathematical Modeling in Physical Sciences)
Open AccessArticle A Novel Air Quality Monitoring Unit Using Cloudino and FIWARE Technologies
Math. Comput. Appl. 2019, 24(1), 15; https://doi.org/10.3390/mca24010015
Received: 27 December 2018 / Revised: 20 January 2019 / Accepted: 22 January 2019 / Published: 24 January 2019
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Abstract
Smart City applications aim to improve the quality of life of citizens. Applying technologies of the Internet of Things (IoT) to urban environments is considered as a key of the development of smart cities. In this context, air pollution is one of the [...] Read more.
Smart City applications aim to improve the quality of life of citizens. Applying technologies of the Internet of Things (IoT) to urban environments is considered as a key of the development of smart cities. In this context, air pollution is one of the most important factors affecting the quality of life and the health of the increasing urban population of industrial societies. For this reason, it is essential to develop applications that allow citizens monitoring the concentration of pollutants and avoid places with high levels of pollution. Due to the increasing use of IoT in different areas, there are arising platforms which deal with the challenges IoT implies, such as FIWARE, which provides technologies to facilitate the development of IoT applications. In this paper, an Air Quality Monitoring Unit using Cloudino and Arduino devices and FIWARE technologies is presented. Through Cloudino and Arduino, the monitoring unit gather data from various sensors and transforms the data in a FIWARE data model. Then, the measurements are sent to the Orion Context Broker (OCB), which is a software component provided by FIWARE. The Orion Context Broker allows to manage and publish the data to be consumed by users and applications. Full article
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Open AccessArticle A Hidden Markov Model for the Linguistic Analysis of the Voynich Manuscript
Math. Comput. Appl. 2019, 24(1), 14; https://doi.org/10.3390/mca24010014
Received: 23 November 2018 / Revised: 14 January 2019 / Accepted: 19 January 2019 / Published: 23 January 2019
Viewed by 187 | PDF Full-text (640 KB) | Supplementary Files
Abstract
Hidden Markov models are a very useful tool in the modeling of time series and any sequence of data. In particular, they have been successfully applied to the field of mathematical linguistics. In this paper, we apply a hidden Markov model to analyze [...] Read more.
Hidden Markov models are a very useful tool in the modeling of time series and any sequence of data. In particular, they have been successfully applied to the field of mathematical linguistics. In this paper, we apply a hidden Markov model to analyze the underlying structure of an ancient and complex manuscript, known as the Voynich manuscript, which remains undeciphered. By assuming a certain number of internal states representations for the symbols of the manuscripts, we train the network by means of the α and β -pass algorithms to optimize the model. By this procedure, we are able to obtain the so-called transition and observation matrices to compare with known languages concerning the frequency of consonant andvowel sounds. From this analysis, we conclude that transitions occur between the two states with similar frequencies to other languages. Moreover, the identification of the vowel and consonant sounds matches some previous tentative bottom-up approaches to decode the manuscript. Full article
(This article belongs to the Special Issue Mathematical Modelling in Engineering & Human Behaviour 2018)
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Open AccessArticle Evolution of an Exponential Polynomial Family of Discrete Dynamical Systems
Math. Comput. Appl. 2019, 24(1), 13; https://doi.org/10.3390/mca24010013
Received: 14 December 2018 / Revised: 15 January 2019 / Accepted: 16 January 2019 / Published: 18 January 2019
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Abstract
In this paper, we introduce and analyze a family of exponential polynomial discrete dynamical systems that can be considered as functional perturbations of a linear dynamical system. The stability analysis of equilibria of this family is performed by considering three different parametric scenarios, [...] Read more.
In this paper, we introduce and analyze a family of exponential polynomial discrete dynamical systems that can be considered as functional perturbations of a linear dynamical system. The stability analysis of equilibria of this family is performed by considering three different parametric scenarios, from which we show the intricate and complex dynamical behavior of their orbits. Full article
(This article belongs to the Special Issue Mathematical Modelling in Engineering & Human Behaviour 2018)
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Open AccessArticle Modeling the Macrophage-Mediated Inflammation Involved in the Bone Fracture Healing Process
Math. Comput. Appl. 2019, 24(1), 12; https://doi.org/10.3390/mca24010012
Received: 14 November 2018 / Revised: 14 January 2019 / Accepted: 15 January 2019 / Published: 17 January 2019
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Abstract
A new mathematical model is presented to study the effects of macrophages on the bone fracture healing process. The model consists of a system of nonlinear ordinary differential equations that represents the interactions among classically and alternatively activated macrophages, mesenchymal stem cells, osteoblasts, [...] Read more.
A new mathematical model is presented to study the effects of macrophages on the bone fracture healing process. The model consists of a system of nonlinear ordinary differential equations that represents the interactions among classically and alternatively activated macrophages, mesenchymal stem cells, osteoblasts, and pro- and anti-inflammatory cytokines. A qualitative analysis of the model is performed to determine the equilibria and their corresponding stability properties. Numerical simulations are also presented to support the theoretical results, and to monitor the evolution of a broken bone for different types of fractures under various medical interventions. The model can be used to guide clinical experiments and to explore possible medical treatments that accelerate the bone fracture healing process, either by surgical interventions or drug administrations. Full article
(This article belongs to the Special Issue Mathematical Modelling in Engineering & Human Behaviour 2018)
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Open AccessEditorial Acknowledgement to Reviewers of MCA in 2018
Math. Comput. Appl. 2019, 24(1), 11; https://doi.org/10.3390/mca24010011
Published: 16 January 2019
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Abstract
Rigorous peer-review is the corner-stone of high-quality academic publishing [...] Full article
Open AccessArticle Chaos and Relativistic Energy-Momentum of the Nonlinear Time Fractional Duffing Equation
Math. Comput. Appl. 2019, 24(1), 10; https://doi.org/10.3390/mca24010010
Received: 20 December 2018 / Revised: 9 January 2019 / Accepted: 11 January 2019 / Published: 16 January 2019
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Abstract
This paper studies the nonlinear fractional undamped Duffing equation. The Duffing equation is one of the fundamental equations in engineering. The geographical areas of this model represent chaos, relativistic energy-momentum, electrodynamics, and electromagnetic interactions. These properties have many benefits in different science fields. [...] Read more.
This paper studies the nonlinear fractional undamped Duffing equation. The Duffing equation is one of the fundamental equations in engineering. The geographical areas of this model represent chaos, relativistic energy-momentum, electrodynamics, and electromagnetic interactions. These properties have many benefits in different science fields. The equation depicts the energy of a point mass, which is well thought out as a periodically-forced oscillator. We employed twelve different techniques to the nonlinear fractional Duffing equation to find explicit solutions and approximate solutions. The stability of the solutions was also examined to show the ability of our obtained solutions in the application. The main goals here were to apply a novel computational method (modified auxiliary equation method) and compare the novel method with other methods via the solutions that were obtained by each of these methods. Full article
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Open AccessFeature PaperArticle Block Preconditioning Matrices for the Newton Method to Compute the Dominant λ-Modes Associated with the Neutron Diffusion Equation
Math. Comput. Appl. 2019, 24(1), 9; https://doi.org/10.3390/mca24010009
Received: 30 November 2018 / Revised: 10 January 2019 / Accepted: 10 January 2019 / Published: 15 January 2019
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Abstract
In nuclear engineering, the λ-modes associated with the neutron diffusion equation are applied to study the criticality of reactors and to develop modal methods for the transient analysis. The differential eigenvalue problem that needs to be solved is discretized using a finite [...] Read more.
In nuclear engineering, the λ -modes associated with the neutron diffusion equation are applied to study the criticality of reactors and to develop modal methods for the transient analysis. The differential eigenvalue problem that needs to be solved is discretized using a finite element method, obtaining a generalized algebraic eigenvalue problem whose associated matrices are large and sparse. Then, efficient methods are needed to solve this problem. In this work, we used a block generalized Newton method implemented with a matrix-free technique that does not store all matrices explicitly. This technique reduces mainly the computational memory and, in some cases, when the assembly of the matrices is an expensive task, the computational time. The main problem is that the block Newton method requires solving linear systems, which need to be preconditioned. The construction of preconditioners such as ILU or ICC based on a fully-assembled matrix is not efficient in terms of the memory with the matrix-free implementation. As an alternative, several block preconditioners are studied that only save a few block matrices in comparison with the full problem. To test the performance of these methodologies, different reactor problems are studied. Full article
(This article belongs to the Special Issue Mathematical Modelling in Engineering & Human Behaviour 2018)
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Open AccessArticle An Efficient Semi-Analytical Solution of a One-Dimensional Curvature Equation that Describes the Human Corneal Shape
Math. Comput. Appl. 2019, 24(1), 8; https://doi.org/10.3390/mca24010008
Received: 8 December 2018 / Revised: 5 January 2019 / Accepted: 8 January 2019 / Published: 10 January 2019
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Abstract
In this paper, a numerical approach is proposed to find a semi analytical solution for a prescribed anisotropic mean curvature equation modeling the human corneal shape. The method is based on an integral operator that is constructed in terms of Green’s function coupled [...] Read more.
In this paper, a numerical approach is proposed to find a semi analytical solution for a prescribed anisotropic mean curvature equation modeling the human corneal shape. The method is based on an integral operator that is constructed in terms of Green’s function coupled with the implementation of Picard’s or Mann’s fixed point iteration schemes. Using the contraction principle, it will be shown that the method is convergent for both fixed point iteration schemes. Numerical examples will be presented to demonstrate the applicability, efficiency, and high accuracy of the proposed method. Full article
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Open AccessArticle Accurate Approximate Solution of Ambartsumian Delay Differential Equation via Decomposition Method
Math. Comput. Appl. 2019, 24(1), 7; https://doi.org/10.3390/mca24010007
Received: 11 November 2018 / Revised: 19 December 2018 / Accepted: 26 December 2018 / Published: 9 January 2019
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Abstract
The Ambartsumian delay equation is used in the theory of surface brightness in the Milky way. The Adomian decomposition method (ADM) is applied in this paper to solve this equation. Two canonical forms are implemented to obtain two types of the approximate solutions. [...] Read more.
The Ambartsumian delay equation is used in the theory of surface brightness in the Milky way. The Adomian decomposition method (ADM) is applied in this paper to solve this equation. Two canonical forms are implemented to obtain two types of the approximate solutions. The first solution is provided in the form of a power series which agrees with the solution in the literature, while the second expresses the solution in terms of exponential functions which is viewed as a new solution. A rapid rate of convergence has been achieved and displayed in several graphs. Furthermore, only a few terms of the new approximate solution (expressed in terms of exponential functions) are sufficient to achieve extremely accurate numerical results when compared with a large number of terms of the first solution in the literature. In addition, the residual error using a few terms approaches zero as the delay parameter increases, hence, this confirms the effectiveness of the present approach over the solution in the literature. Full article
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Open AccessArticle Mathematical Modeling and Characterization of the Spread of Chikungunya in Colombia
Math. Comput. Appl. 2019, 24(1), 6; https://doi.org/10.3390/mca24010006
Received: 23 November 2018 / Revised: 27 December 2018 / Accepted: 28 December 2018 / Published: 3 January 2019
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Abstract
The Chikungunya virus is the cause of an emerging disease in Asia and Africa, and also in America, where the virus was first detected in 2006. In this paper, we present a mathematical model of the Chikungunya epidemic at the population level that [...] Read more.
The Chikungunya virus is the cause of an emerging disease in Asia and Africa, and also in America, where the virus was first detected in 2006. In this paper, we present a mathematical model of the Chikungunya epidemic at the population level that incorporates the transmission vector. The epidemic threshold parameter R 0 for the extinction of disease is computed using the method of the next generation matrix, which allows for insights about what are the most relevant model parameters. Using Lyapunov function theory, some sufficient conditions for global stability of the the disease-free equilibrium are obtained. The proposed mathematical model of the Chikungunya epidemic is used to investigate and understand the importance of some specific model parameters and to give some explanation and understanding about the real infected cases with Chikungunya virus in Colombia for data belonging to the year 2015. In this study, we were able to estimate the value of the basic reproduction number R 0 . We use bootstrapping and Markov chain Monte Carlo techniques in order to study parameters’ identifiability. Finally, important policies and insights are provided that could help government health institutions in reducing the number of cases of Chikungunya in Colombia. Full article
(This article belongs to the Special Issue Mathematical Modelling in Engineering & Human Behaviour 2018)
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Open AccessArticle An Approach for Integrating Uncertainty When Selecting an Anti-Torpedo Decoy in Brand New Warships
Math. Comput. Appl. 2019, 24(1), 5; https://doi.org/10.3390/mca24010005
Received: 27 November 2018 / Revised: 26 December 2018 / Accepted: 27 December 2018 / Published: 3 January 2019
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Abstract
The Spanish Navy has planned that the F-80 frigates will be replaced by the brand new F-110 frigates in 2022. The F-110 program is in the conceptual design phase and one of the objectives is to provide the new F-110 frigate with a [...] Read more.
The Spanish Navy has planned that the F-80 frigates will be replaced by the brand new F-110 frigates in 2022. The F-110 program is in the conceptual design phase and one of the objectives is to provide the new F-110 frigate with a salient anti-submarine capability. Therefore, it is necessary to choose what anti-torpedo decoy should be installed in the warship. The Joint Chiefs of Navy Staff (EMA) established some guidelines and, considering the Navy guidance’s, the Analytic Hierarchy Process (AHP) method was applied. After applying the AHP method, none of the decoys obtained a better score to the other one to make a decision. This paper addresses the problem of the selection of the best anti-torpedo decoy to be installed in the new frigates. This allowed implementing a new approach, the Graphic Method of Measurement of Uncertainty Beyond Objectivity (GMUBO). This approach considers different scenarios from the AHP, quantifies the uncertainty, and evaluates which is the best alternative. The method integrates the uncertainty in the AHP and allows measuring the robustness of the selected alternative, also providing a useful graphical tool. Furthermore, GMUBO has a great ease of use and it is helpful to make decisions under uncertainty conditions. Full article
(This article belongs to the Special Issue Mathematical Modelling in Engineering & Human Behaviour 2018)
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Open AccessArticle Construction of Regular Developable Bézier Patches
Math. Comput. Appl. 2019, 24(1), 4; https://doi.org/10.3390/mca24010004
Received: 3 December 2018 / Revised: 25 December 2018 / Accepted: 25 December 2018 / Published: 2 January 2019
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Abstract
The modeling of developable surfaces is considered a very important application in plat-metal-based industries. Relating to the purpose, this discussion aims to obtain some formulas for constructing the regular developable Bézier patches, in which each boundary curve must be laid in two parallel [...] Read more.
The modeling of developable surfaces is considered a very important application in plat-metal-based industries. Relating to the purpose, this discussion aims to obtain some formulas for constructing the regular developable Bézier patches, in which each boundary curve must be laid in two parallel planes. The results as follows: We find some formulas of the equation systems that are described by the constant, linear, and quadratic control parameters of the regular developable Bézier patches criteria. The new approach is numerically tested for constructing the regular developable Bézier patches, in which their boundary curves are defined, respectively, by the combination of four, five, and six degrees. Full article
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Open AccessArticle Modified Potra-Pták Multi-step Schemes with Accelerated Order of Convergence for Solving Systems of Nonlinear Equations
Math. Comput. Appl. 2019, 24(1), 3; https://doi.org/10.3390/mca24010003
Received: 29 November 2018 / Revised: 22 December 2018 / Accepted: 23 December 2018 / Published: 27 December 2018
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Abstract
In this study, an iterative scheme of sixth order of convergence for solving systems of nonlinear equations is presented. The scheme is composed of three steps, of which the first two steps are that of third order Potra-Pták method and last is weighted-Newton [...] Read more.
In this study, an iterative scheme of sixth order of convergence for solving systems of nonlinear equations is presented. The scheme is composed of three steps, of which the first two steps are that of third order Potra-Pták method and last is weighted-Newton step. Furthermore, we generalize our work to derive a family of multi-step iterative methods with order of convergence 3 r + 6 , r = 0 , 1 , 2 , . The sixth order method is the special case of this multi-step scheme for r = 0 . The family gives a four-step ninth order method for r = 1 . As much higher order methods are not used in practice, so we study sixth and ninth order methods in detail. Numerical examples are included to confirm theoretical results and to compare the methods with some existing ones. Different numerical tests, containing academical functions and systems resulting from the discretization of boundary problems, are introduced to show the efficiency and reliability of the proposed methods. Full article
(This article belongs to the Special Issue Mathematical Modelling in Engineering & Human Behaviour 2018)
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Open AccessFeature PaperArticle Active Optics in Astronomy: Freeform Mirror for the MESSIER Telescope Proposal
Math. Comput. Appl. 2019, 24(1), 2; https://doi.org/10.3390/mca24010002
Received: 28 November 2018 / Revised: 20 December 2018 / Accepted: 20 December 2018 / Published: 27 December 2018
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Abstract
Active optics techniques in astronomy provide high imaging quality. This paper is dedicated to highly deformable active optics that can generate non-axisymmetric aspheric surfaces—or freeform surfaces—by use of a minimum number of actuators. The aspheric mirror is obtained from a single uniform load [...] Read more.
Active optics techniques in astronomy provide high imaging quality. This paper is dedicated to highly deformable active optics that can generate non-axisymmetric aspheric surfaces—or freeform surfaces—by use of a minimum number of actuators. The aspheric mirror is obtained from a single uniform load that acts over the surface of a closed-form substrate whilst under axial reaction to its elliptical perimeter ring during spherical polishing. MESSIER space proposal is a wide-field low-central-obstruction folded-two-mirror-anastigmat or here called briefly three-mirror-anastigmat (TMA) telescope. The optical design is a folded reflective Schmidt. Basic telescope features are 36 cm aperture, f/2.5, with 1.6° × 2.6° field of view and a curved field detector allowing null distortion aberration for drift-scan observations. The freeform mirror is generated by spherical stress polishing that provides super-polished freeform surfaces after elastic relaxation. Preliminary analysis required use of the optics theory of 3rd-order aberrations and elasticity theory of thin elliptical plates. Final cross-optimizations were carried out with Zemax raytracing code and Nastran FEA elasticity code in order to determine the complete geometry of a glass ceramic Zerodur deformable substrate. Full article
(This article belongs to the Special Issue Related Problems of Continuum Mechanics)
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Open AccessArticle Modified Auxiliary Equation Method versus Three Nonlinear Fractional Biological Models in Present Explicit Wave Solutions
Math. Comput. Appl. 2019, 24(1), 1; https://doi.org/10.3390/mca24010001
Received: 29 November 2018 / Revised: 16 December 2018 / Accepted: 18 December 2018 / Published: 20 December 2018
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Abstract
In this article, we present a modified auxiliary equation method. We harness this modification in three fundamental models in the biological branch of science. These models are the biological population model, equal width model and modified equal width equation. The three models represent [...] Read more.
In this article, we present a modified auxiliary equation method. We harness this modification in three fundamental models in the biological branch of science. These models are the biological population model, equal width model and modified equal width equation. The three models represent the population density occurring as a result of population supply, a lengthy wave propagating in the positive x-direction, and the simulation of one-dimensional wave propagation in nonlinear media with dispersion processes, respectively. We discuss these models in nonlinear fractional partial differential equation formulas. We used the conformable derivative properties to convert them into nonlinear ordinary differential equations with integer order. After adapting, we applied our new modification to these models to obtain solitary solutions of them. We obtained many novel solutions of these models, which serve to understand more about their properties. All obtained solutions were verified by putting them back into the original equations via computer software such as Maple, Mathematica, and Matlab. Full article
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Math. Comput. Appl. EISSN 2297-8747 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
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