Mathematical and Computational Applications

18 pages, 2663 KiB

Open AccessArticle

A Novel Method of Optimal Capacitor Placement in the Presence of Harmonics for Power Distribution Network Using NSGA-II Multi-Objective Genetic Optimization Algorithm

by Majid Ebrahimi Moghadam, Hamid Falaghi and Mahdi Farhadi

Math. Comput. Appl. 2020, 25(1), 17; https://doi.org/10.3390/mca25010017 - 19 Mar 2020

Cited by 7 | Viewed by 3220

Abstract

One of the effective ways of reducing power system losses is local compensation of part of the reactive power consumption by deploying shunt capacitor banks. Since the capacitor’s impedance is frequency-dependent and it is possible to generate resonances at harmonic frequencies, it is [...] Read more.

One of the effective ways of reducing power system losses is local compensation of part of the reactive power consumption by deploying shunt capacitor banks. Since the capacitor’s impedance is frequency-dependent and it is possible to generate resonances at harmonic frequencies, it is important to provide an efficient method for the placement of capacitor banks in the presence of nonlinear loads which are the main cause of harmonic generation. This paper proposes a solution for a multi-objective optimization problem to address the optimal placement of capacitor banks in the presence of nonlinear loads, and it establishes a reasonable reconciliation between costs, along with improvement of harmonic distortion and a voltage index. In this paper, while using the harmonic power flow method to calculate the electrical quantities of the grid in terms of harmonic effects, the non-dominated sorting genetic (NSGA)-II multi-objective genetic optimization algorithm was used to obtain a set of solutions named the Pareto front for the problem. To evaluate the effectiveness of the proposed method, the problem was tested for an IEEE 18-bus system. The results were compared with the methods used in eight other studies. The simulation results show the considerable efficiency and superiority of the proposed flexible method over other methods. Full article

(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2019)

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12 pages, 1566 KiB

Open AccessArticle

Chaos Synchronization for Hyperchaotic Lorenz-Type System via Fuzzy-Based Sliding-Mode Observer

by Corina Plata, Pablo J. Prieto, Ramon Ramirez-Villalobos and Luis N. Coria

Math. Comput. Appl. 2020, 25(1), 16; https://doi.org/10.3390/mca25010016 - 14 Mar 2020

Cited by 4 | Viewed by 2487

Abstract

Hyperchaotic systems have applications in multiple areas of science and engineering. The study and development of these type of systems helps to solve diverse problems related to encryption and decryption of information. In order to solve the chaos synchronization problem for a hyperchaotic [...] Read more.

Hyperchaotic systems have applications in multiple areas of science and engineering. The study and development of these type of systems helps to solve diverse problems related to encryption and decryption of information. In order to solve the chaos synchronization problem for a hyperchaotic Lorenz-type system, we propose an observer based synchronization under a master-slave configuration. The proposed methodology consists of designing a sliding-mode observer (SMO) for the hyperchaotic system. In contrast, this type of methodology exhibits high-frequency oscillations, commonly known as chattering. To solve this problem, a fuzzy-based SMO system was designed. Numerical simulations illustrate the effectiveness of the synchronization between the hyperchaotic system obtained and the proposed observer. Full article

(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2019)

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15 pages, 1966 KiB

Open AccessArticle

A Novel Non-Isotonic Statistical Bivariate Regression Method—Application to Stratigraphic Data Modeling and Interpolation

by Daniele Polucci, Michele Marchetti and Simone Fiori

Math. Comput. Appl. 2020, 25(1), 15; https://doi.org/10.3390/mca25010015 - 10 Mar 2020

Cited by 2 | Viewed by 2677

Abstract

The present paper deals with nonlinear, non-monotonic data regression. This paper introduces an efficient algorithm to perform data transformation from non-monotonic to monotonic to be paired with a statistical bivariate regression method. The proposed algorithm is applied to a number of synthetic and [...] Read more.

The present paper deals with nonlinear, non-monotonic data regression. This paper introduces an efficient algorithm to perform data transformation from non-monotonic to monotonic to be paired with a statistical bivariate regression method. The proposed algorithm is applied to a number of synthetic and real-world non-monotonic data sets to test its effectiveness. The proposed novel non-isotonic regression algorithm is also applied to a collection of data about strontium isotope stratigraphy and compared to a LOWESS regression tool. Full article

(This article belongs to the Section Natural Sciences)

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17 pages, 3935 KiB

Open AccessFeature PaperArticle

Investigation of Volumic Permanent-Magnet Eddy-Current Losses in Multi-Phase Synchronous Machines from Hybrid Multi-Layer Model

by Youcef Benmessaoud, Daoud Ouamara, Frédéric Dubas and Mickael Hilairet

Math. Comput. Appl. 2020, 25(1), 14; https://doi.org/10.3390/mca25010014 - 4 Mar 2020

Cited by 4 | Viewed by 2925

Abstract

This paper investigates the permanent-magnet (PM) eddy-current losses in multi-phase PM synchronous machines (PMSM) with concentric winding and surface-mounted PMs. A hybrid multi-layer model, combining a two-dimensional (2-D) generic magnetic equivalent circuit (MEC) with a 2-D analytical model based on the Maxwell–Fourier method [...] Read more.

This paper investigates the permanent-magnet (PM) eddy-current losses in multi-phase PM synchronous machines (PMSM) with concentric winding and surface-mounted PMs. A hybrid multi-layer model, combining a two-dimensional (2-D) generic magnetic equivalent circuit (MEC) with a 2-D analytical model based on the Maxwell–Fourier method (i.e., the formal resolution of Maxwell’s equations by using the separation of variables method and the Fourier’s series), performs the eddy-current loss calculations. First, the magnetic flux density was obtained from the 2-D generic MEC and then subjected to the Fast Fourier Transform (FFT). The semi-analytical model includes the automatic mesh of static/moving zones, the saturation effect and zones connection in accordance with rotor motion based on a new approach called “Air-gap sliding line technic”. The results of the hybrid multi-layer model were compared with those obtained by three-dimensional (3-D) nonlinear finite-element analysis (FEA). The PM eddy-current losses were estimated on different paths for different segmentations as follow: (i) one segment (no segmentation), (ii) five axial segments, and (iii) two circumferential segments, where the non-uniformity loss distribution is shown. The top of PMs presents a higher quantity of losses compared to the bottom. Full article

(This article belongs to the Special Issue Mathematical Models for the Design of Electrical Machines)

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26 pages, 4053 KiB

Open AccessArticle

Hybrid State Constraint Adaptive Disturbance Rejection Controller for a Mobile Worm Bio-Inspired Robot

by Vania Lara-Ortiz, Ivan Salgado, David Cruz-Ortiz, Alejandro Guarneros, Misael Magos-Sanchez and Isaac Chairez

Math. Comput. Appl. 2020, 25(1), 13; https://doi.org/10.3390/mca25010013 - 4 Mar 2020

Cited by 9 | Viewed by 3744

Abstract

This study presents the design of a hybrid active disturbance rejection controller (H-ADRC) which regulates the gait cycle of a worm bio-inspired robotic device (WBRD). The WBRD is designed as a full actuated six rigid link robotic manipulator. The controller considers the state [...] Read more.

This study presents the design of a hybrid active disturbance rejection controller (H-ADRC) which regulates the gait cycle of a worm bio-inspired robotic device (WBRD). The WBRD is designed as a full actuated six rigid link robotic manipulator. The controller considers the state restrictions in the device articulations; this means the maximum and minimum angular ranges, to avoid any possible damage to the structure. The controller uses an active compensation method to estimate the unknown dynamics of the WBRD by means of an extended state observer. The sequence of movements for the gait cycle of a WBRD is represented as a class of hybrid system by alternative reference frameworks placed at the first and the last link. The stability analysis employs a class of Hybrid Barrier Lyapunov Function to ensure the fulfillment of the angular restrictions in the robotic device. The proposed controller is evaluated using a numerical simulation system based on the virtual version of the WBRD. Moreover, experimental results confirmed that the H-ADRC may endorse the realization of the proposed gait cycle despite the presence of perturbations and modeling uncertainties. The H-ADRC is compared against a proportional derivative (PD) controller and a proportional-integral-derivative (PID) controller. The H-ADRC shows a superior performance as a consequence of the estimation provided by the homogeneous extended state observer. Full article

(This article belongs to the Special Issue Advanced Mathematics and Computational Applications in Control Systems Engineering)

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21 pages, 4604 KiB

Open AccessFeature PaperArticle

Certified Motorcycle Helmets: Computational Evaluation of the Efficacy of Standard Requirements with Finite Element Models

by Fábio A. O. Fernandes, Ricardo J. Alves de Sousa, Mariusz Ptak and Johannes Wilhelm

Math. Comput. Appl. 2020, 25(1), 12; https://doi.org/10.3390/mca25010012 - 16 Feb 2020

Cited by 13 | Viewed by 6350

Abstract

Every year, thousands of people die in the European Union as a direct result of road accidents. Helmets are one of the most important types of personal safety gear. The ECE R22.05 standard, adopted in 2000, is responsible for the certification of motorcycle [...] Read more.

Every year, thousands of people die in the European Union as a direct result of road accidents. Helmets are one of the most important types of personal safety gear. The ECE R22.05 standard, adopted in 2000, is responsible for the certification of motorcycle helmets in the European Union and in many other countries. Two decades later, it is still being used with the same requirements, without any update. The aim of this work is to evaluate the efficacy of a motorcycle helmet certified by such standard, using computational models as an assessment tool. First, a finite element model of a motorcycle helmet available on the market was developed and validated by simulating the same impacts required by the standard. Then, a finite element model of the human head is used as an injury prediction tool to assess its safety performance. Results indicate a significant risk of brain injury, which is in accordance with previous studies available in the literature. Therefore, this work underlines and emphasizes the need of improving the requirements of ECE R22.05. Full article

(This article belongs to the Special Issue Numerical Modelling and Simulation Applied to Head Trauma)

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25 pages, 3937 KiB

Open AccessFeature PaperArticle

Reduced-Order Model of Rotor Cage in Multiphase Induction Machines: Application on the Prediction of Torque Pulsations

by Abdelhak Mekahlia, Eric Semail, Franck Scuiller and Hussein Zahr

Math. Comput. Appl. 2020, 25(1), 11; https://doi.org/10.3390/mca25010011 - 29 Jan 2020

Cited by 5 | Viewed by 2942

Abstract

For three-phase induction machines supplied by sinusoidal current, it is usual to model the n-bar squirrel-cage by an equivalent two-phase circuit. For a multiphase induction machine which can be supplied with different harmonics of current, the reduced-order model of the rotor must be [...] Read more.

For three-phase induction machines supplied by sinusoidal current, it is usual to model the n-bar squirrel-cage by an equivalent two-phase circuit. For a multiphase induction machine which can be supplied with different harmonics of current, the reduced-order model of the rotor must be more carefully chosen in order to predict the pulsations of torque. The proposed analysis allows to avoid a wrong design with non-sinusoidal magnetomotive forces. An analytical approach is proposed and confirmed by Finite-Element modelling at first for a three-phase induction machine and secondly for a five-phase induction machine. Full article

(This article belongs to the Special Issue Mathematical Models for the Design of Electrical Machines)

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17 pages, 4595 KiB

Open AccessArticle

Maximal Product of Graphs under Vague Environment

by Behnaz Sheikh Hoseini, Muhammad Akram, Mehrnaz Sheikh Hosseini, Hossein Rashmanlou and Rajab Ali Borzooei

Math. Comput. Appl. 2020, 25(1), 10; https://doi.org/10.3390/mca25010010 - 23 Jan 2020

Cited by 6 | Viewed by 2245

Abstract

Graph models are found everywhere in natural and human made structures, including process dynamics in physical, biological and social systems. The product of graphs are appropriately used in several combinatorial applications and in the formation of different structural models. In this paper, we [...] Read more.

Graph models are found everywhere in natural and human made structures, including process dynamics in physical, biological and social systems. The product of graphs are appropriately used in several combinatorial applications and in the formation of different structural models. In this paper, we present a new product of graphs, namely, maximal product of two vague graphs. Then we describe certain concepts, including strongly, completely, regularity and connectedness on a maximal product of vague graphs. Further, we consider some results of edge regular and totally edge regular in a maximal product of vague graphs. Finally, we present an application for optimization of the biomass based on a maximal product of vague graphs. Full article

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16 pages, 487 KiB

Open AccessArticle

Added-Mass Based Efficient Fluid–Structure Interaction Model for Dynamics of Axially Moving Panels with Thermal Expansion

by Nikolay Banichuk, Svetlana Ivanova, Evgeny Makeev, Juha Jeronen and Tero Tuovinen

Math. Comput. Appl. 2020, 25(1), 9; https://doi.org/10.3390/mca25010009 - 22 Jan 2020

Cited by 1 | Viewed by 2908

Abstract

The paper considers the analysis of a traveling panel, submerged in axially flowing fluid. In order to accurately model the dynamics and stability of a lightweight moving material, the interaction between the material and the surrounding air must be taken into account. The [...] Read more.

The paper considers the analysis of a traveling panel, submerged in axially flowing fluid. In order to accurately model the dynamics and stability of a lightweight moving material, the interaction between the material and the surrounding air must be taken into account. The lightweight material leads to the inertial contribution of the surrounding air to the acceleration of the panel becoming significant. This formulation is novel and the case complements our previous studies on the field. The approach described in this paper allows for an efficient semi-analytical solution, where the reaction pressure of the fluid flow is analytically represented by an added-mass model in terms of the panel displacement. Then, the panel displacement, accounting also for the fluid–structure interaction, is analyzed with the help of the weak form of the governing partial differential equation, using a Galerkin method. In the first part of this paper, we represent the traveling panel by a single partial differential equation in weak form, using an added-mass approximation of the exact fluid reaction. In the second part, we apply a Galerkin method for dynamic stability analysis of the panel, and present an analytical investigation of static stability loss (divergence, buckling) based on the added-mass model. Full article

(This article belongs to the Special Issue Coupled CFD Problems with Moving Boundaries and Interfaces)

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32 pages, 1428 KiB

Open AccessArticle

Spherical Fuzzy Graphs with Application to Decision-Making

by Muhammad Akram, Danish Saleem and Talal Al-Hawary

Math. Comput. Appl. 2020, 25(1), 8; https://doi.org/10.3390/mca25010008 - 20 Jan 2020

Cited by 32 | Viewed by 4001

Abstract

In a network model, the evaluation information given by decision makers are occasionally of types: yes, abstain, no, and refusal. To deal with such problems, we use mathematical models based on picture fuzzy sets. The spherical fuzzy model is more versatile than the [...] Read more.

In a network model, the evaluation information given by decision makers are occasionally of types: yes, abstain, no, and refusal. To deal with such problems, we use mathematical models based on picture fuzzy sets. The spherical fuzzy model is more versatile than the picture fuzzy model as it broadens the space of uncertain and vague information, due to its outstanding feature of vast space of participation of acceptable triplets. Graphs are a mathematical representation of networks. Thus to deal with many real-world phenomena represented by networks, spherical fuzzy graphs can be used to model different practical scenarios in a more flexible manner than picture fuzzy graphs. In this research article, we discuss two operations on spherical fuzzy graphs (SFGs), namely, symmetric difference and rejection; and develop some results regarding their degrees and total degrees. We describe certain concepts of irregular SFGs with several important properties. Further, we present an application of SFGs in decision making. Full article

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28 pages, 2987 KiB

Open AccessArticle

Evolutionary Algorithms Enhanced with Quadratic Coding and Sensing Search for Global Optimization

by Abdel-Rahman Hedar, Wael Deabes, Majid Almaraashi and Hesham H. Amin

Math. Comput. Appl. 2020, 25(1), 7; https://doi.org/10.3390/mca25010007 - 16 Jan 2020

Cited by 4 | Viewed by 2905

Abstract

Enhancing Evolutionary Algorithms (EAs) using mathematical elements significantly contribute to their development and control the randomness they are experiencing. Moreover, the automation of the primary process steps of EAs is still one of the hardest problems. Specifically, EAs still have no robust automatic [...] Read more.

Enhancing Evolutionary Algorithms (EAs) using mathematical elements significantly contribute to their development and control the randomness they are experiencing. Moreover, the automation of the primary process steps of EAs is still one of the hardest problems. Specifically, EAs still have no robust automatic termination criteria. Moreover, the highly random behavior of some evolutionary operations should be controlled, and the methods should invoke advanced learning process and elements. As follows, this research focuses on the problem of automating and controlling the search process of EAs by using sensing and mathematical mechanisms. These mechanisms can provide the search process with the needed memories and conditions to adapt to the diversification and intensification opportunities. Moreover, a new quadratic coding and quadratic search operator are invoked to increase the local search improving possibilities. The suggested quadratic search operator uses both regression and Radial Basis Function (RBF) neural network models. Two evolutionary-based methods are proposed to evaluate the performance of the suggested enhancing elements using genetic algorithms and evolution strategies. Results show that for both the regression, RBFs and quadratic techniques could help in the approximation of high-dimensional functions with the use of a few adjustable parameters for each type of function. Moreover, the automatic termination criteria could allow the search process to stop appropriately. Full article

(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2019)

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4 pages, 233 KiB

Open AccessEditorial

Acknowledgement to Reviewers of MCA in 2019

by MCA Editorial Office

Math. Comput. Appl. 2020, 25(1), 6; https://doi.org/10.3390/mca25010006 - 16 Jan 2020

Viewed by 1656

Abstract

The editorial team greatly appreciates the reviewers who have dedicated their considerable time and expertise to the journal’s rigorous editorial process over the past 12 months, regardless of whether the papers are finally published or not[...] Full article

17 pages, 5837 KiB

Open AccessFeature PaperArticle

Jacob’s Ladder: Prime Numbers in 2D

by Alberto Fraile, Roberto Martínez and Daniel Fernández

Math. Comput. Appl. 2020, 25(1), 5; https://doi.org/10.3390/mca25010005 - 11 Jan 2020

Cited by 2 | Viewed by 5701

Abstract

Prime numbers are one of the most intriguing figures in mathematics. Despite centuries of research, many questions remain still unsolved. In recent years, computer simulations are playing a fundamental role in the study of an immense variety of problems. In this work, we [...] Read more.

Prime numbers are one of the most intriguing figures in mathematics. Despite centuries of research, many questions remain still unsolved. In recent years, computer simulations are playing a fundamental role in the study of an immense variety of problems. In this work, we present a simple representation of prime numbers in two dimensions that allows us to formulate a number of conjectures that may lead to important avenues in the field of research on prime numbers. In particular, although the zeroes in our representation grow in a somewhat erratic, hardly predictable way, the gaps between them present a remarkable and intriguing property: a clear exponential decay in the frequency of gaps vs. gap size. The smaller the gaps, the more frequently they appear. Additionally, the sequence of zeroes, despite being non-consecutive numbers, contains a number of primes approximately equal to

n / \log n

, n being the number of terms in the sequence. Full article

(This article belongs to the Section Natural Sciences)

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23 pages, 398 KiB

Open AccessArticle

Cost Efficiency Evaluation Based on a Data Envelopment Analysis Approach by Considering Undesirable Outputs on the Basis of the Semi-Disposability Assumption

by Mehdi Karami Khorramabadi, Majid Yarahmadi and Mojtaba Ghiyasi

Math. Comput. Appl. 2020, 25(1), 4; https://doi.org/10.3390/mca25010004 - 10 Jan 2020

Viewed by 3178

Abstract

It is considerably important to calculate the cost efficiency in data envelopment analysis for the efficiency evaluation of decision-making units. The present paper develops the classical cost efficiency model in which all the input prices are constant and certain for each decision-making unit, [...] Read more.

It is considerably important to calculate the cost efficiency in data envelopment analysis for the efficiency evaluation of decision-making units. The present paper develops the classical cost efficiency model in which all the input prices are constant and certain for each decision-making unit, considering undesirable outputs under the semi-disposability assumption. The proposed models are interval and uncertain under the constant returns to scale and also variable returns to scale assumptions, for the easy solution of which, their lower and upper bounds are obtained on the basis of the theorem presented in the text. In order to simulate the proposed models and show their scientific capabilities, additionally, 56 electricity producing thermal power plants in Iran were studied in 2015. Results of the present study show that under both assumptions of constant returns to scale and variable returns to scale, the highest cost efficiency bounds belonged to the combined and steam cycle power plants. Moreover, the average of lower and upper cost efficiency bounds of the power plants under study were 34% and 35%, respectively, in 2015, under the constant returns to scale assumption, and 52% and 54%, respectively, under the variable returns to scale assumption. Full article

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25 pages, 1355 KiB

Open AccessFeature PaperArticle

Non-Epsilon Dominated Evolutionary Algorithm for the Set of Approximate Solutions

by Carlos Ignacio Hernández Castellanos, Oliver Schütze, Jian-Qiao Sun and Sina Ober-Blöbaum

Math. Comput. Appl. 2020, 25(1), 3; https://doi.org/10.3390/mca25010003 - 8 Jan 2020

Cited by 8 | Viewed by 2863

Abstract

In this paper, we present a novel evolutionary algorithm for the computation of approximate solutions for multi-objective optimization problems. These solutions are of particular interest to the decision-maker as backup solutions since they can provide solutions with similar quality but in different regions [...] Read more.

In this paper, we present a novel evolutionary algorithm for the computation of approximate solutions for multi-objective optimization problems. These solutions are of particular interest to the decision-maker as backup solutions since they can provide solutions with similar quality but in different regions of the decision space. The novel algorithm uses a subpopulation approach to put pressure towards the Pareto front while exploring promissory areas for approximate solutions. Furthermore, the algorithm uses an external archiver to maintain a suitable representation in both decision and objective space. The novel algorithm is capable of computing an approximation of the set of interest with good quality in terms of the averaged Hausdorff distance. We underline the statements on some academic problems from literature and an application in non-uniform beams. Full article

(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2019)

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13 pages, 4460 KiB

Open AccessFeature PaperArticle

Efficient Methods to Calculate Partial Sphere Surface Areas for a Higher Resolution Finite Volume Method for Diffusion-Reaction Systems in Biological Modeling

by Abigail Bowers, Jared Bunn and Myles Kim

Math. Comput. Appl. 2020, 25(1), 2; https://doi.org/10.3390/mca25010002 - 23 Dec 2019

Viewed by 4050

Abstract

Computational models for multicellular biological systems, in both in vitro and in vivo environments, require solving systems of differential equations to incorporate molecular transport and their reactions such as release, uptake, or decay. Examples can be found from drugs, growth nutrients, and signaling [...] Read more.

Computational models for multicellular biological systems, in both in vitro and in vivo environments, require solving systems of differential equations to incorporate molecular transport and their reactions such as release, uptake, or decay. Examples can be found from drugs, growth nutrients, and signaling factors. The systems of differential equations frequently fall into the category of the diffusion-reaction system due to the nature of the spatial and temporal change. Due to the complexity of equations and complexity of the modeled systems, an analytical solution for the systems of the differential equations is not possible. Therefore, numerical calculation schemes are required and have been used for multicellular biological systems such as bacterial population dynamics or cancer cell dynamics. Finite volume methods in conjunction with agent-based models have been popular choices to simulate such reaction-diffusion systems. In such implementations, the reaction occurs within each finite volume and finite volumes interact with one another following the law of diffusion. The characteristic of the reaction can be determined by the agents in the finite volume. In the case of cancer cell growth dynamics, it is observed that cell behavior can be different by a matter of a few cell size distances because of the chemical gradient. Therefore, in the modeling of such systems, the spatial resolution must be comparable to the cell size. Such spatial resolution poses an extra challenge in the development and execution of the computational model due to the agents sitting over multiple finite volumes. In this article, a few computational methods for cell surface-based reaction for the finite volume method will be introduced and tested for their performance in terms of accuracy and computation speed. Full article

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20 pages, 1036 KiB

Open AccessFeature PaperArticle

Numerical Optimal Control of HIV Transmission in Octave/MATLAB

by Carlos Campos, Cristiana J. Silva and Delfim F. M. Torres

Math. Comput. Appl. 2020, 25(1), 1; https://doi.org/10.3390/mca25010001 - 19 Dec 2019

Cited by 16 | Viewed by 6166

Abstract

We provide easy and readable GNU Octave/MATLAB code for the simulation of mathematical models described by ordinary differential equations and for the solution of optimal control problems through Pontryagin’s maximum principle. For that, we consider a normalized HIV/AIDS transmission dynamics model based on [...] Read more.

We provide easy and readable GNU Octave/MATLAB code for the simulation of mathematical models described by ordinary differential equations and for the solution of optimal control problems through Pontryagin’s maximum principle. For that, we consider a normalized HIV/AIDS transmission dynamics model based on the one proposed in our recent contribution (Silva, C.J.; Torres, D.F.M. A SICA compartmental model in epidemiology with application to HIV/AIDS in Cape Verde. Ecol. Complex. 2017, 30, 70–75), given by a system of four ordinary differential equations. An HIV initial value problem is solved numerically using the ode45 GNU Octave function and three standard methods implemented by us in Octave/MATLAB: Euler method and second-order and fourth-order Runge–Kutta methods. Afterwards, a control function is introduced into the normalized HIV model and an optimal control problem is formulated, where the goal is to find the optimal HIV prevention strategy that maximizes the fraction of uninfected HIV individuals with the least HIV new infections and cost associated with the control measures. The optimal control problem is characterized analytically using the Pontryagin Maximum Principle, and the extremals are computed numerically by implementing a forward-backward fourth-order Runge–Kutta method. Complete algorithms, for both uncontrolled initial value and optimal control problems, developed under the free GNU Octave software and compatible with MATLAB are provided along the article. Full article

(This article belongs to the Special Issue Numerical and Symbolic Computation: Developments and Applications)

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Math. Comput. Appl., Volume 25, Issue 1 (March 2020) – 17 articles

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