Mathematics

Research

31 pages, 6144 KiB

Open AccessArticle

Evaluation and Mathematical Analysis of a Four-Dimensional Lotka–Volterra-like Equation Designed to Describe the Batch Nisin Production System

by Fernando Giménez-Palomares, Pedro Fernández de Córdoba, Juan C. Mejuto, Ricardo J. Bendaña-Jácome and Nelson Pérez-Guerra

Mathematics 2022, 10(5), 677; https://doi.org/10.3390/math10050677 - 22 Feb 2022

Viewed by 2410

Abstract

Nisin, an antibacterial compound produced by Lactococcus lactis strains, has been approved by the US Food and Drug Administration to be used as a safe food additive to control the growth of undesirable pathogenic bacteria. Nisin is commonly described as a pH-dependent primary [...] Read more.

Nisin, an antibacterial compound produced by Lactococcus lactis strains, has been approved by the US Food and Drug Administration to be used as a safe food additive to control the growth of undesirable pathogenic bacteria. Nisin is commonly described as a pH-dependent primary metabolite since its production depends on growth and culture pH evolution. However, the relationships between bacteriocin synthesis (BT), biomass production (X), culture pH, and the consumption of the limiting nutrient (total nitrogen: TN) have not been described until now. Therefore, this study aims to develop a competitive four-dimensional Lotka–Volterra-like Equation (predator-prey system) to describe these complex relationships in three series of batch fermentations with L. lactis CECT 539 in diluted whey (DW)-based media. The developed four-dimensional predator-prey system accurately described each individual culture, providing a good description of the relationships between pH, TN, X, and BT, higher values for R² and F-ratios, lower values (<10%) for the mean relative percentage deviation modulus, with bias and accuracy factor values approximately equal to one. The mathematical analysis of the developed equation showed the existence of one asymptotically stable equilibrium point, and the phase’s diagram obtained did not show the closed elliptic trajectories observed in biological predator-prey systems. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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26 pages, 5901 KiB

Open AccessArticle

A Numerical Method for Computing Double Integrals with Variable Upper Limits

by Olha Chernukha, Yurii Bilushchak, Natalya Shakhovska and Rastislav Kulhánek

Mathematics 2022, 10(1), 108; https://doi.org/10.3390/math10010108 - 30 Dec 2021

Cited by 2 | Viewed by 3939

Abstract

We propose and justify a numerical method for computing the double integral with variable upper limits that leads to the variableness of the region of integration. Imposition of simple variables as functions for upper limits provides the form of triangles of integration region [...] Read more.

We propose and justify a numerical method for computing the double integral with variable upper limits that leads to the variableness of the region of integration. Imposition of simple variables as functions for upper limits provides the form of triangles of integration region and variable in the external limit of integral leads to a continuous set of similar triangles. A variable grid is overlaid on the integration region. We consider three cases of changes of the grid for the division of the integration region into elementary volumes. The first is only the size of the imposed grid changes with the change of variable of the external upper limit. The second case is the number of division elements changes with the change of the external upper limit variable. In the third case, the grid size and the number of division elements change after fixing their multiplication. In these cases, the formulas for computing double integrals are obtained based on the application of cubatures in the internal region of integration and performing triangulation division along the variable boundary. The error of the method is determined by expanding the double integral into the Taylor series using Barrow’s theorem. Test of efficiency and reliability of the obtained formulas of the numerical method for three cases of ways of the division of integration region is carried out on examples of the double integration of sufficiently simple functions. Analysis of the obtained results shows that the smallest absolute and relative errors are obtained in the case of an increase of the number of division elements changes when the increase of variable of the external upper limit and the grid size is fixed. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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19 pages, 333 KiB

Open AccessArticle

Eighth Order Two-Step Methods Trained to Perform Better on Keplerian-Type Orbits

by Vladislav N. Kovalnogov, Ruslan V. Fedorov, Andrey V. Chukalin, Theodore E. Simos and Charalampos Tsitouras

Mathematics 2021, 9(23), 3071; https://doi.org/10.3390/math9233071 - 29 Nov 2021

Cited by 7 | Viewed by 1245

Abstract

The family of Numerov-type methods that effectively uses seven stages per step is considered. All the coefficients of the methods belonging to this family can be expressed analytically with respect to four free parameters. These coefficients are trained through a differential evolution technique [...] Read more.

The family of Numerov-type methods that effectively uses seven stages per step is considered. All the coefficients of the methods belonging to this family can be expressed analytically with respect to four free parameters. These coefficients are trained through a differential evolution technique in order to perform best in a wide range of Keplerian-type orbits. Then it is observed with extended numerical tests that a certain method behaves extremely well in a variety of orbits (e.g., Kepler, perturbed Kepler, Arenstorf, Pleiades) for various steplengths used by the methods and for various intervals of integration. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

19 pages, 12531 KiB

Open AccessArticle

Parallel Algorithms for Solving Inverse Gravimetry Problems: Application for Earth’s Crust Density Models Creation

by Petr Martyshko, Igor Ladovskii and Denis Byzov

Mathematics 2021, 9(22), 2966; https://doi.org/10.3390/math9222966 - 20 Nov 2021

Cited by 8 | Viewed by 1978

Abstract

The paper describes a method of gravity data inversion, which is based on parallel algorithms. The choice of the density model of the initial approximation and the set on which the solution is sought guarantees the stability of the algorithms. We offer a [...] Read more.

The paper describes a method of gravity data inversion, which is based on parallel algorithms. The choice of the density model of the initial approximation and the set on which the solution is sought guarantees the stability of the algorithms. We offer a new upward and downward continuation algorithm for separating the effects of shallow and deep sources. Using separated field of layers, the density distribution is restored in a form of 3D grid. We use the iterative parallel algorithms for the downward continuation and restoration of the density values (by solving the inverse linear gravity problem). The algorithms are based on the ideas of local minimization; they do not require a nonlinear minimization; they are easier to implement and have better stability. We also suggest an optimization of the gravity field calculation, which speeds up the inversion. A practical example of interpretation is presented for the gravity data of the Urals region, Russia. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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16 pages, 560 KiB

Open AccessArticle

Modeling the Context of the Problem Domain of Time Series with Type-2 Fuzzy Sets

by Anton A. Romanov, Aleksey A. Filippov, Valeria V. Voronina, Gleb Guskov and Nadezhda G. Yarushkina

Mathematics 2021, 9(22), 2947; https://doi.org/10.3390/math9222947 - 18 Nov 2021

Cited by 6 | Viewed by 1544

Abstract

Data analysis in the context of the features of the problem domain and the dynamics of processes are significant in various industries. Uncertainty modeling based on fuzzy logic allows building approximators for solving a large class of problems. In some cases, type-2 fuzzy [...] Read more.

Data analysis in the context of the features of the problem domain and the dynamics of processes are significant in various industries. Uncertainty modeling based on fuzzy logic allows building approximators for solving a large class of problems. In some cases, type-2 fuzzy sets in the model are used. The article describes constructing fuzzy time series models of the analyzed processes within the context of the problem domain. An algorithm for fuzzy modeling of the time series was developed. A new time series forecasting scheme is proposed. An illustrative example of the time series modeling is presented. The benefits of contextual modeling are demonstrated. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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23 pages, 1903 KiB

Open AccessArticle

An Approach to Building Decision Support Systems Based on an Ontology Service

by Anton Romanov, Julia Stroeva, Aleksey Filippov and Nadezhda Yarushkina

Mathematics 2021, 9(22), 2946; https://doi.org/10.3390/math9222946 - 18 Nov 2021

Cited by 2 | Viewed by 1855

Abstract

Modern decision support systems (DSSs) need components for storing knowledge. Moreover, DSSs must support fuzzy inference to work with uncertainty. Ontologies are designed to represent knowledge of complex structures and to perform inference tasks. Developers must use the OWLAPI and SWRL API libraries [...] Read more.

Modern decision support systems (DSSs) need components for storing knowledge. Moreover, DSSs must support fuzzy inference to work with uncertainty. Ontologies are designed to represent knowledge of complex structures and to perform inference tasks. Developers must use the OWLAPI and SWRL API libraries to use ontology features. They are impossible to use in DSSs written in programming languages not for Java Virtual Machines. The FuzzyOWL library and the FuzzyDL inference engine are required to work with fuzzy ontologies. The FuzzyOWL library is currently unmaintained and does not have a public Git repository. Thus, it is necessary to develop the ontology service. The ontology service must allow working with ontologies and making fuzzy inferences. The article presents ontology models for decision support, fuzzy inference, and the fuzzy inference algorithm. The article considers examples of DSSs for balancing production capacities and image analysis. The article also describes the architecture of the ontology service. The proposed novel ontology models for decision support make it possible to reduce the time of a knowledge base formation. The ontology service can integrate with external systems with HTTP protocol. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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17 pages, 3479 KiB

Open AccessArticle

Numerical Study of Powder Flow Nozzle for Laser-Assisted Metal Deposition

by Romuald Petkevič, Giedrius Jočbalis, Ada Steponavičiūtė, Karolis Stravinskas, Aleksej Romanov, Rimantas Kačianauskas, Sergejus Borodinas and Genrik Mordas

Mathematics 2021, 9(22), 2913; https://doi.org/10.3390/math9222913 - 16 Nov 2021

Cited by 2 | Viewed by 2663

Abstract

Metal additive manufacturing has received much attention in the past few decades, and it offers a variety of technologies for three-dimensional object production. One of such technologies, allowing large-sized object production, is laser-assisted metal deposition, the limits of which are determined by the [...] Read more.

Metal additive manufacturing has received much attention in the past few decades, and it offers a variety of technologies for three-dimensional object production. One of such technologies, allowing large-sized object production, is laser-assisted metal deposition, the limits of which are determined by the capabilities of the positioning system. The already-existing nozzles have either a relatively low build rate or a poor resolution. The goal of this work is to develop a new nozzle with a centered particle beam at high velocity for the laser-assisted metal additive manufacturing technologies. Scientific challenges are addressed with regards to the fluid dynamics, the particle-substrate contact, and tracking of the thermodynamic state during contact. In this paper, two nozzles based on the de Laval geometry with Witoszynski and Bicubic curves of convergence zone were designed; the results showed that the average flow velocity in a Bicubic outlet curve nozzle is around 615 m/s and in Witoszynski this is 435 m/s. Investigation of particle beam formation for the Bicubic curve geometry revealed that small particles have the highest velocity and the lowest total force at the nozzle outlet. Fine particles have a shorter response time, and therefore, a smaller dispersion area. The elasto-plastic particle-surface contact showed that particles of diameter limited up to 3 μm are able to reach experimentally obtained critical velocity without additional heating. For particle sizes above 10 μm, additional heating is needed for deposition. The maximum coefficient of restitution (COR) is achieved with a particle size of 30 μm; smaller particles are characterized by the values of COR, which are lower due to a relatively high velocity. Particles larger than 30 μm are scalable, characterized by a small change in velocity and a rise in temperature as their mass increases. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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22 pages, 4055 KiB

Open AccessArticle

A Mating Selection Based on Modified Strengthened Dominance Relation for NSGA-III

by Saykat Dutta, Sri Srinivasa Raju M, Rammohan Mallipeddi, Kedar Nath Das and Dong-Gyu Lee

Mathematics 2021, 9(22), 2837; https://doi.org/10.3390/math9222837 - 10 Nov 2021

Cited by 3 | Viewed by 2290

Abstract

In multi/many-objective evolutionary algorithms (MOEAs), to alleviate the degraded convergence pressure of Pareto dominance with the increase in the number of objectives, numerous modified dominance relationships were proposed. Recently, the strengthened dominance relation (SDR) has been proposed, where the dominance area of a [...] Read more.

In multi/many-objective evolutionary algorithms (MOEAs), to alleviate the degraded convergence pressure of Pareto dominance with the increase in the number of objectives, numerous modified dominance relationships were proposed. Recently, the strengthened dominance relation (SDR) has been proposed, where the dominance area of a solution is determined by convergence degree and niche size (

\bar{θ}

). Later, in controlled SDR (CSDR),

\bar{θ}

and an additional parameter

(k)

associated with the convergence degree are dynamically adjusted depending on the iteration count. Depending on the problem characteristics and the distribution of the current population, different situations require different values of

k

, rendering the linear reduction of k based on the generation count ineffective. This is because a particular value of

k

is expected to bias the dominance relationship towards a particular region on the Pareto front (PF). In addition, due to the same reason, using SDR or CSDR in the environmental selection cannot preserve the diversity of solutions required to cover the entire PF. Therefore, we propose an MOEA, referred to as NSGA-III*, where (1) a modified SDR (MSDR)-based mating selection with an adaptive ensemble of parameter k would prioritize parents from specific sections of the PF depending on k, and (2) the traditional weight vector and non-dominated sorting-based environmental selection of NSGA-III would protect the solutions corresponding to the entire PF. The performance of NSGA-III* is favourably compared with state-of-the-art MOEAs on DTLZ and WFG test suites with up to 10 objectives. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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12 pages, 300 KiB

Open AccessArticle

Sixth Order Numerov-Type Methods with Coefficients Trained to Perform Best on Problems with Oscillating Solutions

by Vladislav N. Kovalnogov, Ruslan V. Fedorov, Tamara V. Karpukhina, Theodore E. Simos and Charalampos Tsitouras

Mathematics 2021, 9(21), 2756; https://doi.org/10.3390/math9212756 - 29 Oct 2021

Cited by 7 | Viewed by 2086

Abstract

Numerov-type methods using four stages per step and sharing sixth algebraic order are considered. The coefficients of such methods are depended on two free parameters. For addressing problems with oscillatory solutions, we traditionally try to satisfy some specific properties such as reduce the [...] Read more.

Numerov-type methods using four stages per step and sharing sixth algebraic order are considered. The coefficients of such methods are depended on two free parameters. For addressing problems with oscillatory solutions, we traditionally try to satisfy some specific properties such as reduce the phase-lag error, extend the interval of periodicity or even nullify the amplification. All of these latter properties come from a test problem that poses as a solution to an ideal trigonometric orbit. Here, we propose the training of the coefficients of the selected family of methods in a wide set of relevant problems. After performing this training using the differential evolution technique, we arrive at a certain method that outperforms the other ones from this family in an even wider set of oscillatory problems. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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21 pages, 5924 KiB

Open AccessArticle

Adjustment of Force–Gradient Operator in Symplectic Methods

by Lina Zhang, Xin Wu and Enwei Liang

Mathematics 2021, 9(21), 2718; https://doi.org/10.3390/math9212718 - 27 Oct 2021

Cited by 6 | Viewed by 1730

Abstract

Many force–gradient explicit symplectic integration algorithms have been designed for the Hamiltonian

H = T (p) + V (q)

with kinetic energy

T (p) = p^{2} / 2

in the existing references. When a force–gradient operator [...] Read more.

Many force–gradient explicit symplectic integration algorithms have been designed for the Hamiltonian

H = T (p) + V (q)

with kinetic energy

T (p) = p^{2} / 2

in the existing references. When a force–gradient operator is appropriately adjusted as a new operator, it is still suitable for a class of Hamiltonian problems

H = K (p, q) + V (q)

with integrable part

K (p, q) = \sum_{i = 1}^{n} \sum_{j = 1}^{n} a_{i j} p_{i} p_{j} + \sum_{i = 1}^{n} b_{i} p_{i}

, where

a_{i j} = a_{i j} (q)

and

b_{i} = b_{i} (q)

are functions of coordinates

q

. The newly adjusted operator is not a force–gradient operator but is similar to the momentum-version operator associated to the potential V. The newly extended (or adjusted) algorithms are no longer solvers of the original Hamiltonian, but are solvers of slightly modified Hamiltonians. They are explicit symplectic integrators with symmetry or time reversibility. Numerical tests show that the standard symplectic integrators without the new operator are generally poorer than the corresponding extended methods with the new operator in computational accuracies and efficiencies. The optimized methods have better accuracies than the corresponding non-optimized counterparts. Among the tested symplectic methods, the two extended optimized seven-stage fourth-order methods of Omelyan, Mryglod and Folk exhibit the best numerical performance. As a result, one of the two optimized algorithms is used to study the orbital dynamical features of a modified Hénon–Heiles system and a spring pendulum. These extended integrators allow for integrations in Hamiltonian problems, such as the spiral structure in self-consistent models of rotating galaxies and the spiral arms in galaxies. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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26 pages, 19281 KiB

Open AccessArticle

Conservative Finite-Difference Schemes for Two Nonlinear Schrödinger Equations Describing Frequency Tripling in a Medium with Cubic Nonlinearity: Competition of Invariants

by Vyacheslav Trofimov and Maria Loginova

Mathematics 2021, 9(21), 2716; https://doi.org/10.3390/math9212716 - 26 Oct 2021

Cited by 2 | Viewed by 1862

Abstract

Two 1D nonlinear coupled Schrödinger equations are often used for describing optical frequency conversion possessing a few conservation laws (invariants), for example, the energy’s invariant and the Hamiltonian. Their influence on the properties of the finite-difference schemes (FDSs) may be different. The influence [...] Read more.

Two 1D nonlinear coupled Schrödinger equations are often used for describing optical frequency conversion possessing a few conservation laws (invariants), for example, the energy’s invariant and the Hamiltonian. Their influence on the properties of the finite-difference schemes (FDSs) may be different. The influence of each of both invariants on the computer simulation result accuracy is analyzed while solving the problem describing the third optical harmonic generation process. Two implicit conservative FDSs are developed for a numerical solution of this problem. One of them preserves a difference analog of the energy invariant (or the Hamiltonian) accurately, while the Hamiltonian (or the energy’s invariant) is preserved with the second order of accuracy. Both FDSs possess the second order of approximation at a smooth enough solution of the differential problem. Computer simulations demonstrate advantages of the implicit FDS preserving the Hamiltonian. To illustrate the advantages of the developed FDSs, a comparison of the computer simulation results with those obtained applying the Strang method, based on either an implicit scheme or the Runge–Kutta method, is made. The corresponding theorems, which claim the second order of approximation for preserving invariants for the FDSs under consideration, are stated. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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22 pages, 6577 KiB

Open AccessArticle

Three-Dimensional Numerical Modeling of Internal Ballistics for Solid Propellant Combinations

by Ramón A. Otón-Martínez, Francisco Javier S. Velasco, Francisco Nicolás-Pérez, José R. García-Cascales and Ramón Mur-Sanz de Galdeano

Mathematics 2021, 9(21), 2714; https://doi.org/10.3390/math9212714 - 26 Oct 2021

Cited by 12 | Viewed by 4908

Abstract

The processes that take place within the Internal Ballistics cycle of an artillery round are highly influenced by geometric effects. They are also highly affected by the presence of a combination of energetic materials, such as the propellant, igniter, primer, and the combustible [...] Read more.

The processes that take place within the Internal Ballistics cycle of an artillery round are highly influenced by geometric effects. They are also highly affected by the presence of a combination of energetic materials, such as the propellant, igniter, primer, and the combustible cartridge cases. For a more realistic simulation of these phenomena, a multidimensional and multicomponent numerical model is presented, based on adaptations and improvements of previous models of conservation equations, maintaining a two-phase, Eulerian–Eulerian approximation. A numerical method based on Finite Volumes and conservative flux schemes (Rusanov and AUSM+), with the ability to predict detonation effects, is proposed. As a result, a versatile 3D numerical code was obtained that was tested in the simulation of artillery firing with conventional and modular charges (MACS). Results show the code is able to characterize the heat and mass transfer of the different energetic materials during the combustion of the propellant and the cartridge cases, the gas expansion, and the projectile acceleration. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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11 pages, 2605 KiB

Open AccessArticle

Estimating Gini Coefficient from Grouped Data Based on Shape-Preserving Cubic Hermite Interpolation of Lorenz Curve

by Songpu Shang and Songhao Shang

Mathematics 2021, 9(20), 2551; https://doi.org/10.3390/math9202551 - 12 Oct 2021

Cited by 3 | Viewed by 3777

Abstract

The Lorenz curve and Gini coefficient are widely used to describe inequalities in many fields, but accurate estimation of the Gini coefficient is still difficult for grouped data with fewer groups. We proposed a shape-preserving cubic Hermite interpolation method to approximate the Lorenz [...] Read more.

The Lorenz curve and Gini coefficient are widely used to describe inequalities in many fields, but accurate estimation of the Gini coefficient is still difficult for grouped data with fewer groups. We proposed a shape-preserving cubic Hermite interpolation method to approximate the Lorenz curve by maximizing or minimizing the strain energy or curvature variation energy of the interpolation curve, and a method to estimate the Gini coefficient directly from the coefficients of the interpolation curve. This interpolation method can preserve the essential requirements of the Lorenz curve, i.e., non-negativity, monotonicity, and convexity, and can estimate the derivatives at intermediate points and endpoints at the same time. These methods were tested with 16 grouped quintiles or unequally spaced datasets, and the results were compared with the true Gini coefficients calculated with all census data and results estimated with other methods. Results indicate that the maximum strain energy interpolation method generally performs the best among different methods, which is applicable to both equally and unequally spaced grouped datasets with higher precision, especially for grouped data with fewer groups. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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15 pages, 1714 KiB

Open AccessArticle

Machine Learning Applied to the Oxygen-18 Isotopic Composition, Salinity and Temperature/Potential Temperature in the Mediterranean Sea

by Gonzalo Astray, Benedicto Soto, Enrique Barreiro, Juan F. Gálvez and Juan C. Mejuto

Mathematics 2021, 9(19), 2523; https://doi.org/10.3390/math9192523 - 8 Oct 2021

Cited by 6 | Viewed by 2849

Abstract

This study proposed different techniques to estimate the isotope composition (δ¹⁸O), salinity and temperature/potential temperature in the Mediterranean Sea using five different variables: (i–ii) geographic coordinates (Longitude, Latitude), (iii) year, (iv) month and (v) depth. Three kinds of models based on [...] Read more.

This study proposed different techniques to estimate the isotope composition (δ¹⁸O), salinity and temperature/potential temperature in the Mediterranean Sea using five different variables: (i–ii) geographic coordinates (Longitude, Latitude), (iii) year, (iv) month and (v) depth. Three kinds of models based on artificial neural network (ANN), random forest (RF) and support vector machine (SVM) were developed. According to the results, the random forest models presents the best prediction accuracy for the querying phase and can be used to predict the isotope composition (mean absolute percentage error (MAPE) around 4.98%), salinity (MAPE below 0.20%) and temperature (MAPE around 2.44%). These models could be useful for research works that require the use of past data for these variables. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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20 pages, 4272 KiB

Open AccessArticle

A Quadratic–Exponential Model of Variogram Based on Knowing the Maximal Variability: Application to a Rainfall Time Series

by Francisco Gerardo Benavides-Bravo, Roberto Soto-Villalobos, José Roberto Cantú-González, Mario A. Aguirre-López and Ángela Gabriela Benavides-Ríos

Mathematics 2021, 9(19), 2466; https://doi.org/10.3390/math9192466 - 3 Oct 2021

Cited by 7 | Viewed by 3132

Abstract

Variogram models are a valuable tool used to analyze the variability of a time series; such variability usually entails a spherical or exponential behavior, and so, models based on such functions are commonly used to fit and explain a time series. Variograms have [...] Read more.

Variogram models are a valuable tool used to analyze the variability of a time series; such variability usually entails a spherical or exponential behavior, and so, models based on such functions are commonly used to fit and explain a time series. Variograms have a quasi-periodic structure for rainfall cases, and some extra steps are required to analyze their entire behavior. In this work, we detailed a procedure for a complete analysis of rainfall time series, from the construction of the experimental variogram to curve fitting with well-known spherical and exponential models, and finally proposed a novel model: quadratic–exponential. Our model was developed based on the analysis of 6 out of 30 rainfall stations from our case study: the Río Bravo–San Juan basin, and was constructed from the exponential model while introducing a quadratic behavior near to the origin and taking into account the fact that the maximal variability of the process is known. Considering a sample with diverse Hurst exponents, the stations were selected. The results obtained show robustness in our proposed model, reaching a good fit with and without the nugget effect for different Hurst exponents. This contrasts to previous models, which show good outcomes only without the nugget effect. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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18 pages, 14492 KiB

Open AccessArticle

Generalised S-System-Type Equation: Sensitivity of the Deterministic and Stochastic Models for Bone Mechanotransduction

by Julijana Simonović and Thomas E. Woolley

Mathematics 2021, 9(19), 2422; https://doi.org/10.3390/math9192422 - 29 Sep 2021

Viewed by 1600

Abstract

The formalism of a bone cell population model is generalised to be of the form of an S-System. This is a system of nonlinear coupled ordinary differential equations (ODEs), each with the same structure: the change in a variable is equal to a [...] Read more.

The formalism of a bone cell population model is generalised to be of the form of an S-System. This is a system of nonlinear coupled ordinary differential equations (ODEs), each with the same structure: the change in a variable is equal to a difference in the product of a power-law functions with a specific variable. The variables are the densities of a variety of biological populations involved in bone remodelling. They will be specified concretely in the cases of a specific periodically forced system to describe the osteocyte mechanotransduction activities. Previously, such models have only been deterministically simulated causing the populations to form a continuum. Thus, very little is known about how sensitive the model of mechanotransduction is to perturbations in parameters and noise. Here, we revisit this assumption using a Stochastic Simulation Algorithm (SSA), which allows us to directly simulate the discrete nature of the problem and encapsulate the noisy features of individual cell division and death. Critically, these stochastic features are able to cause unforeseen dynamics in the system, as well as completely change the viable parameter region, which produces biologically realistic results. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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12 pages, 312 KiB

Open AccessArticle

Runge–Kutta Pairs of Orders 5(4) Trained to Best Address Keplerian Type Orbits

by Vladislav N. Kovalnogov, Ruslan V. Fedorov, Tamara V. Karpukhina, Theodore E. Simos and Charalampos Tsitouras

Mathematics 2021, 9(19), 2400; https://doi.org/10.3390/math9192400 - 27 Sep 2021

Cited by 8 | Viewed by 1971

Abstract

The derivation of Runge–Kutta pairs of orders five and four that effectively uses six stages per step is considered. The coefficients provided by such a method are 27 and have to satisfy a system of 25 nonlinear equations. Traditionally, various solutions have been [...] Read more.

The derivation of Runge–Kutta pairs of orders five and four that effectively uses six stages per step is considered. The coefficients provided by such a method are 27 and have to satisfy a system of 25 nonlinear equations. Traditionally, various solutions have been tried. Each of these solutions makes use of some simplified assumptions and offers different families of methods. Here, we make use of the most celebrated family to appear in the literature, where we may use as the last stage the first function evaluation from the next step (FSAL property). The family under consideration has the advantage of being solved explicitly. Actually, we arrive at a subsystem where all the coefficients are found with respect to five free parameters. These free parameters are adjusted (trained) in order to deliver a pair that outperforms other similar pairs of orders

5 (4)

in Keplerian type orbits, e.g., Kepler, perturbed Kepler, Arenstorf orbit or Pleiades. The training uses differential evolution technique. The finally proposed pair has a remarkable performance and offers on average more than a digit of accuracy in a variety of orbits. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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15 pages, 7812 KiB

Open AccessArticle

Generalized Kalman Filter and Ensemble Optimal Interpolation, Their Comparison and Application to the Hybrid Coordinate Ocean Model

by Konstantin Belyaev, Andrey Kuleshov, Ilya Smirnov and Clemente A. S. Tanajura

Mathematics 2021, 9(19), 2371; https://doi.org/10.3390/math9192371 - 24 Sep 2021

Cited by 9 | Viewed by 1780

Abstract

In this paper, we consider a recently developed data assimilation method, the Generalized Kalman Filter (GKF), which is a generalization of the widely-used Ensemble Optimal Interpolation (EnOI) method. Both methods are applied for modeling the Atlantic Ocean circulation using the known Hybrid Coordinate [...] Read more.

In this paper, we consider a recently developed data assimilation method, the Generalized Kalman Filter (GKF), which is a generalization of the widely-used Ensemble Optimal Interpolation (EnOI) method. Both methods are applied for modeling the Atlantic Ocean circulation using the known Hybrid Coordinate Ocean Model. The along-track altimetry data taken from the Archiving, Validating and Interpolating Satellite Oceanography Data (AVISO) were used for data assimilation and other data from independent archives of observations; particularly, the temperature and salinity data from the Pilot Research Array in the Tropical Atlantic were used for independent comparison. Several numerical experiments were performed with their results discussed and analyzed. It is shown that values of the ocean state variables obtained in the calculations using the GKF method are closer to the observations in terms of standard metrics in comparison with the calculations using the standard data assimilation method EnOI. Furthermore, the GKF method requires less computational effort compared to the EnOI method. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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13 pages, 530 KiB

Open AccessArticle

A Coupling between Integral Equations and On-Surface Radiation Conditions for Diffraction Problems by Non Convex Scatterers

by Saleh Mousa Alzahrani, Xavier Antoine and Chokri Chniti

Mathematics 2021, 9(18), 2299; https://doi.org/10.3390/math9182299 - 17 Sep 2021

Cited by 1 | Viewed by 1794

Abstract

The aim of this paper is to introduce an orignal coupling procedure between surface integral equation formulations and on-surface radiation condition (OSRC) methods for solving two-dimensional scattering problems for non convex structures. The key point is that the use of the OSRC introduces [...] Read more.

The aim of this paper is to introduce an orignal coupling procedure between surface integral equation formulations and on-surface radiation condition (OSRC) methods for solving two-dimensional scattering problems for non convex structures. The key point is that the use of the OSRC introduces a sparse block in the surface operator representation of the wave field while the integral part leads to an improved accuracy of the OSRC method in the non convex part of the scattering structure. The procedure is given for both the Dirichlet and Neumann scattering problems. Some numerical simulations show the improvement induced by the coupling method. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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14 pages, 700 KiB

Open AccessArticle

On the Diffuse Interface Models for High Codimension Dispersed Inclusions

by Elizaveta Zipunova and Evgeny Savenkov

Mathematics 2021, 9(18), 2206; https://doi.org/10.3390/math9182206 - 8 Sep 2021

Cited by 4 | Viewed by 1605

Abstract

Diffuse interface models are widely used to describe the evolution of multi-phase systems of various natures. Dispersed inclusions described by these models are usually three-dimensional (3D) objects characterized by phase field distribution. When employed to describe elastic fracture evolution, the dispersed phase elements [...] Read more.

Diffuse interface models are widely used to describe the evolution of multi-phase systems of various natures. Dispersed inclusions described by these models are usually three-dimensional (3D) objects characterized by phase field distribution. When employed to describe elastic fracture evolution, the dispersed phase elements are effectively two-dimensional (2D) objects. An example of the model with effectively one-dimensional (1D) dispersed inclusions is a phase field model for electric breakdown in solids. Any diffuse interface field model is defined by an appropriate free energy functional, which depends on a phase field and its derivatives. In this work we show that codimension of the dispersed inclusions significantly restricts the functional dependency of the free energy on the derivatives of the problem state variables. It is shown that to describe codimension 2 diffuse objects, the free energy of the model necessarily depends on higher order derivatives of the phase field or needs an additional smoothness of the solution, i.e., its first derivatives should be integrable with a power greater than two. Numerical experiments are presented to support our theoretical discussion. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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20 pages, 962 KiB

Open AccessArticle

Weighted Fractional-Order Transform Based on Periodic Matrix

by Tieyu Zhao and Yingying Chi

Mathematics 2021, 9(17), 2073; https://doi.org/10.3390/math9172073 - 27 Aug 2021

Viewed by 1270

Abstract

Tao et al. proposed the definition of the linear summation of fractional-order matrices based on the theory of Yeh and Pei. This definition was further extended and applied to image encryption. In this paper, we propose a reformulation of the definitions of Yeh [...] Read more.

Tao et al. proposed the definition of the linear summation of fractional-order matrices based on the theory of Yeh and Pei. This definition was further extended and applied to image encryption. In this paper, we propose a reformulation of the definitions of Yeh et al. and Tao et al. and analyze them theoretically. The results show that many weighted terms are invalid. Therefore, we use the proposed reformulation to prove that the effective weighted terms depend on the period of the matrix. This also shows that the image encryption methods based on the weighted fractional-order transform will lead to the security risk of key invalidation. Finally, our hypothesis is verified by the unified theoretical framework of multiple-parameter discrete fractional-order transforms. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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32 pages, 1866 KiB

Open AccessArticle

A New Extended Model with Bathtub-Shaped Failure Rate: Properties, Inference, Simulation, and Applications

by Alya Al Mutairi, Muhammad Z. Iqbal, Muhammad Z. Arshad, Badr Alnssyan, Hazem Al-Mofleh and Ahmed Z. Afify

Mathematics 2021, 9(17), 2024; https://doi.org/10.3390/math9172024 - 24 Aug 2021

Cited by 9 | Viewed by 2430

Abstract

Theoretical and applied researchers have been frequently interested in proposing alternative skewed and symmetric lifetime parametric models that provide greater flexibility in modeling real-life data in several applied sciences. To fill this gap, we introduce a three-parameter bounded lifetime model called the exponentiated [...] Read more.

Theoretical and applied researchers have been frequently interested in proposing alternative skewed and symmetric lifetime parametric models that provide greater flexibility in modeling real-life data in several applied sciences. To fill this gap, we introduce a three-parameter bounded lifetime model called the exponentiated new power function (E-NPF) distribution. Some of its mathematical and reliability features are discussed. Furthermore, many possible shapes over certain choices of the model parameters are presented to understand the behavior of the density and hazard rate functions. For the estimation of the model parameters, we utilize eight classical approaches of estimation and provide a simulation study to assess and explore the asymptotic behaviors of these estimators. The maximum likelihood approach is used to estimate the E-NPF parameters under the type II censored samples. The efficiency of the E-NPF distribution is evaluated by modeling three lifetime datasets, showing that the E-NPF distribution gives a better fit over its competing models such as the Kumaraswamy-PF, Weibull-PF, generalized-PF, Kumaraswamy, and beta distributions. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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17 pages, 777 KiB

Open AccessArticle

Mathematical Modelling of Turbulent Combustion of Two-Phase Mixtures of Gas and Solid Particles with a Eulerian–Eulerian Approach: The Case of Hydrogen Combustion in the Presence of Graphite Particles

by Francisco Nicolás-Pérez, F.J.S. Velasco, Ramón A. Otón-Martínez, José R. García-Cascales, Ahmed Bentaib and Nabiha Chaumeix

Mathematics 2021, 9(17), 2017; https://doi.org/10.3390/math9172017 - 24 Aug 2021

Cited by 2 | Viewed by 2398

Abstract

The numerical modelling of turbulent combustion of H₂–air mixtures with solid graphite particles is a challenging and key issue in many industrial problems including nuclear safety. This study presents a Eulerian–Eulerian model based on the resolution of the Navier–Stokes equations via [...] Read more.

The numerical modelling of turbulent combustion of H₂–air mixtures with solid graphite particles is a challenging and key issue in many industrial problems including nuclear safety. This study presents a Eulerian–Eulerian model based on the resolution of the Navier–Stokes equations via large eddy simulation (LES) coupled with a system of ordinary differential equations (ODEs) of the detailed chemical kinetics to simulate the combustion of mixtures of gases and particles. The model was applied to predict the transient evolution of turbulent combustion sequences of mixtures of hydrogen, air and graphite particles under low concentration conditions. When applied to simulate lab-scale combustion experiments, the results showed a good agreement between experimental and numerical data using a detailed chemical kinetic model. Moreover, the model was able to predict some key experimental tendencies and revealed that the presence of a low concentration of graphite particles (~96 g/m³) in the scenario influenced the hydrogen combustion dynamics for mixtures of 20% (in volume) of hydrogen in air. Under these conditions, pressure levels reached at the walls of the sphere were increased and the combustion time was shortened. The results also showed the viability of using this kind of a model for obtaining global combustion parameters such as wall pressure evolution with time. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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22 pages, 1128 KiB

Open AccessArticle

Numerical Investigation of Fuzzy Predator-Prey Model with a Functional Response of the Form Arctan(ax)

by Saed Mallak, Doa’a Farekh and Basem Attili

Mathematics 2021, 9(16), 1919; https://doi.org/10.3390/math9161919 - 12 Aug 2021

Cited by 5 | Viewed by 1737

Abstract

In this paper we study a fuzzy predator-prey model with functional response

a r c t a n (a x) .

The fuzzy derivatives are approximated using the generalized Hukuhara derivative. To execute the numerical simulation, we use the fuzzy Runge-Kutta [...] Read more.

In this paper we study a fuzzy predator-prey model with functional response

a r c t a n (a x) .

The fuzzy derivatives are approximated using the generalized Hukuhara derivative. To execute the numerical simulation, we use the fuzzy Runge-Kutta method. The results obtained over time for the evolution and the population are presented numerically and graphically with some conclusions. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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25 pages, 1231 KiB

Open AccessArticle

A New Family of High-Order Ehrlich-Type Iterative Methods

by Petko D. Proinov and Maria T. Vasileva

Mathematics 2021, 9(16), 1855; https://doi.org/10.3390/math9161855 - 5 Aug 2021

Cited by 5 | Viewed by 1900

Abstract

One of the famous third-order iterative methods for finding simultaneously all the zeros of a polynomial was introduced by Ehrlich in 1967. In this paper, we construct a new family of high-order iterative methods as a combination of Ehrlich’s iteration function and an [...] Read more.

One of the famous third-order iterative methods for finding simultaneously all the zeros of a polynomial was introduced by Ehrlich in 1967. In this paper, we construct a new family of high-order iterative methods as a combination of Ehrlich’s iteration function and an arbitrary iteration function. We call these methods Ehrlich’s methods with correction. The paper provides a detailed local convergence analysis of presented iterative methods for a large class of iteration functions. As a consequence, we obtain two types of local convergence theorems as well as semilocal convergence theorems (with computer verifiable initial condition). As special cases of the main results, we study the convergence of several particular iterative methods. The paper ends with some experiments that show the applicability of our semilocal convergence theorems. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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10 pages, 269 KiB

Open AccessArticle

A Neural Network Technique for the Derivation of Runge–Kutta Pairs Adjusted for Scalar Autonomous Problems

by Vladislav N. Kovalnogov, Ruslan V. Fedorov, Yuri A. Khakhalev, Theodore E. Simos and Charalampos Tsitouras

Mathematics 2021, 9(16), 1842; https://doi.org/10.3390/math9161842 - 4 Aug 2021

Cited by 9 | Viewed by 1937

Abstract

We consider the scalar autonomous initial value problem as solved by an explicit Runge–Kutta pair of orders 6 and 5. We focus on an efficient family of such pairs, which were studied extensively in previous decades. This family comes with 5 coefficients that [...] Read more.

We consider the scalar autonomous initial value problem as solved by an explicit Runge–Kutta pair of orders 6 and 5. We focus on an efficient family of such pairs, which were studied extensively in previous decades. This family comes with 5 coefficients that one is able to select arbitrarily. We set, as a fitness function, a certain measure, which is evaluated after running the pair in a couple of relevant problems. Thus, we may adjust the coefficients of the pair, minimizing this fitness function using the differential evolution technique. We conclude with a method (i.e. a Runge–Kutta pair) which outperforms other pairs of the same two orders in a variety of scalar autonomous problems. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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29 pages, 1143 KiB

Open AccessArticle

Space Analyticity and Bounds for Derivatives of Solutions to the Evolutionary Equations of Diffusive Magnetohydrodynamics

by Vladislav Zheligovsky

Mathematics 2021, 9(15), 1789; https://doi.org/10.3390/math9151789 - 28 Jul 2021

Cited by 3 | Viewed by 1788

Abstract

In 1981, Foias, Guillopé and Temam proved a priori estimates for arbitrary-order space derivatives of solutions to the Navier–Stokes equation. Such bounds are instructive in the numerical investigation of intermittency that is often observed in simulations, e.g., numerical study of vorticity moments by [...] Read more.

In 1981, Foias, Guillopé and Temam proved a priori estimates for arbitrary-order space derivatives of solutions to the Navier–Stokes equation. Such bounds are instructive in the numerical investigation of intermittency that is often observed in simulations, e.g., numerical study of vorticity moments by Donzis et al. (2013) revealed depletion of nonlinearity that may be responsible for smoothness of solutions to the Navier–Stokes equation. We employ an original method to derive analogous estimates for space derivatives of three-dimensional space-periodic weak solutions to the evolutionary equations of diffusive magnetohydrodynamics. Construction relies on space analyticity of the solutions at almost all times. An auxiliary problem is introduced, and a Sobolev norm of its solutions bounds from below the size in

C^{3}

of the region of space analyticity of the solutions to the original problem. We recover the exponents obtained earlier for the hydrodynamic problem. Moreover, the same approach is followed here to derive and prove similar a priori bounds for arbitrary-order space derivatives of the first-order time derivative of the weak MHD solutions. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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14 pages, 381 KiB

Open AccessArticle

An Economic Model for OECD Economies with Truncated M-Derivatives: Exact Solutions and Simulations

by Luis A. Quezada-Téllez, Guillermo Fernández-Anaya, Dominique Brun-Battistini, Benjamín Nuñez-Zavala and Jorge E. Macías-Díaz

Mathematics 2021, 9(15), 1780; https://doi.org/10.3390/math9151780 - 28 Jul 2021

Cited by 8 | Viewed by 2779

Abstract

This article proposes two conformal Solow models (with and without migration), accompanied by simulations for six Organisation for Economic Co-operation and Development economies. The models are proposed by employing suitable Inada conditions on the Cobb–Douglas function and making use of the truncated M [...] Read more.

This article proposes two conformal Solow models (with and without migration), accompanied by simulations for six Organisation for Economic Co-operation and Development economies. The models are proposed by employing suitable Inada conditions on the Cobb–Douglas function and making use of the truncated M-derivative for the Mittag–Leffler function. In the exact solutions derived in this manuscript, two new parameters play an important role in the convergence towards, or the divergence from, the steady state of capital and per capita product. The economical dynamics of these nations are influenced by the intensity of the capital and labor factors, as well as the level of depreciation, the labor force rate and the level of saving. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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14 pages, 4543 KiB

Open AccessArticle

Experimental and Numerical Analysis of Mode I Fracture Process of Rock by Semi-Circular Bend Specimen

by Peng Xiao, Diyuan Li, Guoyan Zhao and Meng Liu

Mathematics 2021, 9(15), 1769; https://doi.org/10.3390/math9151769 - 27 Jul 2021

Cited by 23 | Viewed by 2688

Abstract

The semi-circular bend (SCB) specimen is widely used to measure fracture toughness of brittle materials such as rock. In this work, the stress field, fracture process zone (FPZ), and crack propagation velocity of SCB specimen are analyzed during the fracture process of rock [...] Read more.

The semi-circular bend (SCB) specimen is widely used to measure fracture toughness of brittle materials such as rock. In this work, the stress field, fracture process zone (FPZ), and crack propagation velocity of SCB specimen are analyzed during the fracture process of rock specimens. The FPZ of specimen is obtained by experimental and numerical methods under a three-point bend test. The stress concentration zones of specimen present a heart shape at peak load points. FPZ forms before macro fracture occurs. The macro fracture form inside FPZ in a post-peak region of a load–displacement curve. The crack propagation process of specimen include two stages, namely the rapid crack initial development stage, and the final crack splitting stage. The maximum crack propagation velocity of specimen is about 267 m/s, and the average crack propagation velocity is about 111 m/s. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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31 pages, 1944 KiB

Open AccessArticle

A Mass- and Energy-Conserving Numerical Model for a Fractional Gross–Pitaevskii System in Multiple Dimensions

by Adán J. Serna-Reyes and Jorge E. Macías-Díaz

Mathematics 2021, 9(15), 1765; https://doi.org/10.3390/math9151765 - 26 Jul 2021

Cited by 3 | Viewed by 1744

Abstract

This manuscript studies a double fractional extended p-dimensional coupled Gross–Pitaevskii-type system. This system consists of two parabolic partial differential equations with equal interaction constants, coupling terms, and spatial derivatives of the Riesz type. Associated with the mathematical model, there are energy and [...] Read more.

This manuscript studies a double fractional extended p-dimensional coupled Gross–Pitaevskii-type system. This system consists of two parabolic partial differential equations with equal interaction constants, coupling terms, and spatial derivatives of the Riesz type. Associated with the mathematical model, there are energy and non-negative mass functions which are conserved throughout time. Motivated by this fact, we propose a finite-difference discretization of the double fractional Gross–Pitaevskii system which inherits the energy and mass conservation properties. As the continuous model, the mass is a non-negative constant and the solutions are bounded under suitable numerical parameter assumptions. We prove rigorously the existence of solutions for any set of initial conditions. As in the continuous system, the discretization has a discrete Hamiltonian associated. The method is implicit, multi-consistent, stable and quadratically convergent. Finally, we implemented the scheme computationally to confirm the validity of the mass and energy conservation properties, obtaining satisfactory results. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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25 pages, 3738 KiB

Open AccessArticle

Optimal Control of Insect Populations

by Anderson L. Albuquerque de Araujo, José L. Boldrini, Roberto C. Cabrales, Enrique Fernández-Cara and Milton L. Oliveira

Mathematics 2021, 9(15), 1762; https://doi.org/10.3390/math9151762 - 26 Jul 2021

Cited by 2 | Viewed by 1995

Abstract

We consider some optimal control problems for systems governed by linear parabolic PDEs with local controls that can move along the domain region

Ω

of the plane. We prove the existence of optimal paths and also deduce the first order necessary optimality conditions, [...] Read more.

We consider some optimal control problems for systems governed by linear parabolic PDEs with local controls that can move along the domain region

Ω

of the plane. We prove the existence of optimal paths and also deduce the first order necessary optimality conditions, using the Dubovitskii–Milyutin’s formalism, which leads to an iterative algorithm of the fixed-point kind. This problem may be considered as a model for the control of a mosquito population existing in a given region by using moving insecticide spreading devices. In this situation, an optimal control is any trajectory or path that must follow such spreading device in order to reduce the population as much as possible with a reasonable not too expensive strategy. We illustrate our results by presenting some numerical experiments. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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20 pages, 4208 KiB

Open AccessArticle

Numerical Analysis of Flow Phenomena in Discharge Object with Siphon Using Lattice Boltzmann Method and CFD

by Jiří Fürst, Tomáš Halada, Milan Sedlář, Tomáš Krátký, Pavel Procházka and Martin Komárek

Mathematics 2021, 9(15), 1734; https://doi.org/10.3390/math9151734 - 22 Jul 2021

Cited by 4 | Viewed by 2131

Abstract

This article presents numerical simulation of flow in the discharge object with the welded siphon and the free water level. The main numerical tool used in this study is the lattice Boltzmann method combined with the Volume-of-Fluid approach and the Smagorinski LES model. [...] Read more.

This article presents numerical simulation of flow in the discharge object with the welded siphon and the free water level. The main numerical tool used in this study is the lattice Boltzmann method combined with the Volume-of-Fluid approach and the Smagorinski LES model. Some aspects of the numerical method are discussed, especially the formulation of the outlet boundary condition. The simulations are carried out with in-house software based on the open-source Palabos framework. Presented results are compared with the CFD simulations, based on the ANSYS CFX software applying the SST and SAS turbulence models and the free-surface flow modeling by means of the Volume-of-Fluid method. The evolution and interactions of main flow structures are analyzed using visualizations and the spectral analysis. All numerical simulations are verified by the experimental data obtained in the hydraulic laboratory with water circuit. A stationary flow regime has been visualized by means of PIV. Both the vertical planes and horizontal planes have been examined, focused mainly on the regions below and behind the siphon outlet. The results show a good agreement of calculated and measured complex flow structures, including time-averaged and instantaneous flow fields. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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10 pages, 596 KiB

Open AccessArticle

Modelling Oil Price with Lie Algebras and Long Short-Term Memory Networks

by Melike Bildirici, Nilgun Guler Bayazit and Yasemen Ucan

Mathematics 2021, 9(14), 1708; https://doi.org/10.3390/math9141708 - 20 Jul 2021

Cited by 4 | Viewed by 2379

Abstract

In this paper, we propose hybrid models for modelling the daily oil price during the period from 2 January 1986 to 5 April 2021. The models on

S^{2}

manifolds that we consider, including the reference ones, employ matrix representations rather than differential [...] Read more.

In this paper, we propose hybrid models for modelling the daily oil price during the period from 2 January 1986 to 5 April 2021. The models on

S^{2}

manifolds that we consider, including the reference ones, employ matrix representations rather than differential operator representations of Lie algebras. Firstly, the performance of Lie_NLS model is examined in comparison to the Lie-OLS model. Then, both of these reference models are improved by integrating them with a recurrent neural network model used in deep learning. Thirdly, the forecasting performance of these two proposed hybrid models on the

S^{2}

manifold, namely Lie-LSTM_OLS and Lie-LSTM_NLS, are compared with those of the reference Lie_OLS and Lie_NLS models. The in-sample and out-of-sample results show that our proposed methods can achieve improved performance over Lie_OLS and Lie_NLS models in terms of RMSE and MAE metrics and hence can be more reliably used to assess volatility of time-series data. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

11 pages, 624 KiB

Open AccessArticle

Inverse Problem for Ising Connection Matrix with Long-Range Interaction

by Leonid Litinskii and Boris Kryzhanovsky

Mathematics 2021, 9(14), 1624; https://doi.org/10.3390/math9141624 - 9 Jul 2021

Cited by 2 | Viewed by 1453

Abstract

In the present paper, we examine Ising systems on d-dimensional hypercube lattices and solve an inverse problem where we have to determine interaction constants of an Ising connection matrix when we know a spectrum of its eigenvalues. In addition, we define restrictions [...] Read more.

In the present paper, we examine Ising systems on d-dimensional hypercube lattices and solve an inverse problem where we have to determine interaction constants of an Ising connection matrix when we know a spectrum of its eigenvalues. In addition, we define restrictions allowing a random number sequence to be a connection matrix spectrum. We use the previously obtained analytical expressions for the eigenvalues of Ising connection matrices accounting for an arbitrary long-range interaction and supposing periodic boundary conditions. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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14 pages, 14917 KiB

Open AccessArticle

A Solution of Richards’ Equation by Generalized Finite Differences for Stationary Flow in a Dam

by Carlos Chávez-Negrete, Daniel Santana-Quinteros and Francisco Domínguez-Mota

Mathematics 2021, 9(14), 1604; https://doi.org/10.3390/math9141604 - 7 Jul 2021

Cited by 7 | Viewed by 2841

Abstract

The accurate description of the flow of water in porous media is of the greatest importance due to its numerous applications in several areas (groundwater, soil mechanics, etc.). The nonlinear Richards equation is often used as the governing equation that describes this phenomenon [...] Read more.

The accurate description of the flow of water in porous media is of the greatest importance due to its numerous applications in several areas (groundwater, soil mechanics, etc.). The nonlinear Richards equation is often used as the governing equation that describes this phenomenon and a large number of research studies aimed to solve it numerically. However, due to the nonlinearity of the constitutive expressions for permeability, it remains a challenging modeling problem. In this paper, the stationary form of Richards’ equation used in saturated soils is solved by two numerical methods: generalized finite differences, an emerging method that has been successfully applied to the transient case, and a finite element method, for benchmarking. The nonlinearity of the solution in both cases is handled using a Newtonian iteration. The comparative results show that a generalized finite difference iteration yields satisfactory results in a standard test problem with a singularity at the boundary. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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21 pages, 3372 KiB

Open AccessArticle

Solution of Moore–Gibson–Thompson Equation of an Unbounded Medium with a Cylindrical Hole

by Ahmed E. Abouelregal, Hakan Ersoy and Ömer Civalek

Mathematics 2021, 9(13), 1536; https://doi.org/10.3390/math9131536 - 30 Jun 2021

Cited by 77 | Viewed by 3910

Abstract

In the current article, in the presence of thermal and diffusion processes, the equations governing elastic materials through thermodiffusion are obtained. The Moore–Gibson–Thompson (MGT) equation modifies and defines the equations for thermal conduction and mass diffusion that occur in solids. This modification is [...] Read more.

In the current article, in the presence of thermal and diffusion processes, the equations governing elastic materials through thermodiffusion are obtained. The Moore–Gibson–Thompson (MGT) equation modifies and defines the equations for thermal conduction and mass diffusion that occur in solids. This modification is based on adding heat and diffusion relaxation times in the Green–Naghdi Type III (GN-III) models. In an unbounded medium with a cylindrical hole, the built model has been applied to examine the influence of the coupling between temperature and mass diffusion and responses. At constant concentration as well as intermittent and decaying varying heat, the surrounding cavity surface is traction-free and is filled slowly. Laplace transform and Laplace inversion techniques are applied to obtain the solutions of the studied field variables. In order to explore thermal diffusion analysis and find closed solutions, a suitable numerical approximation technique has been used. Comparisons are made between the results obtained with the results of the corresponding previous models. Additionally, to explain and realize the presented model, tables and figures for various physical fields are presented. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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7 pages, 323 KiB

Open AccessCommunication

Efficient Algorithm for the Computation of the Solution to a Sparse Matrix Equation in Distributed Control Theory

by Leonardo Pedroso and Pedro Batista

Mathematics 2021, 9(13), 1497; https://doi.org/10.3390/math9131497 - 25 Jun 2021

Cited by 6 | Viewed by 2015

Abstract

In this short communication, an algorithm for efficiently solving a sparse matrix equation, which arises frequently in the field of distributed control and estimation theory, is proposed. The efficient algorithm stems from the fact that the sparse equation at hand can be reduced [...] Read more.

In this short communication, an algorithm for efficiently solving a sparse matrix equation, which arises frequently in the field of distributed control and estimation theory, is proposed. The efficient algorithm stems from the fact that the sparse equation at hand can be reduced to a system of linear equations. The proposed algorithm is shown to require significantly fewer floating point operations than the state-of-the-art solution. The proposed solution is applied to a real-life example, which models a wide range of industrial processes. The experimental results show that the solution put forward allows for a significant increase in efficiency in relation to the state-of-the-art solution. The significant increase in efficiency of the presented algorithm allows for a valuable widening of the applications of distributed estimation and control. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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26 pages, 1396 KiB

Open AccessArticle

Integrated Structure-Control Design of a Bipedal Robot Based on Passive Dynamic Walking

by Josué Nathán Martínez-Castelán and Miguel Gabriel Villarreal-Cervantes

Mathematics 2021, 9(13), 1482; https://doi.org/10.3390/math9131482 - 24 Jun 2021

Cited by 5 | Viewed by 3121

Abstract

The design of bipedal robots is generally fulfilled through considering a sequential design approach, where a synergistic relationship between its structure and control features is not promoted. Hence, a novel integrated structure-control design approach is proposed to simultaneously obtain the optimal structural description, [...] Read more.

The design of bipedal robots is generally fulfilled through considering a sequential design approach, where a synergistic relationship between its structure and control features is not promoted. Hence, a novel integrated structure-control design approach is proposed to simultaneously obtain the optimal structural description, the torque magnitudes, and the on/off time intervals for the control signal input of a semi-passive bipedal robot. The proposed approach takes advantage of the natural dynamics of the system and the control signal activation/deactivation for generating stable gait cycles with minimum energy consumption. Consequently, the passive features of the semi-passive bipedal robot are included in the integrated structure-control design process through evaluating the system behavior along consecutive passive and semi-passive walking stages. Then, the proposed design approach is formulated as a nonlinear discontinuous dynamic optimization problem, where the solution search is carried out using the differential evolution algorithm due to the discontinuities of the semi-passive bipedal robot dynamics. The results of the proposal are compared with those obtained by a sequential design process. The integrated structure-control design achieves a reduction of

63.55 %

in the value of the performance function related to the synergy between the walking capability and energetic efficiency, with a reduction in the activation of the control and its magnitude of

95.41 %

. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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17 pages, 927 KiB

Open AccessFeature PaperArticle

An Operator-Based Scheme for the Numerical Integration of FDEs

by Inga Timofejeva, Zenonas Navickas, Tadas Telksnys, Romas Marcinkevicius and Minvydas Ragulskis

Mathematics 2021, 9(12), 1372; https://doi.org/10.3390/math9121372 - 13 Jun 2021

Cited by 2 | Viewed by 2198

Abstract

An operator-based scheme for the numerical integration of fractional differential equations is presented in this paper. The generalized differential operator is used to construct the analytic solution to the corresponding characteristic ordinary differential equation in the form of an infinite power series. The [...] Read more.

An operator-based scheme for the numerical integration of fractional differential equations is presented in this paper. The generalized differential operator is used to construct the analytic solution to the corresponding characteristic ordinary differential equation in the form of an infinite power series. The approximate numerical solution is constructed by truncating the power series, and by changing the point of the expansion. The developed adaptive integration step selection strategy is based on the controlled error of approximation induced by the truncation. Computational experiments are used to demonstrate the efficacy of the proposed scheme. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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23 pages, 410 KiB

Open AccessArticle

Analysis of a New Nonlinear Interpolatory Subdivision Scheme on σ Quasi-Uniform Grids

by Pedro Ortiz and Juan Carlos Trillo

Mathematics 2021, 9(12), 1320; https://doi.org/10.3390/math9121320 - 8 Jun 2021

Cited by 1 | Viewed by 1694

Abstract

In this paper, we introduce and analyze the behavior of a nonlinear subdivision operator called PPH, which comes from its associated PPH nonlinear reconstruction operator on nonuniform grids. The acronym PPH stands for Piecewise Polynomial Harmonic, since the reconstruction is built by using [...] Read more.

In this paper, we introduce and analyze the behavior of a nonlinear subdivision operator called PPH, which comes from its associated PPH nonlinear reconstruction operator on nonuniform grids. The acronym PPH stands for Piecewise Polynomial Harmonic, since the reconstruction is built by using piecewise polynomials defined by means of an adaption based on the use of the weighted Harmonic mean. The novelty of this work lies in the generalization of the already existing PPH subdivision scheme to the nonuniform case. We define the corresponding subdivision scheme and study some important issues related to subdivision schemes such as convergence, smoothness of the limit function, and preservation of convexity. In order to obtain general results, we consider σ quasi-uniform grids. We also perform some numerical experiments to reinforce the theoretical results. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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21 pages, 1889 KiB

Open AccessArticle

A Low Dissipative and Stable Cell-Centered Finite Volume Method with the Simultaneous Approximation Term for Compressible Turbulent Flows

by Myeongseok Kang and Donghyun You

Mathematics 2021, 9(11), 1206; https://doi.org/10.3390/math9111206 - 26 May 2021

Cited by 1 | Viewed by 2564

Abstract

A simultaneous-approximation term is a non-reflecting boundary condition that is usually accompanied by summation-by-parts schemes for provable time stability. While a high-order convective flux based on reconstruction is often employed in a finite-volume method for compressible turbulent flow, finite-volume methods with the summation-by-parts [...] Read more.

A simultaneous-approximation term is a non-reflecting boundary condition that is usually accompanied by summation-by-parts schemes for provable time stability. While a high-order convective flux based on reconstruction is often employed in a finite-volume method for compressible turbulent flow, finite-volume methods with the summation-by-parts property involve either equally weighted averaging or the second-order central flux for convective fluxes. In the present study, a cell-centered finite-volume method for compressible Naiver–Stokes equations was developed by combining a simultaneous-approximation term based on extrapolation and a low-dissipative discretization method without the summation-by-parts property. Direct numerical simulations and a large eddy simulation show that the resultant combination leads to comparable non-reflecting performance to that of the summation-by-parts scheme combined with the simultaneous-approximation term reported in the literature. Furthermore, a characteristic boundary condition was implemented for the present method, and its performance was compared with that of the simultaneous-approximation term for a direct numerical simulation and a large eddy simulation to show that the simultaneous-approximation term better maintained the average target pressure at the compressible flow outlet, which is useful for turbomachinery and aerodynamic applications, while the characteristic boundary condition better preserved the flow field near the outlet. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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34 pages, 6322 KiB

Open AccessArticle

Analysis, Evaluation and Exact Tracking of the Finite Precision Error Generated in Arbitrary Number of Multiplications

by Constantin Papaodysseus, Dimitris Arabadjis, Fotios Giannopoulos, Athanasios Rafail Mamatsis and Constantinos Chalatsis

Mathematics 2021, 9(11), 1199; https://doi.org/10.3390/math9111199 - 25 May 2021

Viewed by 1842

Abstract

In the present paper, a novel approach is introduced for the study, estimation and exact tracking of the finite precision error generated and accumulated during any number of multiplications. It is shown that, as a rule, this operation is very “toxic”, in the [...] Read more.

In the present paper, a novel approach is introduced for the study, estimation and exact tracking of the finite precision error generated and accumulated during any number of multiplications. It is shown that, as a rule, this operation is very “toxic”, in the sense that it may force the finite precision error accumulation to grow arbitrarily large, under specific conditions that are fully described here. First, an ensemble of definitions of general applicability is given for the rigorous determination of the number of erroneous digits accumulated in any quantity of an arbitrary algorithm. Next, the exact number of erroneous digits produced in a single multiplication is given as a function of the involved operands, together with formulae offering the corresponding probabilities. In case the statistical properties of these operands are known, exact evaluation of the aforementioned probabilities takes place. Subsequently, the statistical properties of the accumulated finite precision error during any number of successive multiplications are explicitly analyzed. A method for exact tracking of this accumulated error is presented, together with associated theorems. Moreover, numerous dedicated experiments are developed and the corresponding results that fully support the theoretical analysis are given. Eventually, a number of important, probable and possible applications is proposed, where all of them are based on the methodology and the results introduced in the present work. The proposed methodology is expandable, so as to tackle the round-off error analysis in all arithmetic operations. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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18 pages, 366 KiB

Open AccessArticle

Synchronization in Finite-Time Analysis of Clifford-Valued Neural Networks with Finite-Time Distributed Delays

by Grienggrai Rajchakit, Ramalingam Sriraman, Chee Peng Lim, Panu Sam-ang and Porpattama Hammachukiattikul

Mathematics 2021, 9(11), 1163; https://doi.org/10.3390/math9111163 - 21 May 2021

Cited by 33 | Viewed by 2270

Abstract

In this paper, we explore the finite-time synchronization of Clifford-valued neural networks with finite-time distributed delays. To address the problem associated with non-commutativity pertaining to the multiplication of Clifford numbers, the original n-dimensional Clifford-valued drive and response systems are firstly decomposed into [...] Read more.

In this paper, we explore the finite-time synchronization of Clifford-valued neural networks with finite-time distributed delays. To address the problem associated with non-commutativity pertaining to the multiplication of Clifford numbers, the original n-dimensional Clifford-valued drive and response systems are firstly decomposed into the corresponding

2^{m}

-dimensional real-valued counterparts. On the basis of a new Lyapunov–Krasovskii functional, suitable controller and new computational techniques, finite-time synchronization criteria are formulated for the corresponding real-valued drive and response systems. The feasibility of the main results is verified by a numerical example. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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29 pages, 9113 KiB

Open AccessArticle

Three-Dimensional Numerical Study of the Effect of Protective Barrier on the Dispersion of the Contaminant in a Building

by Chemseddine Maatki

Mathematics 2021, 9(10), 1125; https://doi.org/10.3390/math9101125 - 16 May 2021

Cited by 2 | Viewed by 1669

Abstract

The finite volume method and potential-vorticity vector formalism in their three-dimensional form were used to numerically study the impact of an adiabatic and impermeable vertical barrier on the dispersion of a local aero-contaminant due to the double-diffusive Rayleigh–Benard convection inside a cubic container. [...] Read more.

The finite volume method and potential-vorticity vector formalism in their three-dimensional form were used to numerically study the impact of an adiabatic and impermeable vertical barrier on the dispersion of a local aero-contaminant due to the double-diffusive Rayleigh–Benard convection inside a cubic container. Different governing parameters such as the Rayleigh number, buoyancy ratio and barrier height were analyzed for Le = 1.2 and Pr = 0.7, representing an air-contaminant mixture. The potential-vector-vorticity formalism in the three-dimensional form allowed the elimination of the pressure terms appearing in the Navier–Stokes equations. It was found that the heat and mass transfer as well as the effectiveness of the barrier in reducing contaminant dispersion are strongly influenced by the buoyancy ratio, the barrier size and the Rayleigh number. In addition, the barrier effectiveness is more than 70% for a height of half the building height. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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17 pages, 8634 KiB

Open AccessArticle

Mindlin-Reissner Analytical Model with Curvature for Tunnel Ventilation Shafts Analysis

by José Álvarez-Pérez and Fernando Peña

Mathematics 2021, 9(10), 1096; https://doi.org/10.3390/math9101096 - 13 May 2021

Cited by 2 | Viewed by 2516

Abstract

The formulation and analytic solution of a new mathematical model with constitutive curvature for analysis of tunnel ventilation shaft wall is proposed. Based on the Mindlin–Reissner theory for thick shells, this model also takes into account the shell constitutive curvature and considers an [...] Read more.

The formulation and analytic solution of a new mathematical model with constitutive curvature for analysis of tunnel ventilation shaft wall is proposed. Based on the Mindlin–Reissner theory for thick shells, this model also takes into account the shell constitutive curvature and considers an expression of the shear correction factor variable

(α_{n})

in terms of the thickness

(h)

and the radius of curvature

(R)

. The main advantage of the proposed model is that it has the possibility to analyze thin, medium and thick tunnel ventilation shafts. As a result, two comparisons were made: the first one, between the new model and the Mindlin–Reissner model without constitutive curvature with the shear correction factor

α_{n} = 5 /

6 as a constant, and the other, between the new model and the tridimensional numerical models (solids and shells) obtained by finite element method for different slenderness ratios

(h / R)

. The limitation of the proposed model is that it is to be formulated for a general linear-elastic and axial-symmetrical state with continuous distribution of the mass. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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15 pages, 537 KiB

Open AccessArticle

High-Order Accurate Flux-Splitting Scheme for Conservation Laws with Discontinuous Flux Function in Space

by Tingting Xiang, Guodong Wang and Suping Zhang

Mathematics 2021, 9(10), 1079; https://doi.org/10.3390/math9101079 - 11 May 2021

Cited by 1 | Viewed by 1993

Abstract

A new modified Engquist–Osher-type flux-splitting scheme is proposed to approximate the scalar conservation laws with discontinuous flux function in space. The fact that the discontinuity of the fluxes in space results in the jump of the unknown function may be the reason why [...] Read more.

A new modified Engquist–Osher-type flux-splitting scheme is proposed to approximate the scalar conservation laws with discontinuous flux function in space. The fact that the discontinuity of the fluxes in space results in the jump of the unknown function may be the reason why it is difficult to design a high-order scheme to solve this hyperbolic conservation law. In order to implement the WENO flux reconstruction, we apply the new modified Engquist–Osher-type flux to compensate for the discontinuity of fluxes in space. Together the third-order TVD Runge–Kutta time discretization, we can obtain the high-order accurate scheme, which keeps equilibrium state across the discontinuity in space, to solve the scalar conservation laws with discontinuous flux function. Some examples are given to demonstrate the good performance of the new high-order accurate scheme. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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16 pages, 1124 KiB

Open AccessArticle

Brain Signals Classification Based on Fuzzy Lattice Reasoning

by Eleni Vrochidou, Chris Lytridis, Christos Bazinas, George A. Papakostas, Hiroaki Wagatsuma and Vassilis G. Kaburlasos

Mathematics 2021, 9(9), 1063; https://doi.org/10.3390/math9091063 - 9 May 2021

Cited by 2 | Viewed by 2903

Abstract

Cyber-Physical System (CPS) applications including human-robot interaction call for automated reasoning for rational decision-making. In the latter context, typically, audio-visual signals are employed. Τhis work considers brain signals for emotion recognition towards an effective human-robot interaction. An ElectroEncephaloGraphy (EEG) signal here is represented [...] Read more.

Cyber-Physical System (CPS) applications including human-robot interaction call for automated reasoning for rational decision-making. In the latter context, typically, audio-visual signals are employed. Τhis work considers brain signals for emotion recognition towards an effective human-robot interaction. An ElectroEncephaloGraphy (EEG) signal here is represented by an Intervals’ Number (IN). An IN-based, optimizable parametric k Nearest Neighbor (kNN) classifier scheme for decision-making by fuzzy lattice reasoning (FLR) is proposed, where the conventional distance between two points is replaced by a fuzzy order function (σ) for reasoning-by-analogy. A main advantage of the employment of INs is that no ad hoc feature extraction is required since an IN may represent all-order data statistics, the latter are the features considered implicitly. Four different fuzzy order functions are employed in this work. Experimental results demonstrate comparably the good performance of the proposed techniques. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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11 pages, 3301 KiB

Open AccessArticle

Dynamic Analysis of a Fiber-Reinforced Composite Beam under a Moving Load by the Ritz Method

by Şeref D. Akbaş, Hakan Ersoy, Bekir Akgöz and Ömer Civalek

Mathematics 2021, 9(9), 1048; https://doi.org/10.3390/math9091048 - 6 May 2021

Cited by 103 | Viewed by 4856

Abstract

This paper presents the dynamic responses of a fiber-reinforced composite beam under a moving load. The Timoshenko beam theory was employed to analyze the kinematics of the composite beam. The constitutive equations for motion were obtained by utilizing the Lagrange procedure. The Ritz [...] Read more.

This paper presents the dynamic responses of a fiber-reinforced composite beam under a moving load. The Timoshenko beam theory was employed to analyze the kinematics of the composite beam. The constitutive equations for motion were obtained by utilizing the Lagrange procedure. The Ritz method with polynomial functions was employed to solve the resulting equations in conjunction with the Newmark average acceleration method (NAAM). The influence of fiber orientation angle, volume fraction, and velocity of the moving load on the dynamic responses of the fiber-reinforced nonhomogeneous beam is presented and discussed. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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7 pages, 409 KiB

Open AccessArticle

Modified Hybrid Method with Four Stages for Second Order Ordinary Differential Equations

by Faieza Samat and Eddie Shahril Ismail

Mathematics 2021, 9(9), 1028; https://doi.org/10.3390/math9091028 - 1 May 2021

Cited by 3 | Viewed by 1570

Abstract

A modified explicit hybrid method with four stages is presented, with the first stage exactly integrating exp(wx), while the remaining stages exactly integrate sin(wx) and cos(wx). Special attention is paid to the phase properties of the method [...] Read more.

A modified explicit hybrid method with four stages is presented, with the first stage exactly integrating exp(wx), while the remaining stages exactly integrate sin(wx) and cos(wx). Special attention is paid to the phase properties of the method during the process of parameter selection. Numerical comparisons of the proposed and existing hybrid methods for several second-order problems show that the proposed method gives high accuracy in solving the Duffing equation and Kramarz’s system. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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14 pages, 1678 KiB

Open AccessArticle

Sixth-Order Combined Compact Finite Difference Scheme for the Numerical Solution of One-Dimensional Advection-Diffusion Equation with Variable Parameters

by Gurhan Gurarslan

Mathematics 2021, 9(9), 1027; https://doi.org/10.3390/math9091027 - 1 May 2021

Cited by 4 | Viewed by 2493

Abstract

A high-accuracy numerical method based on a sixth-order combined compact difference scheme and the method of lines approach is proposed for the advection–diffusion transport equation with variable parameters. In this approach, the partial differential equation representing the advection-diffusion equation is converted into many [...] Read more.

A high-accuracy numerical method based on a sixth-order combined compact difference scheme and the method of lines approach is proposed for the advection–diffusion transport equation with variable parameters. In this approach, the partial differential equation representing the advection-diffusion equation is converted into many ordinary differential equations. These time-dependent ordinary differential equations are then solved using an explicit fourth order Runge–Kutta method. Three test problems are studied to demonstrate the accuracy of the present methods. Numerical solutions obtained by the proposed method are compared with the analytical solutions and the available numerical solutions given in the literature. In addition to requiring less CPU time, the proposed method produces more accurate and more stable results than the numerical methods given in the literature. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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17 pages, 618 KiB

Open AccessArticle

Numerical Solution of Bending of the Beam with Given Friction

by Michaela Bobková and Lukáš Pospíšil

Mathematics 2021, 9(8), 898; https://doi.org/10.3390/math9080898 - 18 Apr 2021

Cited by 1 | Viewed by 2059

Abstract

We are interested in a contact problem for a thin fixed beam with an internal point obstacle with possible rotation and shift depending on a given swivel and sliding friction. This problem belongs to the most basic practical problems in, for instance, the [...] Read more.

We are interested in a contact problem for a thin fixed beam with an internal point obstacle with possible rotation and shift depending on a given swivel and sliding friction. This problem belongs to the most basic practical problems in, for instance, the contact mechanics in the sustainable building construction design. The analysis and the practical solution plays a crucial role in the process and cannot be ignored. In this paper, we consider the classical Euler–Bernoulli beam model, which we formulate, analyze, and numerically solve. The objective function of the corresponding optimization problem for finding the coefficients in the finite element basis combines a quadratic function and an additional non-differentiable part with absolute values representing the influence of considered friction. We present two basic algorithms for the solution: the regularized primal solution, where the non-differentiable part is approximated, and the dual formulation. We discuss the disadvantages of the methods on the solution of the academic benchmarks. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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6 pages, 502 KiB

Open AccessArticle

Symmetries and Invariant Solutions for the Coagulation of Aerosols

by Mingliang Zheng

Mathematics 2021, 9(8), 876; https://doi.org/10.3390/math9080876 - 16 Apr 2021

Viewed by 1669

Abstract

The coagulation of aerosol particles plays an important role in the structural morphological changes of suspended particles at any time and in any space. In this study, based on the Smoluchowski equation of population balance, a kinetic model of aerosol coalescence considering Brownian [...] Read more.

The coagulation of aerosol particles plays an important role in the structural morphological changes of suspended particles at any time and in any space. In this study, based on the Smoluchowski equation of population balance, a kinetic model of aerosol coalescence considering Brownian motion collision is established. By applying the developed Lie group method, we derive the allowed infinitesimal symmetries and group-invariant solutions of the integro-differential equation, as well as the exact solution under some special conditions. We also provide detailed steps and a discussion of the properties. The content and results provide an effective analytic solution for the progressive evolution of aerosol particle size considering boundary and initial conditions. This solution reveals the self-conservative phenomena in the process of aerosol coalescence and also provides validation for the numerical algorithms of general dynamics equations. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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19 pages, 1442 KiB

Open AccessArticle

Automated Generation of EQ-Algebras through Genetic Algorithms

by Hashim Habiballa, Eva Volna and Martin Kotyrba

Mathematics 2021, 9(8), 861; https://doi.org/10.3390/math9080861 - 14 Apr 2021

Cited by 1 | Viewed by 1759

Abstract

This article introduces an approach to the automated generation of special algebras through genetic algorithms. These algorithms can be also used for a broader variety of applications in mathematics. We describe the results of research aiming at automated production of such algebras with [...] Read more.

This article introduces an approach to the automated generation of special algebras through genetic algorithms. These algorithms can be also used for a broader variety of applications in mathematics. We describe the results of research aiming at automated production of such algebras with the help of evolutionary techniques. Standard approach is not relevant due to the time complexity of the task, which is superexponential. Our research concerning the usage of genetic algorithms enabled the problem to be solvable in reasonable time and we were able to produce finite algebras with special properties called EQ-algebras. EQ-algebras form an alternate truth–value structure for new fuzzy logics. We present the algorithms and special versions of genetic operators suitable for this task. Then we performed experiments with application EQ-Creator are discussed with proper statistical analysis through ANOVA. The genetic approach enables to automatically generate algebras of sufficient extent without superexponential complexity. Our main results include: that elitism is necessary at least for several parent members, a high mutation ratio must be set, optional axioms fulfilment increases computing time significantly, optional properties negatively affect convergence, and colorfulness was defined to prevent trivial solutions (evolution tends to the simplest way of achieving results). Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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13 pages, 406 KiB

Open AccessArticle

Multigrid Method for Optimal Control Problem Constrained by Stochastic Stokes Equations with Noise

by Muhammad Munir Butt

Mathematics 2021, 9(7), 738; https://doi.org/10.3390/math9070738 - 29 Mar 2021

Cited by 3 | Viewed by 2039

Abstract

Optimal control problems governed by stochastic partial differential equations have become an important field in applied mathematics. In this article, we investigate one such important optimization problem, that is, the stochastic Stokes control problem with forcing term perturbed by noise. A multigrid scheme [...] Read more.

Optimal control problems governed by stochastic partial differential equations have become an important field in applied mathematics. In this article, we investigate one such important optimization problem, that is, the stochastic Stokes control problem with forcing term perturbed by noise. A multigrid scheme with three-factor coarsening to solve the corresponding discretized control problem is presented. On staggered grids, a three-factor coarsening strategy helps in simplifying the inter-grid transfer operators and reduction in computation (CPU time). For smoothing, a distributive Gauss–Seidel scheme with a line search strategy is employed. To validate the proposed multigrid staggered grid framework, numerical results are presented with white noise at the end. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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9 pages, 300 KiB

Open AccessArticle

Real-Time Estimation of R₀ for COVID-19 Spread

by Theodore E. Simos, Charalampos Tsitouras, Vladislav N. Kovalnogov, Ruslan V. Fedorov and Dmitry A. Generalov

Mathematics 2021, 9(6), 664; https://doi.org/10.3390/math9060664 - 20 Mar 2021

Cited by 13 | Viewed by 2699

Abstract

We propose a real-time approximation of

R_{0}

in an SIR-type model that applies to the COVID-19 epidemic outbreak. A very useful direct formula expressing

R_{0}

is found. Then, various type of models are considered, namely, finite differences, cubic splines, Piecewise Cubic [...] Read more.

We propose a real-time approximation of

R_{0}

in an SIR-type model that applies to the COVID-19 epidemic outbreak. A very useful direct formula expressing

R_{0}

is found. Then, various type of models are considered, namely, finite differences, cubic splines, Piecewise Cubic Hermite interpolation and linear least squares approximation. Preserving the monotonicity of the formula under consideration proves to be of crucial importance. This latter property is preferred over accuracy, since it maintains positive

R_{0}

. Only the Linear Least Squares technique guarantees this, and is finally proposed here. Tests on real COVID-19 data confirm the usefulness of our approach. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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20 pages, 5535 KiB

Open AccessArticle

Unsteady MHD Mixed Convection Flow in Hybrid Nanofluid at Three-Dimensional Stagnation Point

by Nurul Amira Zainal, Roslinda Nazar, Kohilavani Naganthran and Ioan Pop

Mathematics 2021, 9(5), 549; https://doi.org/10.3390/math9050549 - 5 Mar 2021

Cited by 23 | Viewed by 2529

Abstract

There has been significant interest in exploring a stagnation point flow due to its numerous potential uses in engineering applications such as cooling of nuclear reactors. Hence, this study proposed a numerical analysis on the unsteady magnetohydrodynamic (MHD) mixed convection at three-dimensional stagnation [...] Read more.

There has been significant interest in exploring a stagnation point flow due to its numerous potential uses in engineering applications such as cooling of nuclear reactors. Hence, this study proposed a numerical analysis on the unsteady magnetohydrodynamic (MHD) mixed convection at three-dimensional stagnation point flow in Al₂O₃–Cu/H₂O hybrid nanofluid over a permeable sheet. The ordinary differential equations are accomplished by simplifying the governing partial differential equations through suitable similarity transformation. The numerical computation is established by the MATLAB system software using the bvp4c technique. The bvp4c procedure is excellent in providing more than one solution once sufficient predictions are visible. The influence of certain functioning parameters is inspected, and notable results exposed that the rate of heat transfer is exaggerated along with the skin friction coefficient while the suction/injection and magnetic parameters are intensified. The results also signified that the rise in the volume fraction of the nanoparticle and the decline of the unsteadiness parameter demonstrates a downward attribution towards the heat transfer performance and skin friction coefficient. Conclusively, the observations are confirmed to have multiple solutions, which eventually contribute to an investigation of the analysis of the solution stability, thereby justifying the viability of the first solution. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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15 pages, 1293 KiB

Open AccessFeature PaperArticle

On the Application of the Generalized Means to Construct Multiresolution Schemes Satisfying Certain Inequalities Proving Stability

by Sergio Amat, Alberto Magreñan, Juan Ruiz, Juan Carlos Trillo and Dionisio F. Yañez

Mathematics 2021, 9(5), 533; https://doi.org/10.3390/math9050533 - 4 Mar 2021

Cited by 1 | Viewed by 1435

Abstract

Multiresolution representations of data are known to be powerful tools in data analysis and processing, and they are particularly interesting for data compression. In order to obtain a proper definition of the edges, a good option is to use nonlinear reconstructions. These nonlinear [...] Read more.

Multiresolution representations of data are known to be powerful tools in data analysis and processing, and they are particularly interesting for data compression. In order to obtain a proper definition of the edges, a good option is to use nonlinear reconstructions. These nonlinear reconstruction are the heart of the prediction processes which appear in the definition of the nonlinear subdivision and multiresolution schemes. We define and study some nonlinear reconstructions based on the use of nonlinear means, more in concrete the so-called Generalized means. These means have two interesting properties that will allow us to get associated reconstruction operators adapted to the presence of discontinuities, and having the maximum possible order of approximation in smooth areas. Once we have these nonlinear reconstruction operators defined, we can build the related nonlinear subdivision and multiresolution schemes and prove more accurate inequalities regarding the contractivity of the scheme for the first differences and in turn the results about stability. In this paper, we also define a new nonlinear two-dimensional multiresolution scheme as non-separable, i.e., not based on tensor product. We then present the study of the stability issues for the scheme and numerical experiments reinforcing the proven theoretical results and showing the usefulness of the algorithm. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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19 pages, 372 KiB

Open AccessArticle

A Piecewise Polynomial Harmonic Nonlinear Interpolatory Reconstruction Operator on Non Uniform Grids—Adaptation around Jump Discontinuities and Elimination of Gibbs Phenomenon

by Pedro Ortiz and Juan Carlos Trillo

Mathematics 2021, 9(4), 335; https://doi.org/10.3390/math9040335 - 8 Feb 2021

Cited by 4 | Viewed by 1826

Abstract

In this paper, we analyze the behavior of a nonlinear reconstruction operator called PPH around discontinuities. The acronym PPH stands for Piecewise Polynomial Harmonic, since it uses piecewise polynomials defined by means of an adaption based on the use of the weighted Harmonic [...] Read more.

In this paper, we analyze the behavior of a nonlinear reconstruction operator called PPH around discontinuities. The acronym PPH stands for Piecewise Polynomial Harmonic, since it uses piecewise polynomials defined by means of an adaption based on the use of the weighted Harmonic mean. This study is carried out in the general case of nonuniform grids, although for some results we restrict to

σ

quasi-uniform grids. In particular we analyze the numerical order of approximation close to jump discontinuities and the elimination of the Gibbs effects. We show, both theoretically and with numerical examples, that the numerical order is reduced but not completely lost as it is the case in their linear counterparts. Moreover we observe that the reconstruction is free of any Gibbs effects for sufficiently small grid sizes. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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Review

Jump to: Research, Other

14 pages, 517 KiB

Open AccessEditor’s ChoiceReview

A Survey on Software Defect Prediction Using Deep Learning

by Elena N. Akimova, Alexander Yu. Bersenev, Artem A. Deikov, Konstantin S. Kobylkin, Anton V. Konygin, Ilya P. Mezentsev and Vladimir E. Misilov

Mathematics 2021, 9(11), 1180; https://doi.org/10.3390/math9111180 - 24 May 2021

Cited by 53 | Viewed by 9639

Abstract

Defect prediction is one of the key challenges in software development and programming language research for improving software quality and reliability. The problem in this area is to properly identify the defective source code with high accuracy. Developing a fault prediction model is [...] Read more.

Defect prediction is one of the key challenges in software development and programming language research for improving software quality and reliability. The problem in this area is to properly identify the defective source code with high accuracy. Developing a fault prediction model is a challenging problem, and many approaches have been proposed throughout history. The recent breakthrough in machine learning technologies, especially the development of deep learning techniques, has led to many problems being solved by these methods. Our survey focuses on the deep learning techniques for defect prediction. We analyse the recent works on the topic, study the methods for automatic learning of the semantic and structural features from the code, discuss the open problems and present the recent trends in the field. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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Other

Jump to: Research, Review

11 pages, 3680 KiB

Open AccessCase Report

A Climate-Mathematical Clustering of Rainfall Stations in the Río Bravo-San Juan Basin (Mexico) by Using the Higuchi Fractal Dimension and the Hurst Exponent

by Francisco Gerardo Benavides-Bravo, Dulce Martinez-Peon, Ángela Gabriela Benavides-Ríos, Otoniel Walle-García, Roberto Soto-Villalobos and Mario A. Aguirre-López

Mathematics 2021, 9(21), 2656; https://doi.org/10.3390/math9212656 - 20 Oct 2021

Cited by 7 | Viewed by 2059

Abstract

When conducting an analysis of nature’s time series, such as meteorological ones, an important matter is a long-range dependence to quantify the global behavior of the series and connect it with other physical characteristics of the region of study. In this paper, we [...] Read more.

When conducting an analysis of nature’s time series, such as meteorological ones, an important matter is a long-range dependence to quantify the global behavior of the series and connect it with other physical characteristics of the region of study. In this paper, we applied the Higuchi fractal dimension and the Hurst exponent (rescaled range) to quantify the relative trend underlying the time series of historical data from 17 of the 34 weather stations located in the Río Bravo-San Juan Basin, Mexico; these data were provided by the National Water Commission CONAGUA) in Mexico. In this way, this work aims to perform a comparative study about the level of persistency obtained by using the Higuchi fractal dimension and Hurst exponent for each station of the basin. The comparison is supported by a climate clustering of the stations, according to the Köppen classification. Results showed a better fitting between the climate of each station and its Higuchi fractal dimension obtained than when using the Hurst exponent. In fact, we found that the more the aridity of the zone the more the persistency of rainfall, according to Higuchi’s values. In turn, we found more relation between the Hurst exponent and the accumulated amount of rainfall. These are relations between the climate and the long-term persistency of rainfall in the basin that could help to better understand and complete the climatological models of the study region. Trends between the fractal exponents used and the accumulated annual rainfall were also analyzed. Full article

(This article belongs to the Special Issue Numerical Analysis and Scientific Computing)

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