Mathematical and Computational Applications

17 pages, 5343 KiB

Open AccessArticle

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

by Nourddine Azzaoui, Tomoko Matsui and Daisuke Murakami

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

Viewed by 1139

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 [...] Read more.

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

by Beichao Hu and Dwayne McDaniel

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

Viewed by 2016

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 [...] Read more.

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

by Haojie Lin and Xuyang Lou

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

Viewed by 1248

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 [...] Read more.

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

by Kalyanmoy Deb and Matthias Ehrgott

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

Viewed by 1673

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 [...] Read more.

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

by Mohammed Tadj, Lakhdar Chaib, Abdelghani Choucha, Al-Motasem Aldaoudeyeh, Ahmed Fathy, Hegazy Rezk, Mohamed Louzazni and Attia El-Fergany

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

Viewed by 1596

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 [...] Read more.

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

by Bibi Fatima, Mehmet Yavuz, Mati ur Rahman, Ali Althobaiti and Saad Althobaiti

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

Cited by 7 | Viewed by 1350

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 [...] Read more.

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

by Adel M. Al-Mahdi

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

Viewed by 1189

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 [...] Read more.

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

by 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

Viewed by 1083

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

by Guilmer Ferdinand González Flores and Pablo Barrera Sánchez

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

Viewed by 915

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 [...] Read more.

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

by 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

Cited by 1 | Viewed by 908

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 [...] Read more.

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

by Carlos-Iván Páez-Rueda, Arturo Fajardo, Manuel Pérez, German Yamhure and Gabriel Perilla

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

Viewed by 1273

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 [...] Read more.

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

by Molahlehi Charles Kakuli, Winter Sinkala and Phetogo Masemola

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

Viewed by 1013

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, [...] Read more.

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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Math. Comput. Appl., Volume 28, Issue 5 (October 2023) – 12 articles

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