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18 pages, 633 KiB

Open AccessArticle

Local Convergence Analysis of a One Parameter Family of Simultaneous Methods with Applications to Real-World Problems

by Tsonyo M. Pavkov, Valentin G. Kabadzhov, Ivan K. Ivanov and Stoil I. Ivanov

Algorithms 2023, 16(2), 103; https://doi.org/10.3390/a16020103 - 10 Feb 2023

Cited by 4 | Viewed by 1660

In this paper, we provide a detailed local convergence analysis of a one-parameter family of iteration methods for the simultaneous approximation of polynomial zeros due to Ivanov (Numer. Algor. 75(4): 1193–1204, 2017). Thus, we obtain two local convergence theorems that provide sufficient conditions [...] Read more.

In this paper, we provide a detailed local convergence analysis of a one-parameter family of iteration methods for the simultaneous approximation of polynomial zeros due to Ivanov (Numer. Algor. 75(4): 1193–1204, 2017). Thus, we obtain two local convergence theorems that provide sufficient conditions to guarantee the Q-cubic convergence of all members of the family. Among the other contributions, our results unify the latest such kind of results of the well known Dochev–Byrnev and Ehrlich methods. Several practical applications are further given to emphasize the advantages of the studied family of methods and to show the applicability of the theoretical results. Full article

(This article belongs to the Special Issue Iterative Algorithms for Nonlinear Problems: Convergence and Stability 2021-2022)

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14 pages, 1125 KiB

Open AccessArticle

Solving of the Inverse Boundary Value Problem for the Heat Conduction Equation in Two Intervals of Time

by Bashar Talib Al-Nuaimi, H.K. Al-Mahdawi, Zainalabideen Albadran, Hussein Alkattan, Mostafa Abotaleb and El-Sayed M. El-kenawy

Algorithms 2023, 16(1), 33; https://doi.org/10.3390/a16010033 - 6 Jan 2023

Cited by 19 | Viewed by 2356

Abstract

The boundary value problem, BVP, for the PDE heat equation is studied and explained in this article. The problem declaration comprises two intervals; the (0, T) is the first interval and labels the heating of the inside burning chamber, and the second (T, [...] Read more.

The boundary value problem, BVP, for the PDE heat equation is studied and explained in this article. The problem declaration comprises two intervals; the (0, T) is the first interval and labels the heating of the inside burning chamber, and the second (T, ∞) interval defines the normal cooling of the chamber wall when the chamber temperature concurs with the ambient temperature. It is necessary to prove the boundary function of this problem has its place in the space

H^{1} [0, \infty]

in order to successfully apply the Fourier transform method. The applicability of the Fourier transform for time to this problem is verified. The method of projection regularization is used to solve the inverse boundary value problem for the heat equation and to obtain an evaluation for the error between the approximate and the real solution. These results are new and of practical interest as shown in the numerical case study. Full article

(This article belongs to the Special Issue Iterative Algorithms for Nonlinear Problems: Convergence and Stability 2021-2022)

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15 pages, 443 KiB

Open AccessArticle

Some Inexact Version of the Iterative Exponential Method for Solving Nonsmooth Equations

by Marek J. Śmietański

Algorithms 2023, 16(1), 27; https://doi.org/10.3390/a16010027 - 3 Jan 2023

Viewed by 1494

Abstract

This paper presents an inexact version of an exponential iterative method designed for solving nonlinear equations

F (x) = 0

, where the function F is only locally Lipschitz continuous. The proposed algorithm is completely new as an essential extension of [...] Read more.

This paper presents an inexact version of an exponential iterative method designed for solving nonlinear equations

F (x) = 0

, where the function F is only locally Lipschitz continuous. The proposed algorithm is completely new as an essential extension of the iterative exponential method for solving nonsmooth equations. The method with backtracking is globally and superlinearly convergent under some mild assumptions imposed on F. The presented results of the numerical computations confirm both the theoretical properties of the new method and its practical effectiveness. Full article

(This article belongs to the Special Issue Iterative Algorithms for Nonlinear Problems: Convergence and Stability 2021-2022)

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16 pages, 1710 KiB

Open AccessArticle

Overlapping Grid-Based Optimized Single-Step Hybrid Block Method for Solving First-Order Initial Value Problems

by Sandile Motsa

Algorithms 2022, 15(11), 427; https://doi.org/10.3390/a15110427 - 14 Nov 2022

Cited by 6 | Viewed by 2051

Abstract

This study presents a new variant of the hybrid block methods (HBMs) for solving initial value problems (IVPs). The overlapping hybrid block technique is developed by changing each integrating block of the HBM to incorporate the penultimate intra-step point of the previous block. [...] Read more.

This study presents a new variant of the hybrid block methods (HBMs) for solving initial value problems (IVPs). The overlapping hybrid block technique is developed by changing each integrating block of the HBM to incorporate the penultimate intra-step point of the previous block. In this paper, we present preliminary results obtained by applying the overlapping HBM to IVPs of the first order, utilizing equally spaced grid points and optimal points that maximize the local truncation errors of the main formulas at the intersection of each integration block. It is proven that the novel method reduces the local truncation error by at least one order of the integration step size, O(h). In order to demonstrate the superiority of the suggested method, numerical experimentation results were compared to the corresponding HBM based on the standard non-overlapping grid. It is established that the proposed method is more accurate than HBM versions of the same order that have been published in the literature. Full article

(This article belongs to the Special Issue Iterative Algorithms for Nonlinear Problems: Convergence and Stability 2021-2022)

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13 pages, 5012 KiB

Open AccessArticle

Dynamics and Stability on a Family of Optimal Fourth-Order Iterative Methods

by Alicia Cordero, Miguel A. Leonardo Sepúlveda and Juan R. Torregrosa

Algorithms 2022, 15(10), 387; https://doi.org/10.3390/a15100387 - 21 Oct 2022

Cited by 9 | Viewed by 1896

Abstract

In this manuscript, we propose a parametric family of iterative methods of fourth-order convergence, and the stability of the class is studied through the use of tools of complex dynamics. We obtain the fixed and critical points of the rational operator associated with [...] Read more.

In this manuscript, we propose a parametric family of iterative methods of fourth-order convergence, and the stability of the class is studied through the use of tools of complex dynamics. We obtain the fixed and critical points of the rational operator associated with the family. A stability analysis of the fixed points allows us to find sets of values of the parameter for which the behavior of the corresponding method is stable or unstable; therefore, we can select the regions of the parameter in which the methods behave more efficiently when they are applied for solving nonlinear equations or the regions in which the schemes have chaotic behavior. Full article

(This article belongs to the Special Issue Iterative Algorithms for Nonlinear Problems: Convergence and Stability 2021-2022)

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14 pages, 3066 KiB

Open AccessArticle

Numerical Simulation of Micro-Bubbles Dispersion by Surface Waves

by Oleg A. Druzhinin and Wu-Ting Tsai

Algorithms 2022, 15(4), 110; https://doi.org/10.3390/a15040110 - 24 Mar 2022

Cited by 4 | Viewed by 2526

Abstract

This paper presents an algorithm for numerical modeling of bubble dispersion occurring in the near-surface layer of the upper ocean under the action of non-breaking two-dimensional (2D) surface waves. The algorithm is based on a Eulerian-Lagrangian approach where full, 3D Navier-Stokes equations for [...] Read more.

This paper presents an algorithm for numerical modeling of bubble dispersion occurring in the near-surface layer of the upper ocean under the action of non-breaking two-dimensional (2D) surface waves. The algorithm is based on a Eulerian-Lagrangian approach where full, 3D Navier-Stokes equations for the carrier flow induced by a waved water surface are solved in a Eulerian frame, and the trajectories of individual bubbles are simultaneously tracked in a Lagrangian frame, taking into account the impact of the bubbles on the carrier flow. The bubbles diameters are considered in the range from 200 to 400 microns (thus, micro-bubbles), and the effects related to the bubbles deformation and dissolution in water are neglected. The algorithm allows evaluation of the instantaneous as well as statistically stationary, phase-averaged profiles of the carrier-flow turbulence, bubble concentration (void fraction) and void-fraction fluxes for different flow regimes, both with and without wind-induced surface drift. The simulations results show that bubbles are capable of enhancing the carrier-flow turbulence, as compared to the bubble-free flow, and that the vertical water velocity fluctuations are mostly augmented, and increasingly so by larger bubbles. The results also show that the bubbles dynamics are governed by buoyancy, the surrounding fluid acceleration force and the drag force whereas the impact of the lift force remains negligible. Full article

(This article belongs to the Special Issue Iterative Algorithms for Nonlinear Problems: Convergence and Stability 2021-2022)

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13 pages, 843 KiB

Open AccessArticle

A Mathematical Model of Universal Basic Income and Its Numerical Simulations

by Maria Letizia Bertotti

Algorithms 2021, 14(11), 331; https://doi.org/10.3390/a14110331 - 11 Nov 2021

Viewed by 2736

Abstract

In this paper, an elementary mathematical model describing the introduction of a universal basic income in a closed market society is constructed. The model is formulated in terms of a system of nonlinear ordinary differential equations, each of which gives account of how [...] Read more.

In this paper, an elementary mathematical model describing the introduction of a universal basic income in a closed market society is constructed. The model is formulated in terms of a system of nonlinear ordinary differential equations, each of which gives account of how the number of individuals in a certain income class changes in time. Societies ruled by different fiscal systems (with no taxes, with taxation and redistribution, with a welfare system) are considered and the effect of the presence of a basic income in the various cases is analysed by means of numerical simulations. The main findings are that basic income effectively acts as a tool of poverty alleviation: indeed, in its presence the portion of individuals in the poorest classes and economic inequality diminish. Of course, the issue of a universal basic income in the real world is more complex and involves a variety of aspects. The goal here is simply to show how mathematical models can help in forecasting scenarios resulting from one or the other policy. Full article

(This article belongs to the Special Issue Iterative Algorithms for Nonlinear Problems: Convergence and Stability 2021-2022)

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20 pages, 400 KiB

Open AccessArticle

A Unified Formulation of Analytical and Numerical Methods for Solving Linear Fredholm Integral Equations

by Efthimios Providas

Algorithms 2021, 14(10), 293; https://doi.org/10.3390/a14100293 - 10 Oct 2021

Cited by 1 | Viewed by 2563

Abstract

This article is concerned with the construction of approximate analytic solutions to linear Fredholm integral equations of the second kind with general continuous kernels. A unified treatment of some classes of analytical and numerical classical methods, such as the Direct Computational Method (DCM), [...] Read more.

This article is concerned with the construction of approximate analytic solutions to linear Fredholm integral equations of the second kind with general continuous kernels. A unified treatment of some classes of analytical and numerical classical methods, such as the Direct Computational Method (DCM), the Degenerate Kernel Methods (DKM), the Quadrature Methods (QM) and the Projection Methods (PM), is proposed. The problem is formulated as an abstract equation in a Banach space and a solution formula is derived. Then, several approximating schemes are discussed. In all cases, the method yields an explicit, albeit approximate, solution. Several examples are solved to illustrate the performance of the technique. Full article

(This article belongs to the Special Issue Iterative Algorithms for Nonlinear Problems: Convergence and Stability 2021-2022)

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