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Volume 27, June
 
 

Math. Comput. Appl., Volume 27, Issue 4 (August 2022) – 17 articles

Cover Story (view full-size image): Primitives are specific trajectories that can be concatenated to move a vehicle from a starting point to a desired state. The accurate selection of a library of primitives is essential to building an automaton, representing the relationships between them, which generates feasible trajectories rapidly. A new method that combines data with a dynamic model to optimally select primitives is proposed and is tested in a simulation of a real-world scenario. View this paper
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Editorial
Numerical and Evolutionary Optimization 2020
Math. Comput. Appl. 2022, 27(4), 70; https://doi.org/10.3390/mca27040070 - 17 Aug 2022
Viewed by 97
Abstract
Solving scientific and engineering problems from the real world is a very complicated task, currently; hence, the development of powerful search and optimization techniques is of great importance [...] Full article
(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2020)
Article
On Some Numerical Methods for Solving Large Differential Nonsymmetric Stein Matrix Equations
Math. Comput. Appl. 2022, 27(4), 69; https://doi.org/10.3390/mca27040069 - 12 Aug 2022
Viewed by 172
Abstract
In this paper, we propose a new numerical method based on the extended block Arnoldi algorithm for solving large-scale differential nonsymmetric Stein matrix equations with low-rank right-hand sides. This algorithm is based on projecting the initial problem on the extended block Krylov subspace [...] Read more.
In this paper, we propose a new numerical method based on the extended block Arnoldi algorithm for solving large-scale differential nonsymmetric Stein matrix equations with low-rank right-hand sides. This algorithm is based on projecting the initial problem on the extended block Krylov subspace to obtain a low-dimensional differential Stein matrix equation. The obtained reduced-order problem is solved by the backward differentiation formula (BDF) method or the Rosenbrock (Ros) method, the obtained solution is used to build the low-rank approximate solution of the original problem. We give some theoretical results and report some numerical experiments. Full article
(This article belongs to the Special Issue Computing and Green Technology)
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Article
Odd Exponential-Logarithmic Family of Distributions: Features and Modeling
Math. Comput. Appl. 2022, 27(4), 68; https://doi.org/10.3390/mca27040068 - 08 Aug 2022
Viewed by 213
Abstract
This paper introduces a general family of continuous distributions, based on the exponential-logarithmic distribution and the odd transformation. It is called the “odd exponential logarithmic family”. We intend to create novel distributions with desired qualities for practical applications, using the unique properties of [...] Read more.
This paper introduces a general family of continuous distributions, based on the exponential-logarithmic distribution and the odd transformation. It is called the “odd exponential logarithmic family”. We intend to create novel distributions with desired qualities for practical applications, using the unique properties of the exponential-logarithmic distribution as an initial inspiration. Thus, we present some special members of this family that stand out for the versatile shape properties of their corresponding functions. Then, a comprehensive mathematical treatment of the family is provided, including some asymptotic properties, the determination of the quantile function, a useful sum expression of the probability density function, tractable series expressions for the moments, moment generating function, Rényi entropy and Shannon entropy, as well as results on order statistics and stochastic ordering. We estimate the model parameters quite efficiently by the method of maximum likelihood, with discussions on the observed information matrix and a complete simulation study. As a major interest, the odd exponential logarithmic models reveal how to successfully accommodate various kinds of data. This aspect is demonstrated by using three practical data sets, showing that an odd exponential logarithmic model outperforms two strong competitors in terms of data fitting. Full article
(This article belongs to the Special Issue Statistical Inference in Linear Models)
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Article
An Efficient Numerical Scheme Based on Radial Basis Functions and a Hybrid Quasi-Newton Method for a Nonlinear Shape Optimization Problem
Math. Comput. Appl. 2022, 27(4), 67; https://doi.org/10.3390/mca27040067 - 04 Aug 2022
Viewed by 175
Abstract
The purpose of this work is to construct a robust numerical scheme for a class of nonlinear free boundary identification problems. First, a shape optimization problem is constructed based on a least square functional. Schauder’s fixed point theorem is manipulated to show the [...] Read more.
The purpose of this work is to construct a robust numerical scheme for a class of nonlinear free boundary identification problems. First, a shape optimization problem is constructed based on a least square functional. Schauder’s fixed point theorem is manipulated to show the existence solution for the state solution. The existence of an optimal solution of the optimization problem is proved. The proposed numerical scheme is based on the Radial Basis Functions method as a discretization approach, the minimization process is a hybrid Differential Evolution heuristic method and the quasi-Newton method. At the end we establish some numerical examples to show the validity of the theoretical results and robustness of the proposed scheme. Full article
(This article belongs to the Special Issue Computing and Green Technology)
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Article
Analytical Solutions of Microplastic Particles Dispersion Using a Lotka–Volterra Predator–Prey Model with Time-Varying Intraspecies Coefficients
Math. Comput. Appl. 2022, 27(4), 66; https://doi.org/10.3390/mca27040066 - 03 Aug 2022
Viewed by 321
Abstract
Discarded plastic is subjected to weather effects from different ecosystems and becomes microplastic particles. Due to their small size, they have spread across the planet. Their presence in living organisms can have several harmful consequences, such as altering the interaction between prey and [...] Read more.
Discarded plastic is subjected to weather effects from different ecosystems and becomes microplastic particles. Due to their small size, they have spread across the planet. Their presence in living organisms can have several harmful consequences, such as altering the interaction between prey and predator. Huang et al. successfully modeled this system presenting numerical results of ecological relevance. Here, we have rewritten their equations and solved a set of them analytically, confirming that microplastic particles accumulate faster in predators than in prey and calculating the time values from which it happens. Using these analytical solutions, we have retrieved the Lotka–Volterra predator–prey model with time-varying intraspecific coefficients, allowing us to interpret ecological quantities referring to microplastics dispersion. After validating our equations, we solved analytically particular situations of ecological interest, characterized by extreme effects on predatory performance, and proposed a second-order differential equation as a possible next step to address this model. Our results open space for further refinement in the study of predator–prey models under the effects of microplastic particles, either exploring the second-order equation that we propose or modify the Huang et al. model to reduce the number of parameters, embedding in the time-varying intraspecies coefficients all the adverse effects caused by microplastic particles. Full article
(This article belongs to the Collection Feature Papers in Mathematical and Computational Applications)
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Article
Modeling the Adaptive Immune Response in an HBV Infection Model with Virus to Cell Transmission in Both Liver with CTL Immune Response and the Extrahepatic Tissue
Math. Comput. Appl. 2022, 27(4), 65; https://doi.org/10.3390/mca27040065 - 28 Jul 2022
Viewed by 195
Abstract
The objective of this paper is to investigate a mathematical model describing the infection of hepatitis B virus (HBV) in intrahepatic and extrahepatic tissues. Additionally, the model includes the effect of the cytotoxic T cell (CTL) immunity, which is described by a linear [...] Read more.
The objective of this paper is to investigate a mathematical model describing the infection of hepatitis B virus (HBV) in intrahepatic and extrahepatic tissues. Additionally, the model includes the effect of the cytotoxic T cell (CTL) immunity, which is described by a linear activation rate by infected cells. The positivity and boundedness of solutions for non-negative initial data are proven, which is consistent with the biological studies. The local stability of the equilibrium is established. In addition to this, the global stability of the disease-free equilibrium and the endemic equilibrium is fulfilled by using appropriate Lyapanov functions. Finally, numerical simulations are performed to support our theoretical findings. It has been revealed that the fractional-order derivatives have no influence on the stability but only on the speed of convergence toward the equilibria. Full article
(This article belongs to the Special Issue Applied Mathematics (IOCAM 22))
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Article
Using the Theory of Functional Connections to Solve Boundary Value Geodesic Problems
Math. Comput. Appl. 2022, 27(4), 64; https://doi.org/10.3390/mca27040064 - 27 Jul 2022
Viewed by 176
Abstract
This study provides a least-squares-based numerical approach to estimate the boundary value geodesic trajectory and associated parametric velocity on curved surfaces. The approach is based on the Theory of Functional Connections, an analytical framework to perform functional interpolation. Numerical examples are provided for [...] Read more.
This study provides a least-squares-based numerical approach to estimate the boundary value geodesic trajectory and associated parametric velocity on curved surfaces. The approach is based on the Theory of Functional Connections, an analytical framework to perform functional interpolation. Numerical examples are provided for a set of two-dimensional quadrics, including ellipsoid, elliptic hyperboloid, elliptic paraboloid, hyperbolic paraboloid, torus, one-sheeted hyperboloid, Moëbius strips, as well as on a generic surface. The estimated geodesic solutions for the tested surfaces are obtained with residuals at the machine-error level. In principle, the proposed approach can be applied to solve boundary value problems in more complex scenarios, such as on Riemannian manifolds. Full article
(This article belongs to the Collection Feature Papers in Mathematical and Computational Applications)
Article
On the Use of High-Order Shape Functions in the SAFE Method and Their Performance in Wave Propagation Problems
Math. Comput. Appl. 2022, 27(4), 63; https://doi.org/10.3390/mca27040063 - 25 Jul 2022
Viewed by 239
Abstract
In this research, high-order shape functions commonly used in different finite element implementations are investigated with a special focus on their applicability in the semi-analytical finite element (SAFE) method being applied to wave propagation problems. Hierarchical shape functions (p-version of the [...] Read more.
In this research, high-order shape functions commonly used in different finite element implementations are investigated with a special focus on their applicability in the semi-analytical finite element (SAFE) method being applied to wave propagation problems. Hierarchical shape functions (p-version of the finite element method), Lagrange polynomials defined over non-equidistant nodes (spectral element method), and non-uniform rational B-splines (isogeometric analysis) are implemented in an in-house SAFE code, along with different refinement strategies such as h-, p-, and k-refinement. Since the numerical analysis of wave propagation is computationally quite challenging, high-order shape functions and local mesh refinement techniques are required to increase the accuracy of the solution, while at the same time decreasing the computational costs. The obtained results reveal that employing a suitable high-order basis in combination with one of the mentioned mesh refinement techniques has a notable effect on the performance of the SAFE method. This point becomes especially beneficial when dealing with applications in the areas of structural health monitoring or material property identification, where a model problem has to be solved repeatedly. Full article
Article
The Binomial–Natural Discrete Lindley Distribution: Properties and Application to Count Data
Math. Comput. Appl. 2022, 27(4), 62; https://doi.org/10.3390/mca27040062 - 20 Jul 2022
Viewed by 272
Abstract
In this paper, a new discrete distribution called Binomial–Natural Discrete Lindley distribution is proposed by compounding the binomial and natural discrete Lindley distributions. Some properties of the distribution are discussed including the moment-generating function, moments and hazard rate function. Estimation of the distribution’s [...] Read more.
In this paper, a new discrete distribution called Binomial–Natural Discrete Lindley distribution is proposed by compounding the binomial and natural discrete Lindley distributions. Some properties of the distribution are discussed including the moment-generating function, moments and hazard rate function. Estimation of the distribution’s parameter is studied by methods of moments, proportions and maximum likelihood. A simulation study is performed to compare the performance of the different estimates in terms of bias and mean square error. SO2 data applications are also presented to see that the new distribution is useful in modeling data. Full article
(This article belongs to the Special Issue Computational Mathematics and Applied Statistics)
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Article
A Bivariate Beta from Gamma Ratios for Determining a Potential Variance Change Point: Inspired from a Process Control Scenario
Math. Comput. Appl. 2022, 27(4), 61; https://doi.org/10.3390/mca27040061 - 16 Jul 2022
Viewed by 303
Abstract
Within statistical process control (SPC), normality is often assumed as the underlying probabilistic generator where the process variance is assumed equal for all rational subgroups. The parameters of the underlying process are usually assumed to be known—if this is not the case, some [...] Read more.
Within statistical process control (SPC), normality is often assumed as the underlying probabilistic generator where the process variance is assumed equal for all rational subgroups. The parameters of the underlying process are usually assumed to be known—if this is not the case, some challenges arise in the estimation of unknown parameters in the SPC environment especially in the case of few observations. This paper proposes a bivariate beta type distribution to guide the user in the detection of a permanent upward or downward step shift in the process’ variance that does not directly rely on parameter estimates, and as such presents itself as an attractive and intuitive approach for not only potentially identifying the magnitude of the shift, but also the position in time where this shift is most likely to occur. Certain statistical properties of this distribution are derived and simulation illustrates the theoretical results. In particular, some insights are gained by comparing the newly proposed model’s performance with an existing approach. A multivariate extension is described, and useful relationships between the derived model and other bivariate beta distributions are also included. Full article
(This article belongs to the Special Issue Computational Mathematics and Applied Statistics)
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Article
Prony Method for Two-Generator Sparse Expansion Problem
Math. Comput. Appl. 2022, 27(4), 60; https://doi.org/10.3390/mca27040060 - 15 Jul 2022
Viewed by 236
Abstract
In data analysis and signal processing, the recovery of structured functions from the given sampling values is a fundamental problem. Many methods generalized from the Prony method have been developed to solve this problem; however, the current research mainly deals with the functions [...] Read more.
In data analysis and signal processing, the recovery of structured functions from the given sampling values is a fundamental problem. Many methods generalized from the Prony method have been developed to solve this problem; however, the current research mainly deals with the functions represented in sparse expansions using a single generating function. In this paper, we generalize the Prony method to solve the sparse expansion problem for two generating functions, so that more types of functions can be recovered by Prony-type methods. The two-generator sparse expansion problem has some special properties. For example, the two sets of frequencies need to be separated from the zeros of the Prony polynomial. We propose a two-stage least-square detection method to solve this problem effectively. Full article
(This article belongs to the Special Issue Computational Mathematics and Applied Statistics)
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Article
On the Convergence of the Damped Additive Schwarz Methods and the Subdomain Coloring
Math. Comput. Appl. 2022, 27(4), 59; https://doi.org/10.3390/mca27040059 - 13 Jul 2022
Viewed by 215
Abstract
In this paper, we consider that the subdomains of the domain decomposition are colored such that the subdomains with the same color do not intersect and introduce and analyze the convergence of a damped additive Schwarz method related to such a subdomain coloring [...] Read more.
In this paper, we consider that the subdomains of the domain decomposition are colored such that the subdomains with the same color do not intersect and introduce and analyze the convergence of a damped additive Schwarz method related to such a subdomain coloring for the resolution of variational inequalities and equations. In this damped method, a single damping value is associated with all the subdomains having the same color. We first make this analysis both for variational inequalities and, as a special case, for equations in an abstract framework. By introducing an assumption on the decomposition of the convex set of the variational inequality, we theoretically analyze in a reflexive Banach space the convergence of the damped additive Schwarz method. The introduced assumption contains a constant C0, and we explicitly write the expression of the convergence rates, depending on the number of colors and the constant C0, and find the values of the damping constants which minimize them. For problems in the finite element spaces, we write the constant C0 as a function of the overlap parameter of the domain decomposition and the number of colors of the subdomains. We show that, for a fixed overlap parameter, the convergence rate, as a function of the number of subdomains has an upper limit which depends only on the number of the colors of the subdomains. Obviously, this limit is independent of the total number of subdomains. Numerical results are in agreement with the theoretical ones. They have been performed for an elasto-plastic problem to verify the theoretical predictions concerning the choice of the damping parameter, the dependence of the convergence on the overlap parameter and on the number of subdomains. Full article
(This article belongs to the Special Issue Domain Decomposition Methods)
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Article
Numerical Study of the Effect of a Heated Cylinder on Natural Convection in a Square Cavity in the Presence of a Magnetic Field
Math. Comput. Appl. 2022, 27(4), 58; https://doi.org/10.3390/mca27040058 - 11 Jul 2022
Viewed by 351
Abstract
The present research was developed to find out the effect of heated cylinder configurations in accordance with the magnetic field on the natural convective flow within a square cavity. In the cavity, four types of configurations—left bottom heated cylinder (LBC), right bottom heated [...] Read more.
The present research was developed to find out the effect of heated cylinder configurations in accordance with the magnetic field on the natural convective flow within a square cavity. In the cavity, four types of configurations—left bottom heated cylinder (LBC), right bottom heated cylinder (RBC), left top heated cylinder (LTC) and right top heated cylinder (RTC)—were considered in the investigation. The current mathematical problem was formulated using the non-linear governing equations and then solved by engaging the process of Galerkin weighted residuals based on the finite element scheme (FES). The investigation of the present problem was conducted using numerous parameters: the Rayleigh number (Ra = 103–105), the Hartmann number (Ha = 0–200) at Pr = 0.71 on the flow field, thermal pattern and the variation of heat inside the enclosure. The clarifications of the numerical result were exhibited in the form of streamlines, isotherms, velocity profiles and temperature profiles, local and mean Nusselt number, along with heated cylinder configurations. From the obtained outcomes, it was observed that the rate of heat transport, as well as the local Nusselt number, decreased for the LBC and LTC configurations, but increased for the RBC and RTC configurations with the increase of the Hartmann number within the square cavity. In addition, the mean Nusselt number for the LBC, RBC, LTC and RTC configurations increased when the Hartmann number was absent, but decreased when the Hartmann number increased in the cavity. The computational results were verified in relation to a published work and were found to be in good agreement. Full article
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Article
Resolving Boundary Layers with Harmonic Extension Finite Elements
Math. Comput. Appl. 2022, 27(4), 57; https://doi.org/10.3390/mca27040057 - 08 Jul 2022
Viewed by 264
Abstract
In recent years, the standard numerical methods for partial differential equations have been extended with variants that address the issue of domain discretisation in complicated domains. Sometimes similar requirements are induced by local parameter-dependent features of the solutions, for instance, boundary or internal [...] Read more.
In recent years, the standard numerical methods for partial differential equations have been extended with variants that address the issue of domain discretisation in complicated domains. Sometimes similar requirements are induced by local parameter-dependent features of the solutions, for instance, boundary or internal layers. The adaptive reference elements are one way with which harmonic extension elements, an extension of the p-version of the finite element method, can be implemented. In combination with simple replacement rule-based mesh generation, the performance of the method is shown to be equivalent to that of the standard p-version in problems where the boundary layers dominate the solution. The performance over a parameter range is demonstrated in an application of computational asymptotic analysis, where known estimates are recovered via computational means only. Full article
(This article belongs to the Collection Feature Papers in Mathematical and Computational Applications)
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Article
Double-Diffusive Convection in Bidispersive Porous Medium with Coriolis Effect
Math. Comput. Appl. 2022, 27(4), 56; https://doi.org/10.3390/mca27040056 - 30 Jun 2022
Viewed by 390
Abstract
In this paper, the thermal instability of rotating convection in a bidispersive porous layer is analyzed. The linear stability analysis is employed to examine the stability of the system. The neutral curves for different values of the physical parameters are shown graphically. The [...] Read more.
In this paper, the thermal instability of rotating convection in a bidispersive porous layer is analyzed. The linear stability analysis is employed to examine the stability of the system. The neutral curves for different values of the physical parameters are shown graphically. The critical Rayleigh number is evaluated for appropriate values of the other governing parameters. Among the obtained results, we find: the Taylor number has a stabilizing effect on the onset of convection; the Soret number does not show any effect on oscillatory convection, as the oscillatory Rayleigh number is independent of the Soret number; there exists a threshold, Rc* ∈ (0.45, 0.46), for the solute Rayleigh number, such that, if RC > Rc*, then the convection arises via an oscillatory mode; and the oscillatory convection sets in and as soon as the value of the Soret number reaches a critical value, (∈(0.6, 0.7)), and the convection arises via stationary convection. Full article
(This article belongs to the Special Issue Advances in Computational Fluid Dynamics and Heat & Mass Transfer)
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Article
The Generalized Odd Linear Exponential Family of Distributions with Applications to Reliability Theory
Math. Comput. Appl. 2022, 27(4), 55; https://doi.org/10.3390/mca27040055 - 23 Jun 2022
Cited by 1 | Viewed by 505
Abstract
A new family of continuous distributions called the generalized odd linear exponential family is proposed. The probability density and cumulative distribution function are expressed as infinite linear mixtures of exponentiated-F distribution. Important statistical properties such as quantile function, moment generating function, distribution of [...] Read more.
A new family of continuous distributions called the generalized odd linear exponential family is proposed. The probability density and cumulative distribution function are expressed as infinite linear mixtures of exponentiated-F distribution. Important statistical properties such as quantile function, moment generating function, distribution of order statistics, moments, mean deviations, asymptotes and the stress–strength model of the proposed family are investigated. The maximum likelihood estimation of the parameters is presented. Simulation is carried out for two of the mentioned sub-models to check the asymptotic behavior of the maximum likelihood estimates. Two real-life data sets are used to establish the credibility of the proposed model. This is achieved by conducting data fitting of two of its sub-models and then comparing the results with suitable competitive lifetime models to generate conclusive evidence. Full article
(This article belongs to the Special Issue Computational Mathematics and Applied Statistics)
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Article
Learning Motion Primitives Automata for Autonomous Driving Applications
Math. Comput. Appl. 2022, 27(4), 54; https://doi.org/10.3390/mca27040054 - 21 Jun 2022
Viewed by 476
Abstract
Motion planning methods often rely on libraries of primitives. The selection of primitives is then crucial for assuring feasible solutions and good performance within the motion planner. In the literature, the library is usually designed by either learning from demonstration, relying entirely on [...] Read more.
Motion planning methods often rely on libraries of primitives. The selection of primitives is then crucial for assuring feasible solutions and good performance within the motion planner. In the literature, the library is usually designed by either learning from demonstration, relying entirely on data, or by model-based approaches, with the advantage of exploiting the dynamical system’s property, e.g., symmetries. In this work, we propose a method combining data with a dynamical model to optimally select primitives. The library is designed based on primitives with highest occurrences within the data set, while Lie group symmetries from a model are analysed in the available data to allow for structure-exploiting primitives. We illustrate our technique in an autonomous driving application. Primitives are identified based on data from human driving, with the freedom to build libraries of different sizes as a parameter of choice. We also compare the extracted library with a custom selection of primitives regarding the performance of obtained solutions for a street layout based on a real-world scenario. Full article
(This article belongs to the Special Issue Set Oriented Numerics 2022)
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