Convergence Of Solutions Research Articles

A result of convergence for a mono-dimensional two-velocities lattice Boltzmann scheme

We consider a mono-dimensional two-velocities scheme used to approximate the solutions of a scalar hyperbolic conservative partial differential equation. We prove the convergence of the discrete solution towards the unique entropy solution by first estimating the supremum norm and the total variation of the discrete solution, and second by constructing a discrete kinetic entropy-entropy flux pair being given a continuous entropy-entropy flux pair of the hyperbolic system. We finally illustrate our results with numerical simulations of the advection equation and the Burgers equation.

Solving the reaction-diffusion Brusselator system using Generalized Finite Difference Method

<abstract><p>In this paper, we investigate the numerical solution of the Brusselator system using a meshless method. A numerical scheme is derived employing the formulas of the Generalized Finite Difference Method, and the convergence of the approximate solution to the exact solution is examined. In order to demonstrate the applicability and accuracy of the method, several examples are proposed.</p></abstract>

Convergence of finite element solution of stochastic Burgers equation

<abstract><p>We explore the numerical approximation of the stochastic Burgers equation driven by fractional Brownian motion with Hurst index $ H\in(1/4, 1/2) $ and $ H\in(1/2, 1) $, respectively. The spatial and temporal regularity properties for the solution are obtained. The given problem is discretized in time with the implicit Euler scheme and in space with the standard finite element method. We obtain the strong convergence of semidiscrete and fully discrete schemes, performing the error estimates on a subset $ \Omega_{k, h} $ of the sample space $ \Omega $ with the Gronwall argument being used to overcome the difficulties, caused by the subtle interplay of the nonlinear convection term. Numerical examples confirm our theoretical findings.</p></abstract>

Study of numerical treatment of functional first-kind Volterra integral equations

<abstract><p>First-kind Volterra integral equations have ill-posed nature in comparison to the second-kind of these equations such that a measure of ill-posedness can be described by $ \nu $-smoothing of the integral operator. A comprehensive study of the convergence and super-convergence properties of the piecewise polynomial collocation method for the second-kind Volterra integral equations (VIEs) with constant delay has been given in <sup>[<xref ref-type="bibr" rid="b1">1</xref>]</sup>. However, convergence analysis of the collocation method for first-kind delay VIEs appears to be a research problem. Here, we investigated the convergence of the collocation solution as a research problem for a first-kind VIE with constant delay. Three test problems have been fairly well-studied for the sake of verifying theoretical achievements in practice.</p></abstract>

Convergence of smooth solutions to parabolic equations with an oblique derivative boundary condition

<abstract><p>In this paper, the parabolic equation with oblique derivative boundary condition is considered. The long time behavior of the solution is derived by selecting the appropriate auxiliary functions and making priori estimates. Through blow up analysis, time-dependent gradient estimates are obtained, followed by second-order derivative estimates. Then, the convergence of smooth solution to parabolic equations with the oblique derivative boundary condition is obtained using standard theory.</p></abstract>

Solution of the SIR epidemic model of arbitrary orders containing Caputo-Fabrizio, Atangana-Baleanu and Caputo derivatives

<abstract><p>The main aim of this study was to apply an analytical method to solve a nonlinear system of fractional differential equations (FDEs). This method is the Adomian decomposition method (ADM), and a comparison between its results was made by using a numerical method: Runge-Kutta 4 (RK4). It is proven that there is a unique solution to the system. The convergence of the series solution is given, and the error estimate is also proven. After that, the susceptible-infected-recovered (SIR) model was taken as an real phenomenon with such systems. This system is discussed with three different fractional derivatives (FDs): the Caputo-Fabrizio derivative (CFD), the Atangana-Baleanu derivative (ABD), and the Caputo derivative (CD). A comparison between these three different derivatives is given. We aimed to see which one of the new definitions (ABD and CFD) is close to one of the most important classical definitions (CD).</p></abstract>

On the stability and asymptotic behavior for a quasi-linear parabolic flow

In this paper, we study the stability property and asymptotic behavior for a quasi-linear parabolic flow in the whole line. We first show the existence and uniqueness of global solutions of the problem. Then we study the stability of the solution to the straight line. We prove the asymptotic behavior or the convergence of the global solution. Similar to the behavior of solutions to heat equation, we prove that the stationary line attracts the graphical curves which surround it.

On the steady-state interfacial waves with two-dimensional type-A double exact resonance

Steady-state interfacial waves under two-dimensional (2D) type-A exact triad resonance and other related resonances are researched in a two-layer liquid model with a free surface in contact with air. Five groups (groups 1–5) of convergent series solutions are achieved via the homotopy analysis method. It is found that the phenomenon of double exact resonance could exist in periodic interfacial waves if physical parameters correspond to the intersection of two exact resonance curves. The double exact resonance considered here contains a 2D type-A triad resonance and an other resonance. Under the 2D type-A exact triad resonance, the other resonant triad could obviously enlarge or reduce the wave amplitudes and energy proportions of primary and resonant components. Nevertheless, other resonant quartet, quintet, sextet, and septet all produce no influence on interfacial waves when the 2D type-A exact triad resonance occurs. The above-mentioned results indicate that in the neighborhood of the double exact triad resonance, small perturbations of wave vector of a primary component can cause huge changes on wave profiles of free surface and interface, wave amplitude spectrum, and energy distribution of internal waves in real ocean. In addition, the closer the interfacial waves are to the double exact triad resonance, the more possible energy combinations exist in the wave system, and the greater the number of steady-state interfacial wave solutions. All of this should deepen our understanding of nonlinear resonance interactions in short-crested internal waves.

Economic Load Dispatch using IYSGA

The Economic Load Dispatch (ELD) problem is a pivotal aspect of power system management, focusing on the efficient allocation of power generation among various units to meet the demand while minimizing costs. This research paper presents an Improved Yellow Saddle Goat Fish Algorithm (IYSGA) based method for resolving ELD issues. The key objective of proposed IYSGA method is to reduce error between demanded and generated load along with its unit cost. This objective is accomplished by using YSGA whose exploration ability is improved by exploring ability of Grasshopper Optimization Algorithm (GOA). By implementing IYSGA in given ELD problem, the convergence rate, exploring ability and solution quality is enhanced. The fitness function is determined by IYSGA in terms of error and cost reduction, which should be as minimum as possible. The simulations are performed on standardized IEEE bus system with 3-unit and 6-units to meet load demand of 850MW to 1263MW respectively. The experimental simulations conducted provide evidence that the proposed approach met the load demand with zero error. Furthermore, proposed method attained best cost of $8197.633 and $15,285.7055 for the 3-unit and 6-unit generation unit. These outcomes underscore the robustness and superiority of the proposed method in addressing the Economic Load Dispatch (ELD) problem, emphasizing its capacity to optimize power generation with unparalleled precision and cost-effectiveness.

Efficient approximate analytical technique to solve nonlinear coupled Jaulent–Miodek system within a time-fractional order

<abstract><p>In this article, we considered the nonlinear time-fractional Jaulent–Miodek model (FJMM), which is applied to modeling many applications in basic sciences and engineering, especially physical phenomena such as plasma physics, fluid dynamics, electromagnetic waves in nonlinear media, and many other applications. The Caputo fractional derivative (CFD) was applied to express the fractional operator in the mathematical formalism of the FJMM. We implemented the modified generalized Mittag-Leffler method (MGMLFM) to show the analytical approximate solution of FJMM, which is represented by a set of coupled nonlinear fractional partial differential equations (FPDEs) with suitable initial conditions. The suggested method produced convergent series solutions with easily computable components. To demonstrate the accuracy and efficiency of the MGMLFM, a comparison was made between the solutions obtained by MGMLFM and the known exact solutions in some tables. Also, the absolute error was compared with the absolute error provided by some of the other famous methods found in the literature. Our findings confirmed that the presented method is easy, simple, reliable, competitive, and did not require complex calculations. Thus, it can be extensively applied to solve more linear and nonlinear FPDEs that have applications in various areas such as mathematics, engineering, and physics.</p></abstract>

SOLVING THE TASK OF LOCAL OPTIMA TRAPS IN DATA MINING APPLICATIONS THROUGH INTELLIGENT MULT-AGENT SWARM AND ORTHOPAIR FUZZY SETS

Local optima traps pose a significant challenge in optimizing complex problems, particularly in data mining applications, where traditional algorithms may get stuck in suboptimal solutions. This study addresses this issue by combining the power of intelligent multi-agent swarm algorithms and orthopair fuzzy sets to enhance optimization processes. We propose a novel approach that leverages the collective intelligence of a multi-agent swarm system, enabling effective exploration and exploitation of solution spaces. Additionally, orthopair fuzzy sets are introduced to model and represent uncertainties inherent in data mining tasks, providing a more robust optimization framework. Our work contributes to the advancement of optimization techniques in data mining by offering a synergistic solution to local optima traps. The integration of intelligent multi-agent swarms and orthopair fuzzy sets enhances the algorithm’s adaptability and resilience, leading to improved convergence and better solutions. Experimental results demonstrate the efficacy of our proposed approach in overcoming local optima traps, showcasing superior performance compared to traditional algorithms. The hybrid system exhibits increased convergence rates and consistently discovers more accurate and diverse solutions across various data mining scenarios.

Optimalno podešavanje parametara simuliranog kaljenja

Introduction/purpose: Simulated annealing is a powerful technique widely used in optimization problems. One critical aspect of using simulated annealing effectively is a proper and optimal adjustment of its parameters. This paper presents a novel approach to efficiently adjust the parameters of simulated annealing to enhance its performance and convergence speed. Methods: Since the simulated algorithm is inspired by the cooling Metropolis process, the basic idea is to simulate and analyze this process using a mathematical model. The proposed work tends to properly imitate the Metropolis cooling process in the algorithmic field. By intelligently adjusting the temperature schedule, temperature reduction and cooling rate, the algorithm optimizes the balance between exploration and exploitation, leading to improved convergence and higher-quality solutions. Results: To evaluate the effectiveness of this approach, it was applied first on a chosen sample function to be minimized, and then on some usual known optimization functions. The results demonstrate that our approach, called Optimal Adjusting of Simulated Annealing parameters (OASA), achieves superior performance compared to traditional static parameter settings and other existing approaches, showing how to well adjust the parameters of the simulated annealing algorithm to improve its efficiency in terms of solution quality and processing time. Conclusion: Adjusting the algorithm parameters could have a significant contribution in the optimization field even for other metaheuristics.

A hybrid framework for mean-CVaR portfolio selection under jump-diffusion processes: Combining cross-entropy method with beluga whale optimization

<abstract><p>In this paper, a new hybrid meta-heuristic algorithm called CEBWO (cross-entropy method and beluga whale optimization) is presented to solve the mean-CVaR portfolio optimization problem based on jump-diffusion processes. The proposed CEBWO algorithm combines the advantages of the cross-entropy method and beluga whale optimization algorithm with the help of co-evolution technology to enhance the performance of portfolio selection. The method is evaluated on 29 unconstrained benchmark functions from CEC 2017, where its performance is compared against several state-of-the-art algorithms. The results demonstrate the superiority of the hybrid method in terms of solution quality and convergence speed. Finally, Monte Carlo simulation is employed to generate scenario paths based on the jump-diffusion model. Empirical results further confirm the effectiveness of the hybrid meta-heuristic algorithm for mean-CVaR portfolio selection, highlighting its potential for real-world applications.</p></abstract>

Моделювання логістичних процесів імпорту морепродуктів в умовах пошкодженої інфраструктури

The article is devoted to issues of modeling the processes of finding solutions to logistical problems in fish and seafood supply to the population of Ukraine. The paper analyzes the numerous publications of Ukrainian and foreign researches on topics related to issues of marketing and logistics of fish products. The modern conditions’ assessment in which the supply of seafood is carried out is given. The problems faced by product carriers are considered, among which the main ones are hostilities, constant shelling, damage and destruction of infrastructure and energy networks of Ukraine. The Gray Wolf Optimization (GWO) metaheuristic algorithm is used to solve the given problem, which helps to solve NP-complex problems that are difficult to solve by any traditional exact methods, or these methods do not exist at all. Productivity of such type algorithms is evaluated based on five indicators: solution quality, convergence speed, computational efficiency, robustness, and flexibility. The proposed GWO-based model enabled the optimization of several objectives, including the minimization of total supply chain costs, maximization of customer satisfaction, and reduction of harmful environmental impacts, enabling an integrated approach to supply chain management.

Axisymmetric Incompressible Viscous Plasmas: Global Well-Posedness and Asymptotics

Abstract This paper is devoted to the global analysis of the three-dimensional axisymmetric Navier–Stokes–Maxwell equations. More precisely, we are able to prove that, for large values of the speed of light $c\in (c_0, \infty )$ , for some threshold $c_0>0$ depending only on the initial data, the system in question admits a unique global solution. The ensuing bounds on the solutions are uniform with respect to the speed of light, which allows us to study the singular regime $c\rightarrow \infty $ and rigorously derive the limiting viscous magnetohydrodynamic (MHD) system in the axisymmetric setting. The strategy of our proofs draws insight from recent results on the two-dimensional incompressible Euler–Maxwell system to exploit the dissipative–dispersive structure of Maxwell’s system in the axisymmetric setting. Furthermore, a detailed analysis of the asymptotic regime $c\to \infty $ allows us to derive a robust nonlinear energy estimate which holds uniformly in c. As a byproduct of such refined uniform estimates, we are able to describe the global strong convergence of solutions toward the MHD system. This collection of results seemingly establishes the first available global well-posedness of three-dimensional viscous plasmas, where the electric and magnetic fields are governed by the complete Maxwell equations, for large initial data as $c\to \infty $ .

Convergence of solutions of a kind of convex feasibility problem on Hadamard manifolds

Convergence of solutions of a kind of convex feasibility problem on Hadamard manifolds

Stability and Convergence of Solutions to Stochastic Inverse Problems Using Approximate Probability Densities

Stability and Convergence of Solutions to Stochastic Inverse Problems Using Approximate Probability Densities

An almost second order uniformly convergent method for a two-parameter singularly perturbed problem with a discontinuous convection coefficient and source term

<p>In this paper, we discuss a higher-order convergent numerical method for a two-parameter singularly perturbed differential equation with a discontinuous convection coefficient and a discontinuous source term. The presence of perturbation parameters generates boundary layers, and the discontinuous terms produce interior layers on both sides of the discontinuity. In order to obtain a higher-order convergent solution, a hybrid monotone finite difference scheme is constructed on a piecewise uniform Shishkin mesh, which is adapted inside the boundary and interior layers. On this mesh (including the point of discontinuity), the present method is almost second-order parameter-uniform convergent. The current scheme is compared with the standard upwind scheme, which is used at the point of discontinuity. The numerical experiments based on the proposed scheme show higher-order (almost second-order) accuracy compared to the standard upwind scheme, which provides almost first-order parameter-uniform convergence.</p>

Approximate analytical solutions of fractional order differential equations by conformable Sumudu decomposition method

This work offers a fast, approximate analytical technique for solving time fractional differential equations. CSTADM is a gentle combination of two powerful approaches to solve fractional PDE’s. The hybrid approach suggested here produces more accurate findings and convergent solution series faster. Also, the proposed approach is used to get the analytical solution of nonlinear fractional PDE’s. The fractional derivative’s type is assumed to be conformable. Compared to standard fractional derivative definitions, this approach is effective and straightforward in applications involving fractional differential equations with complicated solutions. Furthermore, the results show that the acquired results increase the precision and efficacy of the proposed technique.

DEQ-MPI: A Deep Equilibrium Reconstruction With Learned Consistency for Magnetic Particle Imaging.

Magnetic particle imaging (MPI) offers unparalleled contrast and resolution for tracing magnetic nanoparticles. A common imaging procedure calibrates a system matrix (SM) that is used to reconstruct data from subsequent scans. The ill-posed reconstruction problem can be solved by simultaneously enforcing data consistency based on the SM and regularizing the solution based on an image prior. Traditional hand-crafted priors cannot capture the complex attributes of MPI images, whereas recent MPI methods based on learned priors can suffer from extensive inference times or limited generalization performance. Here, we introduce a novel physics-driven method for MPI reconstruction based on a deep equilibrium model with learned data consistency (DEQ-MPI). DEQ-MPI reconstructs images by augmenting neural networks into an iterative optimization, as inspired by unrolling methods in deep learning. Yet, conventional unrolling methods are computationally restricted to few iterations resulting in non-convergent solutions, and they use hand-crafted consistency measures that can yield suboptimal capture of the data distribution. DEQ-MPI instead trains an implicit mapping to maximize the quality of a convergent solution, and it incorporates a learned consistency measure to better account for the data distribution. Demonstrations on simulated and experimental data indicate that DEQ-MPI achieves superior image quality and competitive inference time to state-of-the-art MPI reconstruction methods.