Mathematical and Computational Applications

21 pages, 5633 KiB

Open AccessArticle

Polynomial Approximation over Arbitrary Shape Domains

by Mohammad J. Mahtabi, Arash Ghasemi, Amirehsan Ghasemi and James C. Newman III

Math. Comput. Appl. 2024, 29(6), 110; https://doi.org/10.3390/mca29060110 - 25 Nov 2024

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In spectral/finite element methods, a robust and stable high-order polynomial approximation method for the solution can significantly reduce the required number of degrees of freedom (DOFs) to achieve a certain level of accuracy. In this work, a closed-form relation is proposed to approximate [...] Read more.

In spectral/finite element methods, a robust and stable high-order polynomial approximation method for the solution can significantly reduce the required number of degrees of freedom (DOFs) to achieve a certain level of accuracy. In this work, a closed-form relation is proposed to approximate the Fekete points (AFPs) on arbitrary shape domains based on the singular value decomposition (SVD) of the Vandermonde matrix. In addition, a novel method is derived to compute the moments on highly complex domains, which may include discontinuities. Then, AFPs are used to generate compatible basis functions using SVD. Equations are derived and presented to determine orthogonal/orthonormal modal basis functions, as well as the Lagrange basis. Furthermore, theorems are proved to show the convergence and accuracy of the proposed method, together with an explicit form of the Weierstrass theorem for polynomial approximation. The method was implemented and some classical cases were analyzed. The results show the superior performance of the proposed method in terms of convergence and accuracy using many fewer DOFs and, thus, a much lower computational cost. It was shown that the orthogonal modal basis is the best choice to decrease the DOFs while maintaining a small Lebesgue constant when very high degree of polynomial is employed. Full article

(This article belongs to the Special Issue Mathematical and Computational Approaches in Applied Mechanics: A Themed Issue Dedicated to Professor J.N. Reddy)

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3 pages, 194 KiB

Open AccessEditorial

Mathematical and Computational Modelling in Mechanics of Materials and Structures

by Nicholas Fantuzzi, Francesco Fabbrocino, Marco Montemurro, Francesca Nanni, Qun Huang, José António Correia, Leonardo Dassatti and Michele Bacciocchi

Math. Comput. Appl. 2024, 29(6), 109; https://doi.org/10.3390/mca29060109 - 25 Nov 2024

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Abstract

The intersection of mathematics and computational modeling with the mechanics of materials and structural engineering continues to yield substantial advancements in both theoretical and applied domains [...] Full article

(This article belongs to the Special Issue Mathematical and Computational Modelling in Mechanics of Materials and Structures)

25 pages, 1578 KiB

Open AccessReview

Optimizing Power Flow and Stability in Hybrid AC/DC Microgrids: AC, DC, and Combined Analysis

by Ghanshyam Meena, Veerpratap Meena, Akhilesh Mathur, Vinay Pratap Singh, Ahmad Taher Azar and Ibrahim A. Hameed

Math. Comput. Appl. 2024, 29(6), 108; https://doi.org/10.3390/mca29060108 - 24 Nov 2024

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Abstract

A microgrid (MG) is a unique area of a power distribution network that combines distributed generators (conventional as well as renewable power sources) and energy storage systems. Due to the integration of renewable generation sources, microgrids have become more unpredictable. MGs can operate [...] Read more.

A microgrid (MG) is a unique area of a power distribution network that combines distributed generators (conventional as well as renewable power sources) and energy storage systems. Due to the integration of renewable generation sources, microgrids have become more unpredictable. MGs can operate in two different modes, namely, grid-connected and islanded modes. MGs face various challenges of voltage variations, frequency deviations, harmonics, unbalances, etc., due to the uncertain behavior of renewable sources. To study the impact of these issues, it is necessary to analyze the behavior of the MG system under normal and abnormal operating conditions. Two different tools are used for the analysis of microgrids under normal and abnormal conditions, namely, power flow and short-circuit analysis, respectively. Power flow analysis is used to determine the voltages, currents, and real and reactive power flow in the MG system under normal operating conditions. Short-circuit analysis is carried out to analyze the behavior of MGs under faulty conditions. In this paper, a review of power flow and short-circuit analysis algorithms for MG systems under two different modes of operation, grid-connected and islanded, is presented. This paper also presents a comparison of various power flow as well as short-circuit analysis techniques for MGs in tabular form. The modeling of different components of MGs is also discussed in this paper. Full article

(This article belongs to the Special Issue Applied Optimization in Automatic Control and Systems Engineering)

18 pages, 2638 KiB

Open AccessArticle

Radical Petrov–Galerkin Approach for the Time-Fractional KdV–Burgers’ Equation

by Youssri Hassan Youssri and Ahmed Gamal Atta

Math. Comput. Appl. 2024, 29(6), 107; https://doi.org/10.3390/mca29060107 - 21 Nov 2024

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Abstract

This paper presents a novel numerical spectral scheme to handle the time-fractional KdV–Burgers’ equation, which is very important in both physics and engineering. The scheme basically uses the tau approach combined with Gegenbauer polynomials to provide accurate and stable numerical solutions. Instead of [...] Read more.

This paper presents a novel numerical spectral scheme to handle the time-fractional KdV–Burgers’ equation, which is very important in both physics and engineering. The scheme basically uses the tau approach combined with Gegenbauer polynomials to provide accurate and stable numerical solutions. Instead of solving the differential problem together with the conditions, we solve a system of algebraic equations. The method can handle complex boundary conditions. Some numerical experiments are exhibited to demonstrate that this approach is highly efficient and produces results that are better than some existing numerical methods in the literature. This technique offers more advanced solutions for time-fractional problems in various fields. Full article

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23 pages, 463 KiB

Open AccessArticle

Semantic Categories: Uncertainty and Similarity

by Ares Fabregat-Hernández, Javier Palanca and Vicent Botti

Math. Comput. Appl. 2024, 29(6), 106; https://doi.org/10.3390/mca29060106 - 16 Nov 2024

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Abstract

This paper addresses understanding and categorizing language by using Markov categories to establish a mathematical framework for semantic concepts. This framework enables us to measure the semantic similarity between linguistic expressions within a given text. Furthermore, this approach enables the measurement and control [...] Read more.

This paper addresses understanding and categorizing language by using Markov categories to establish a mathematical framework for semantic concepts. This framework enables us to measure the semantic similarity between linguistic expressions within a given text. Furthermore, this approach enables the measurement and control of uncertainty in language categorization and the creation of metrics for evaluating semantic similarity. We provide use cases to demonstrate how the proposed methods can be applied and computed, focusing on their interpretability and the universality of categorical constructions. This work contributes to the field by offering a novel perspective on semantic similarity and uncertainty metrics in language processing, generating criteria to automate their computation. Full article

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13 pages, 436 KiB

Open AccessArticle

Structural Stability of Pseudo-Parabolic Equations for Basic Data

by Yanping Wang and Yuanfei Li

Math. Comput. Appl. 2024, 29(6), 105; https://doi.org/10.3390/mca29060105 - 15 Nov 2024

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Abstract

This article investigates the spatial decay properties and continuous dependence on the basic geometric structure. Assuming that the total potential energy is bounded and the homogeneous Dirichlet condition is satisfied on the side of the solution within the cylindrical domain, we establish an [...] Read more.

This article investigates the spatial decay properties and continuous dependence on the basic geometric structure. Assuming that the total potential energy is bounded and the homogeneous Dirichlet condition is satisfied on the side of the solution within the cylindrical domain, we establish an auxiliary function related to the solution. By extending the data at the finite end forward, we can establish the continuous dependence on the perturbation of base data. Full article

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22 pages, 56577 KiB

Open AccessArticle

Resolving Contrast and Detail Trade-Offs in Image Processing with Multi-Objective Optimization

by Daniel Molina-Pérez and Alam Gabriel Rojas-López

Math. Comput. Appl. 2024, 29(6), 104; https://doi.org/10.3390/mca29060104 - 11 Nov 2024

Viewed by 480

Abstract

This article addresses the complex challenge of simultaneously enhancing contrast and detail in an image, where improving one property often compromises the other. This trade-off is tackled using a multi-objective optimization approach. Specifically, the proposal’s model integrates the sigmoid transformation function and unsharp [...] Read more.

This article addresses the complex challenge of simultaneously enhancing contrast and detail in an image, where improving one property often compromises the other. This trade-off is tackled using a multi-objective optimization approach. Specifically, the proposal’s model integrates the sigmoid transformation function and unsharp masking highboost filtering with the NSGA-II algorithm. Additionally, a posterior preference articulation is introduced to select three key solutions from the Pareto front: the maximum contrast solution, the maximum detail solution, and the knee point solution. The proposed technique is evaluated on a range of image types, including medical and natural scenes. The final solutions demonstrated significant superiority in terms of contrast and detail compared to the original images. The three selected solutions, although all are optimal, captured distinct characteristics within the images, offering different solutions according to field preferences. This highlights the method’s effectiveness across different types and enhancement requirements and emphasizes the importance of the proposed preferences in different contexts. Full article

(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2024)

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16 pages, 504 KiB

Open AccessArticle

An Experimental Comparison of Self-Adaptive Differential Evolution Algorithms to Induce Oblique Decision Trees

by Rafael Rivera-López, Efrén Mezura-Montes, Juana Canul-Reich and Marco-Antonio Cruz-Chávez

Math. Comput. Appl. 2024, 29(6), 103; https://doi.org/10.3390/mca29060103 - 9 Nov 2024

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Abstract

This study addresses the challenge of generating accurate and compact oblique decision trees using self-adaptive differential evolution algorithms. Although traditional decision tree induction methods create explainable models, they often fail to achieve optimal classification accuracy. To overcome these limitations, other strategies, such as [...] Read more.

This study addresses the challenge of generating accurate and compact oblique decision trees using self-adaptive differential evolution algorithms. Although traditional decision tree induction methods create explainable models, they often fail to achieve optimal classification accuracy. To overcome these limitations, other strategies, such as those based on evolutionary computation, have been proposed in the literature. In particular, we evaluate the use of self-adaptive differential evolution variants to evolve a population of oblique decision trees encoded as real-valued vectors. Our proposal includes (1) an alternative initialization strategy that reduces redundant nodes and (2) a fitness function that penalizes excessive leaf nodes, promoting smaller and more accurate decision trees. We perform a comparative performance analysis of these differential evolution variants, showing that while they exhibit similar statistical behavior, the Single-Objective real-parameter optimization (jSO) method produces the most accurate oblique decision trees and is second best in compactness. The findings highlight the potential of self-adaptive differential evolution algorithms to improve the effectiveness of oblique decision trees in machine learning applications. Full article

(This article belongs to the Collection Feature Papers in Mathematical and Computational Applications 2024)

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21 pages, 3240 KiB

Open AccessArticle

A Mathematical Lens on the Zoonotic Transmission of Lassa Virus Infections Leading to Disabilities in Severe Cases

by Yasir Ramzan, Hanadi Alzubadi, Aziz Ullah Awan, Kamel Guedri, Mohammed Alharthi and Bandar M. Fadhl

Math. Comput. Appl. 2024, 29(6), 102; https://doi.org/10.3390/mca29060102 - 7 Nov 2024

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Abstract

This study aims to analyze the dynamics of Lassa fever transmission and its impact on the brain and spinal cord then devise and analyze preventive actions. The stability of the infection-free equilibrium point is evaluated; the model’s precision is examined using empirical data; [...] Read more.

This study aims to analyze the dynamics of Lassa fever transmission and its impact on the brain and spinal cord then devise and analyze preventive actions. The stability of the infection-free equilibrium point is evaluated; the model’s precision is examined using empirical data; and all parameters are estimated and fitted. Subsequently, the basic reproductive number is determined, and subpopulation trends are observed over time. Sensitivity analysis is conducted to identify critical drivers influencing transmission dynamics. Two-dimensional plots visualize the impact of crucial parameters on the reproductive number. Through a comprehensive literature review and case analysis, an association between Lassa fever and various disabilities is established, including conditions such as encephalitis, hearing loss, ataxia, neuropsychiatric manifestations, meningitis, seizures, and coma. Solutions are devised and analyzed to enhance early detection, treatment, and mitigation of disease. Full article

(This article belongs to the Special Issue New Trends in Biomathematics)

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19 pages, 5947 KiB

Open AccessArticle

Analytical Solutions of PBTK Models for Evaluating the Impact of Surface Diffusion Characteristics on the Leaching Profile of Implant Bioproducts

by Matheos Giakoumi, Konstantinos Kapnisis, Andreas Anayiotos and Pavlos S. Stephanou

Math. Comput. Appl. 2024, 29(6), 101; https://doi.org/10.3390/mca29060101 - 4 Nov 2024

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Abstract

Toxicokinetic or pharmacokinetic models, physiologically based or not, offer a unique avenue to understand the transport of toxins or pharmaceuticals in living organisms. The availability of analytical solutions to such models offers the means to engage in a plethora of applications. In the [...] Read more.

Toxicokinetic or pharmacokinetic models, physiologically based or not, offer a unique avenue to understand the transport of toxins or pharmaceuticals in living organisms. The availability of analytical solutions to such models offers the means to engage in a plethora of applications. In the present work, we provide the framework to solve analytically such models using the matrix exponential, and we then apply this method to derive an explicit solution to four-to-five-compartment physiologically based toxicokinetic (PBTK) models considering a single- and an infinite-exponential expression for the amount of mass released from an implantable device. We also offer the conditions that need to be met for analytical solutions to be obtained when the kinetic rates are time-dependent functions. Our analysis compares the computation time between analytical and numerical solutions and characterizes the dependency of the maximum substance mass value and the time it occurs in the various tissue compartments from the material surface diffusion characteristics. Our analytical solutions, which have several advantages over the solutions obtained using numerical solvers, can be incorporated into in silico tools and provide valuable information for human health risk assessment. Full article

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17 pages, 532 KiB

Open AccessArticle

Numerical Study of Multi-Term Time-Fractional Sub-Diffusion Equation Using Hybrid L1 Scheme with Quintic Hermite Splines

by Priyanka Priyanka, Shelly Arora, Saroj Sahani and Sharandeep Singh

Math. Comput. Appl. 2024, 29(6), 100; https://doi.org/10.3390/mca29060100 - 2 Nov 2024

Viewed by 435

Abstract

Anomalous diffusion of particles has been described by the time-fractional reaction–diffusion equation. A hybrid formulation of numerical technique is proposed to solve the time-fractional-order reaction–diffusion (FRD) equation numerically. The technique comprises the semi-discretization of the time variable using an L1 finite-difference scheme and [...] Read more.

Anomalous diffusion of particles has been described by the time-fractional reaction–diffusion equation. A hybrid formulation of numerical technique is proposed to solve the time-fractional-order reaction–diffusion (FRD) equation numerically. The technique comprises the semi-discretization of the time variable using an L1 finite-difference scheme and space discretization using the quintic Hermite spline collocation method. The hybrid technique reduces the problem to an iterative scheme of an algebraic system of equations. The stability analysis of the proposed numerical scheme and the optimal error bounds for the approximate solution are also studied. A comparative study of the obtained results and an error analysis of approximation show the efficiency, accuracy, and effectiveness of the technique. Full article

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3 pages, 154 KiB

Open AccessEditorial

Feature Paper Collection of Mathematical and Computational Applications—2023

by Gianluigi Rozza, Oliver Schütze and Nicholas Fantuzzi

Math. Comput. Appl. 2024, 29(6), 99; https://doi.org/10.3390/mca29060099 - 1 Nov 2024

Viewed by 347

Abstract

This Special Issue comprises the second collection of papers submitted by both the Editorial Board Members (EBMs) of the journal Mathematical and Computational Applications (MCA) and the outstanding scholars working in the core research fields of MCA [...] Full article

(This article belongs to the Collection Feature Papers in Mathematical and Computational Applications 2023)

24 pages, 6455 KiB

Open AccessArticle

Using Artificial Neural Network Analysis to Study Jeffrey Nanofluid Flow in Cone–Disk Systems

by Nasser Nammas Albaqami

Math. Comput. Appl. 2024, 29(6), 98; https://doi.org/10.3390/mca29060098 - 31 Oct 2024

Viewed by 432

Abstract

Artificial intelligence (AI) is employed in fluid flow models to enhance the simulation’s accuracy, to more effectively optimize the fluid flow models, and to realize reliable fluid flow systems with improved performance. Jeffery fluid flow through the interstice of a cone-and-disk system is [...] Read more.

Artificial intelligence (AI) is employed in fluid flow models to enhance the simulation’s accuracy, to more effectively optimize the fluid flow models, and to realize reliable fluid flow systems with improved performance. Jeffery fluid flow through the interstice of a cone-and-disk system is considered in this study. The mathematical description of this flow involves converting a partial differential system into a nonlinear ordinary differential system and solving it using a neurocomputational technique. The fluid streaming through the disk–cone gap is investigated under four contrasting frameworks, i.e., (i) passive cone and spinning disk, (ii) spinning cone and passive disk, (iii) cone and disk rotating in the same direction, and (iv) cone and disk rotating in opposite directions. Employing the recently developed technique of artificial neural networks (ANNs) can be effective for handling and optimizing fluid flow exploits. The proposed approach integrates training, testing and analysis, and authentication based on a locus dataset to address various aspects of fluid problems. The mean square error, regression plots, curve-fitting graphs, and error histograms are used to evaluate the performance of the least mean square neural network algorithm (LMS-NNA). The results show that these equations are consistently aligned, and agreement is, on average, in the order of 10⁻⁸. While the resting parameters were kept static, the transverse velocity distribution, in all four cases, exhibited an incremental decreasing behavior in the estimates of magnetic and Jeffery fluid factors. Furthermore, the results obtained were compared with those in the literature, and the close agreement confirms our results. To train the model, 80% of the data were used for LMS-NNA, with 10% used for testing and the remaining 10% for validation. The quantitative and qualitative outputs obtained from the neural network strategy and parameter variation were thoroughly examined and discussed. Full article

(This article belongs to the Special Issue Symmetry Methods for Solving Differential Equations)

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13 pages, 7092 KiB

Open AccessArticle

Design and Optimization of Microfluidic Vortex Diode

by Krzysztof Tadyszak, Alessandro Jäger, Jiří Pánek and Martin Hrubý

Math. Comput. Appl. 2024, 29(6), 97; https://doi.org/10.3390/mca29060097 - 30 Oct 2024

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Abstract

The performed research presents modeling results for designing microfluidic vortex diodes. These devices rectify fluid flow and can be used in many applications on micro and macro scales. The modeling, utilizing computational fluid dynamics (CFD) with the turbulence model RANS k-ε in COMSOL [...] Read more.

The performed research presents modeling results for designing microfluidic vortex diodes. These devices rectify fluid flow and can be used in many applications on micro and macro scales. The modeling, utilizing computational fluid dynamics (CFD) with the turbulence model RANS k-ε in COMSOL Multiphysics, has led to optimizing diodicity—the reversed-to-forward flow pressure drop ratio. The goal was to find the best flow-rectifying geometry within the 2D vortex-type design by changing the wall geometry, diode shape, and inflow velocities, identifying significant parameters and dependencies. Improving diodicity can be achieved by increasing the radius r₁ of the central channel, increasing the entire diode radius r₂, decreasing the width w of the rectangular channel, and reducing its length L. Additionally, changing the circular shape of the diode to an elliptical one can improve diodicity. The significance of this research is evident in the potential applications of these devices in microfluidic setups where fixed-geometry unidirectional flow is required, e.g., mixing, filtration, cell separation, and drug delivery, or on industrial scales, e.g., energy harvesting, wastewater treatment, and water sterilization. Full article

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30 pages, 858 KiB

Open AccessArticle

Sliding Mode Fault-Tolerant Control for Nonlinear LPV Systems with Variable Time-Delay

by Omayma Mansouri, Ali Ben Brahim, Fayçal Ben Hmida and Anis Sellami

Math. Comput. Appl. 2024, 29(6), 96; https://doi.org/10.3390/mca29060096 - 26 Oct 2024

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Abstract

This paper presents a robust sliding mode fault-tolerant control (FTC) strategy for a class of linear parameter variant (LPV) systems with variable time-delays and uncertainties. First fault estimation (FE) is conducted using a robust sliding mode observer, synthesized to simultaneously estimate the states [...] Read more.

This paper presents a robust sliding mode fault-tolerant control (FTC) strategy for a class of linear parameter variant (LPV) systems with variable time-delays and uncertainties. First fault estimation (FE) is conducted using a robust sliding mode observer, synthesized to simultaneously estimate the states and actuator faults of LPV polytopic delayed systems. Second, a sliding mode FTC is developed, ensuring all states of the closed-loop system converge to the origin. This paper presents an integrated sliding mode FTC strategy to achieve optimal robustness between the observer and controller models. The integrated design approach offers several advantages over traditional separated FTC methods. Our novel approach is based on incorporating adaptive law into the design of the Lyapunov–Krasovskii functional to improve both robustness and performance. This is achieved by combining the concept of sliding mode control (SMC) with the Lyapunov–Krasovskii function under the

H_{\infty}

criteria, which plays a key role in guaranteeing the stability of this class of system. The effectiveness of the proposed method is demonstrated through a diesel engine example, which highlights the validity and benefits of the integrated and separated FTC strategy for uncertain nonlinear systems with time delays and the sliding mode control. Full article

(This article belongs to the Special Issue Applied Optimization in Automatic Control and Systems Engineering)

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Math. Comput. Appl., Volume 29, Issue 6 (December 2024) – 15 articles

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