On Generalized Dominance Structures for Multi-Objective Optimization
FE² Computations with Deep Neural Networks: Algorithmic Structure, Data Generation, and Implementation

Journal Description

Mathematical and Computational Applications

Mathematical and Computational Applications is an international, peer-reviewed, open access journal on applications of mathematical and/or computational techniques, published bimonthly online by MDPI.

Open Access— free for readers, with article processing charges (APC) paid by authors or their institutions.
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Rapid Publication: manuscripts are peer-reviewed and a first decision is provided to authors approximately 20.1 days after submission; acceptance to publication is undertaken in 3.5 days (median values for papers published in this journal in the first half of 2023).
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Impact Factor: 1.9 (2022); 5-Year Impact Factor: 1.9 (2022)

Imprint Information Journal Flyer Open Access ISSN: 2297-8747

Latest Articles

19 pages, 443 KiB

Open AccessArticle

Bayes Inference of Structural Safety under Extreme Wind Loads Based upon a Peak-Over-Threshold Process of Exceedances

and

Math. Comput. Appl. 2023, 28(6), 111; https://doi.org/10.3390/mca28060111 - 30 Nov 2023

Abstract

In the present paper, the process of estimating the important statistical properties of extreme wind loads on structures is investigated by considering the effect of large variability. In fact, for the safety design and operating conditions of structures such as the ones characterizing tall buildings, wind towers, and offshore structures, it is of interest to obtain the best possible estimates of extreme wind loads on structures, the recurrence frequency, the return periods, and other stochastic properties, given the available statistical data. In this paper, a Bayes estimation of extreme load values is investigated in the framework of structural safety analysis. The evaluation of extreme values of the wind loads on the structures is performed via a combined employment of a Poisson process model for the peak-over-threshold characterization and an adequate characterization of the parent distribution which generates the base wind load values. In particular, the present investigation is based upon a key parameter for assessing the safety of structures, i.e., a proper safety index referred to a given extreme value of wind speed. The attention is focused upon the estimation process, for which the presented procedure proposes an adequate Bayesian approach based upon prior assumptions regarding (1) the Weibull probability that wind speed is higher than a prefixed threshold value, and (2) the frequency of the Poisson process of gusts. In the last part of the investigation, a large set of numerical simulations is analyzed to evaluate the feasibility and efficiency of the above estimation method and with the objective to analyze and compare the presented approach with the classical Maximum Likelihood method. Moreover, the robustness of the proposed Bayes estimation is also investigated with successful results, both with respect to the assumed parameter prior distributions and with respect to the Weibull distribution of the wind speed values. Full article

(This article belongs to the Special Issue Advanced Numerical Methods and Structural Complex Systems Monitoring Process)

8 pages, 763 KiB

Open AccessArticle

Explicit Integrating Factor Runge–Kutta Method for the Extended Fisher–Kolmogorov Equation

Yanan Wang

and

Shuying Zhai

Math. Comput. Appl. 2023, 28(6), 110; https://doi.org/10.3390/mca28060110 - 22 Nov 2023

Abstract

The extended Fisher–Kolmogorov (EFK) equation is an important model for phase transitions and bistable phenomena. This paper presents some fast explicit numerical schemes based on the integrating factor Runge–Kutta method and the Fourier spectral method to solve the EFK equation. The discrete global convergence of these new schemes is analyzed rigorously. Three numerical examples are presented to verify the theoretical analysis and the efficiency of the proposed schemes. Full article

(This article belongs to the Special Issue Recent Advances and New Challenges in Coupled Systems and Networks: Theory, Modelling, and Applications)

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16 pages, 754 KiB

Open AccessArticle

Asymptotic Consideration of Rayleigh Waves on a Coated Orthorhombic Elastic Half-Space Reinforced Using an Elastic Winkler Foundation

Ali M. Mubaraki

Math. Comput. Appl. 2023, 28(6), 109; https://doi.org/10.3390/mca28060109 - 15 Nov 2023

Abstract

This article derives approximate formulations for Rayleigh waves on a coated orthorhombic elastic half-space with a prescribed vertical load acting as an elastic Winkler foundation. In addition, perfect continuity conditions are imposed between the coating layer and the substrate, while suitable decaying conditions are slated along the infinite depth of the half-space. The effect of the thin layer is modeled using appropriate effective boundary conditions within the long-wave limit. By applying the Radon transform and using the perturbation method, the derived model successfully captures the physical characteristics of elastic surface waves in coated half-spaces. The model consists of a pesudo-static elliptic equation decaying over the interior of the half-space and a singularly perturbed hyperbolic equation with a pseudo-differential operator. The pseudo-differential equation gives the approximate dispersion of surface waves on the coated half-space structure and is analyzed numerically at the end. Full article

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

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20 pages, 352 KiB

Open AccessArticle

Fractional Hermite–Hadamard-Type Inequalities for Differentiable Preinvex Mappings and Applications to Modified Bessel and q-Digamma Functions

and

Math. Comput. Appl. 2023, 28(6), 108; https://doi.org/10.3390/mca28060108 - 09 Nov 2023

Abstract

The theory of convexity pertaining to fractional calculus is a well-established concept that has attracted significant attention in mathematics and various scientific disciplines for over a century. In the realm of applied mathematics, convexity, particularly in relation to fractional analysis, finds extensive and remarkable applications. In this manuscript, we establish new fractional identities. Employing these identities, some extensions of the fractional H-H type inequality via generalized preinvexities are explored. Finally, we discuss some applications to the q-digamma and Bessel functions via the established results. We believe that the methodologies and approaches presented in this work will intrigue and spark the researcher’s interest even more. Full article

(This article belongs to the Special Issue Significance of Mathematical Modelling and Control in Real-World Problems: New Developments and Applications)

23 pages, 7673 KiB

Open AccessArticle

Comparative Study of Metaheuristic Optimization of Convolutional Neural Networks Applied to Face Mask Classification

and

Math. Comput. Appl. 2023, 28(6), 107; https://doi.org/10.3390/mca28060107 - 01 Nov 2023

Abstract

The preventive measures taken to curb the spread of COVID-19 have emphasized the importance of wearing face masks to prevent potential infection with serious diseases during daily activities or for medical professionals working in hospitals. Due to the mandatory use of face masks, various methods employing artificial intelligence and deep learning have emerged to detect whether individuals are wearing masks. In this paper, we utilized convolutional neural networks (CNNs) to classify the use of face masks into three categories: no mask, incorrect mask, and proper mask. Establishing the appropriate CNN architecture can be a demanding task. This study compares four swarm intelligent metaheuristics: particle swarm optimization (PSO), grey wolf optimizer (GWO), bat algorithm (BA), and whale optimization algorithm (WOA). The CNN architecture design involves determining the essential hyperparameters of the CNNs. The results indicate the effectiveness of the PSO and BA in achieving an accuracy of 100% when using 10% of the images for testing. Meanwhile, when 90% of the images were used for testing, the results were as follows: PSO 97.15%, WOA 97.14%, BA 97.23%, and GWO 97.18%. These statistically significant differences demonstrate that the BA allows better results than the other metaheuristics analyzed in this study. Full article

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

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21 pages, 1445 KiB

Open AccessArticle

Model-Based Assessment of Elastic Material Parameters in Rheumatic Heart Disease Patients and Healthy Subjects

and

Math. Comput. Appl. 2023, 28(6), 106; https://doi.org/10.3390/mca28060106 - 01 Nov 2023

Abstract

Non-invasive measurements are important for the development of new treatments for heart failure, which is one of the leading causes of death worldwide. This study aimed to develop realistic subject-specific computational models of human biventricles using clinical data. Three-dimensional finite element models of the human ventricles were created using cardiovascular magnetic resonance images of rheumatic heart disease (RHD) patients and healthy subjects. The material parameter optimization uses inverse modeling based on the finite element method combined with the Levenberg–Marquardt method (LVM) by targeting subject-specific hemodynamics. The study of elastic myocardial parameters between healthy subjects and RHD patients showed an elevated stiffness in diseased hearts. In particular, the anisotropic material behavior of the healthy and diseased cardiac tissue significantly differed from one another. Furthermore, as the LVEF decreased, the stiffness and its orientation-dependent parameters increased. The simulation-derived LV myocardial circumferential and longitudinal stresses were negatively associated with the LVEF. The sensitivity analysis result demonstrated that the observed significant difference between the elastic material parameters of diseased and healthy myocardium was not exclusively attributable to an increased LVEDP in the diseased heart. These results could be applied to future computational studies for developing heart failure treatment. Full article

(This article belongs to the Special Issue Current Problems and Advances in Computational and Applied Mechanics)

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16 pages, 968 KiB

Open AccessArticle

Modelization of Low-Cost Maneuvers for an Areostationary Preliminary Mission Design

Marta M. Sánchez-García

Gonzalo Barderas

and

Pilar Romero

Math. Comput. Appl. 2023, 28(6), 105; https://doi.org/10.3390/mca28060105 - 27 Oct 2023

Abstract

The aim of this paper is to analyze the determination of interplanetary trajectories from Earth to Mars to evaluate the cost of the required impulse magnitudes for an areostationary orbiter mission design. Such analysis is first conducted by solving the Lambert orbital boundary value problem and studying the launch and arrival conditions for various date combinations. Then, genetic algorithms are applied to investigate the minimum-energy transfer orbit. Afterwards, an iterative procedure is used to determine the heliocentric elliptic transfer orbit that matches at the entry point of Mars’s sphere of influence with an areocentric hyperbolic orbit imposing specific conditions on inclination and periapsis radius. Finally, the maneuvers needed to obtain an areostationary orbit are numerically computed for different objective condition values at the Mars entry point to evaluate an areostationary preliminary mission cost for further study and characterization. Results show that, for the dates of the minimum-energy Earth–Mars transfer trajectory, a low value for the maneuvers to achieve an areostationary orbit is obtained for an arrival hyperbola with the minimum possible inclination and a capture into an elliptical trajectory with a low periapsis radius and an apoapsis at the stationary orbit. For a 2026 mission with a TOF of 304 for the minimum-energy Earth–Mars transfer trajectory, for a capture with a periapsis of 300 km above the Mars surface the value achieved will be 2.083 km/s. Full article

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

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18 pages, 2852 KiB

Open AccessArticle

Observer-Based State Estimation for Recurrent Neural Networks: An Output-Predicting and LPV-Based Approach

Wanlin Wang

Jinxiong Chen

and

Zhenkun Huang

Math. Comput. Appl. 2023, 28(6), 104; https://doi.org/10.3390/mca28060104 - 25 Oct 2023

Abstract

An innovative cascade predictor is presented in this study to forecast the state of recurrent neural networks (RNNs) with delayed output. This cascade predictor is a chain-structured observer, as opposed to the conventional single observer, and is made up of several sub-observers that individually estimate the state of the neurons at various periods. This new cascade predictor is more useful than the conventional single observer in predicting neural network states when the output delay is arbitrarily large but known. In contrast to examining the stability of error systems solely employing the Lyapunov–Krasovskii functional (LKF), several new global asymptotic stability standards are obtained by combining the application of the Linear Parameter Varying (LPV) approach, LKF and convex principle. Finally, a series of numerical simulations verify the efficacy of the obtained results. Full article

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

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17 pages, 5343 KiB

Open AccessArticle

Data-Driven Framework for Uncovering Hidden Control Strategies in Evolutionary Analysis

Nourddine Azzaoui

Tomoko Matsui

and

Daisuke Murakami

Math. Comput. Appl. 2023, 28(5), 103; https://doi.org/10.3390/mca28050103 - 20 Oct 2023

Abstract

We devised a data-driven framework for uncovering hidden control strategies used by an evolutionary system described by an evolutionary probability distribution. This innovative framework enables deciphering of the concealed mechanisms that contribute to the progression or mitigation of such situations as the spread of COVID-19. Novel algorithms are used to estimate the optimal control in tandem with the parameters for evolution in general dynamical systems, thereby extending the concept of model predictive control. This marks a significant departure from conventional control methods, which require knowledge of the system to manipulate its evolution and of the controller’s strategy or parameters. We use a generalized additive model, supplemented by extensive statistical testing, to identify a set of predictor covariates closely linked to the control. Using real-world COVID-19 data, we delineate the descriptive behaviors of the COVID-19 epidemics in five prefectures in Japan and nine countries. We compare these nine countries and group them on the basis of shared profiles, providing valuable insights into their pandemic responses. Our findings underscore the potential of our framework as a powerful tool for understanding and managing complex evolutionary processes. Full article

(This article belongs to the Collection Mathematical Modelling of COVID-19)

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15 pages, 5254 KiB

Open AccessArticle

Applying Physics-Informed Neural Networks to Solve Navier–Stokes Equations for Laminar Flow around a Particle

Beichao Hu

and

Dwayne McDaniel

Math. Comput. Appl. 2023, 28(5), 102; https://doi.org/10.3390/mca28050102 - 13 Oct 2023

Abstract

In recent years, Physics-Informed Neural Networks (PINNs) have drawn great interest among researchers as a tool to solve computational physics problems. Unlike conventional neural networks, which are black-box models that “blindly” establish a correlation between input and output variables using a large quantity of labeled data, PINNs directly embed physical laws (primarily partial differential equations) within the loss function of neural networks. By minimizing the loss function, this approach allows the output variables to automatically satisfy physical equations without the need for labeled data. The Navier–Stokes equation is one of the most classic governing equations in thermal fluid engineering. This study constructs a PINN to solve the Navier–Stokes equations for a 2D incompressible laminar flow problem. Flows passing around a 2D circular particle are chosen as the benchmark case, and an elliptical particle is also examined to enrich the research. The velocity and pressure fields are predicted by the PINNs, and the results are compared with those derived from Computational Fluid Dynamics (CFD). Additionally, the particle drag force coefficient is calculated to quantify the discrepancy in the results of the PINNs as compared to CFD outcomes. The drag coefficient maintained an error within 10% across all test scenarios. Full article

(This article belongs to the Topic Numerical Methods for Partial Differential Equations)

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16 pages, 613 KiB

Open AccessArticle

Data-Driven Active Learning Control for Bridge Cranes

Haojie Lin

and

Xuyang Lou

Math. Comput. Appl. 2023, 28(5), 101; https://doi.org/10.3390/mca28050101 - 09 Oct 2023

Abstract

For positioning and anti-swing control of bridge cranes, the active learning control method can reduce the dependence of controller design on the model and the influence of unmodeled dynamics on the controller’s performance. By only using the real-time online input and output data of the bridge crane system, the active learning control method consists of the finite-dimensional approximation of the Koopman operator and the design of an active learning controller based on the linear quadratic optimal tracking control. The effectiveness of the control strategy for positioning and anti-swing of bridge cranes is verified through numerical simulations. Full article

(This article belongs to the Special Issue Advanced Numerical Methods and Structural Complex Systems Monitoring Process)

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31 pages, 1242 KiB

Open AccessFeature PaperArticle

On Generalized Dominance Structures for Multi-Objective Optimization

Kalyanmoy Deb

and

Matthias Ehrgott

Math. Comput. Appl. 2023, 28(5), 100; https://doi.org/10.3390/mca28050100 - 07 Oct 2023

Abstract

Various dominance structures have been proposed in the multi-objective optimization literature. However, a systematic procedure to understand their effect in determining the resulting optimal set for generic domination principles, besides the standard Pareto-dominance principle, is lacking. In this paper, we analyze and lay out properties of generalized dominance structures which help provide insights for resulting optimal solutions. We introduce the concept of the anti-dominance structure, derived from the chosen dominance structure, to explain how the resulting non-dominated or optimal set can be identified easily compared to using the dominance structure directly. The concept allows a unified explanation of optimal solutions for both single- and multi-objective optimization problems. The anti-dominance structure is applied to analyze respective optimal solutions for most popularly used static and spatially changing dominance structures. The theoretical and deductive results of this study can be utilized to create more meaningful dominance structures for practical problems, understand and identify resulting optimal solutions, and help develop better test problems and algorithms for multi-objective optimization. Full article

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

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26 pages, 6386 KiB

Open AccessArticle

Enhanced MPPT-Based Fractional-Order PID for PV Systems Using Aquila Optimizer

Mohammed Tadj

Lakhdar Chaib

Abdelghani Choucha

Al-Motasem Aldaoudeyeh

and

Math. Comput. Appl. 2023, 28(5), 99; https://doi.org/10.3390/mca28050099 - 03 Oct 2023

Abstract

This paper proposes a controller to track the maximum power point (MPP) of a photovoltaic (PV) system using a fractional-order proportional integral derivative (FOPID) controller. The employed MPPT is operated based on a dp/dv feedback approach. The designed FOPID-MPPT method includes a differentiator of order (μ) and integrator of order (λ), meaning it is an extension of the conventional PID controller. FOPID has more flexibility and achieves dynamical tuning, which leads to an efficient control system. The contribution of our paper lies is optimizing FOPID-MPPT parameters using Aquila optimizer (AO). The obtained results with the proposed AO-based FOPID-MPPT are contrasted with those acquired with moth flame optimizer (MFO). The performance of our FOPID-MPPT controller with the conventional technique perturb and observe (P&O) and the classical PID controller is analyzed. In addition, a robustness test is used to assess the performance of the FOPID-MPPT controller under load variations, providing valuable insights into its practical applicability and robustness. The simulation results clearly prove the superiority and high performance of the proposed control system to track the MPP of PV systems. Full article

(This article belongs to the Special Issue Recent Advances in Mathematical Modeling, Analysis and Optimization of Photovoltaic/Thermal System)

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19 pages, 2887 KiB

Open AccessArticle

Predictive Modeling and Control Strategies for the Transmission of Middle East Respiratory Syndrome Coronavirus

and

Math. Comput. Appl. 2023, 28(5), 98; https://doi.org/10.3390/mca28050098 - 30 Sep 2023

Abstract

The Middle East respiratory syndrome coronavirus (MERS-CoV) is a highly infectious respiratory illness that poses a significant threat to public health. Understanding the transmission dynamics of MERS-CoV is crucial for effective control and prevention strategies. In this study, we develop a precise mathematical model to capture the transmission dynamics of MERS-CoV. We incorporate some novel parameters related to birth and mortality rates, which are essential factors influencing the spread of the virus. We obtain epidemiological data from reliable sources to estimate the model parameters. We compute its basic reproduction number (R0). Stability theory is employed to analyze the local and global properties of the model, providing insights into the system’s equilibrium states and their stability. Sensitivity analysis is conducted to identify the most critical parameter affecting the transmission dynamics. Our findings revealed important insights into the transmission dynamics of MERS-CoV. The stability analysis demonstrated the existence of stable equilibrium points, indicating the long-term behavior of the epidemic. Through the evaluation of optimal control strategies, we identify effective intervention measures to mitigate the spread of MERS-CoV. Our simulations demonstrate the impact of time-dependent control variables, such as supportive care and treatment, in reducing the number of infected individuals and controlling the epidemic. The model can serve as a valuable tool for public health authorities in designing effective control and prevention strategies, ultimately reducing the burden of MERS-CoV on global health. Full article

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

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29 pages, 1311 KiB

Open AccessArticle

Preconditioning Technique for an Image Deblurring Problem with the Total Fractional-Order Variation Model

Adel M. Al-Mahdi

Math. Comput. Appl. 2023, 28(5), 97; https://doi.org/10.3390/mca28050097 - 22 Sep 2023

Abstract

Total fractional-order variation (TFOV) in image deblurring problems can reduce/remove the staircase problems observed with the image deblurring technique by using the standard total variation (TV) model. However, the discretization of the Euler–Lagrange equations associated with the TFOV model generates a saddle point system of equations where the coefficient matrix of this system is dense and ill conditioned (it has a huge condition number). The ill-conditioned property leads to slowing of the convergence of any iterative method, such as Krylov subspace methods. One treatment for the slowness property is to apply the preconditioning technique. In this paper, we propose a block triangular preconditioner because we know that using the exact triangular preconditioner leads to a preconditioned matrix with exactly two distinct eigenvalues. This means that we need at most two iterations to converge to the exact solution. However, we cannot use the exact preconditioner because the Shur complement of our system is of the form

S = K^{*} K + λ L_{α}

which is a huge and dense matrix. The first matrix,

K^{*} K

, comes from the blurred operator, while the second one is from the TFOV regularization model. To overcome this difficulty, we propose two preconditioners based on the circulant and standard TV matrices. In our algorithm, we use the flexible preconditioned GMRES method for the outer iterations, the preconditioned conjugate gradient (PCG) method for the inner iterations, and the fixed point iteration (FPI) method to handle the nonlinearity. Fast convergence was found in the numerical results by using the proposed preconditioners. Full article

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

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38 pages, 665 KiB

Open AccessArticle

Lie Symmetry Classification, Optimal System, and Conservation Laws of Damped Klein–Gordon Equation with Power Law Non-Linearity

Fiazuddin D. Zaman

Fazal M. Mahomed

and

Faiza Arif

Math. Comput. Appl. 2023, 28(5), 96; https://doi.org/10.3390/mca28050096 - 12 Sep 2023

Abstract

We used the classical Lie symmetry method to study the damped Klein–Gordon equation (Kge) with power law non-linearity

u_{t t} + α (u) u_{t} = {(u^{β} u_{x})}_{x} + f (u)

[...] Read more.

We used the classical Lie symmetry method to study the damped Klein–Gordon equation (Kge) with power law non-linearity

u_{t t} + α (u) u_{t} = {(u^{β} u_{x})}_{x} + f (u)

. We carried out a complete Lie symmetry classification by finding forms for

α (u)

and

f (u)

. This led to various cases. Corresponding to each case, we obtained one-dimensional optimal systems of subalgebras. Using the subalgebras, we reduced the Kge to ordinary differential equations and determined some invariant solutions. Furthermore, we obtained conservation laws using the partial Lagrangian approach. Full article

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

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16 pages, 4977 KiB

Open AccessArticle

New Quality Measures for Quadrilaterals and New Discrete Functionals for Grid Generation

Guilmer Ferdinand González Flores

and

Pablo Barrera Sánchez

Math. Comput. Appl. 2023, 28(5), 95; https://doi.org/10.3390/mca28050095 - 09 Sep 2023

Abstract

In this paper, we review some grid quality metrics and define some new quality measures for quadrilateral elements. The curved elements are not discussed. Usually, the maximum value of a quality measure corresponds to the minimum value of the energy density over the grid. We also define new discrete functionals, which are implemented as objective functions in an optimization-based method for quadrilateral grid generation and improvement. These functionals are linearly combined with a discrete functional whose domain has an infinite barrier at the boundary of the set of unfolded grids to preserve convex grid cells in each step of the optimization process. Full article

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

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17 pages, 382 KiB

Open AccessArticle

Asymptotic Behavior of Solutions to a Nonlinear Swelling Soil System with Time Delay and Variable Exponents

Mohammad M. Kafini

Mohammed M. Al-Gharabli

and

Adel M. Al-Mahdi

Math. Comput. Appl. 2023, 28(5), 94; https://doi.org/10.3390/mca28050094 - 06 Sep 2023

Abstract

In this research work, we investigate the asymptotic behavior of a nonlinear swelling (also called expansive) soil system with a time delay and nonlinear damping of variable exponents. We should note here that swelling soils contain clay minerals that absorb water, which may lead to increases in pressure. In architectural and civil engineering, swelling soils are considered sources of problems and harm. The presence of the delay is used to create more realistic models since many processes depend on past history, and the delays are frequently added by sensors, actuators, and field networks that travel through feedback loops. The appearance of variable exponents in the delay and damping terms in this system allows for a more flexible and accurate modeling of this physical phenomenon. This can lead to more realistic and precise descriptions of the behavior of fluids in different media. In fact, with the advancements of science and technology, many physical and engineering models require more sophisticated mathematical tools to study and understand. The Lebesgue and Sobolev spaces with variable exponents proved to be efficient tools for studying such problems. By constructing a suitable Lyapunov functional, we establish exponential and polynomial decay results. We noticed that the energy decay of the system depends on the value of the variable exponent. These results improve on some existing results in the literature. Full article

(This article belongs to the Special Issue Advanced Numerical Methods and Structural Complex Systems Monitoring Process)

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35 pages, 5613 KiB

Open AccessArticle

Exploring the Potential of Mixed Fourier Series in Signal Processing Applications Using One-Dimensional Smooth Closed-Form Functions with Compact Support: A Comprehensive Tutorial

Carlos-Iván Páez-Rueda

and

Math. Comput. Appl. 2023, 28(5), 93; https://doi.org/10.3390/mca28050093 - 01 Sep 2023

Abstract

This paper studies and analyzes the approximation of one-dimensional smooth closed-form functions with compact support using a mixed Fourier series (i.e., a combination of partial Fourier series and other forms of partial series). To explore the potential of this approach, we discuss and revise its application in signal processing, especially because it allows us to control the decreasing rate of Fourier coefficients and avoids the Gibbs phenomenon. Therefore, this method improves the signal processing performance in a wide range of scenarios, such as function approximation, interpolation, increased convergence with quasi-spectral accuracy using the time domain or the frequency domain, numerical integration, and solutions of inverse problems such as ordinary differential equations. Moreover, the paper provides comprehensive examples of one-dimensional problems to showcase the advantages of this approach. Full article

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

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11 pages, 269 KiB

Open AccessArticle

Conservation Laws and Symmetry Reductions of the Hunter–Saxton Equation via the Double Reduction Method

Molahlehi Charles Kakuli

Winter Sinkala

and

Phetogo Masemola

Math. Comput. Appl. 2023, 28(5), 92; https://doi.org/10.3390/mca28050092 - 22 Aug 2023

Abstract

This study investigates via Lie symmetry analysis the Hunter–Saxton equation, an equation relevant to the theoretical analysis of nematic liquid crystals. We employ the multiplier method to obtain conservation laws of the equation that arise from first-order multipliers. Conservation laws of the equation, combined with the admitted Lie point symmetries, enable us to perform symmetry reductions by employing the double reduction method. The method exploits the relationship between symmetries and conservation laws to reduce both the number of variables and the order of the equation. Five nontrivial conservation laws of the Hunter–Saxton equation are derived, four of which are found to have associated Lie point symmetries. Applying the double reduction method to the equation results in a set of first-order ordinary differential equations, the solutions of which represent invariant solutions for the equation. While the double reduction method may be more complex to implement than the classical method, since it involves finding Lie point symmetries and deriving conservation laws, it has some advantages over the classical method of reducing PDEs. Firstly, it is more efficient in that it can reduce the number of variables and order of the equation in a single step. Secondly, by incorporating conservation laws, physically meaningful solutions that satisfy important physical constraints can be obtained. Full article

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

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