Axioms

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19 pages, 313 KiB

Open AccessArticle

Superconvergence of Mixed Finite Element Method with Bernstein Polynomials for Stokes Problem

by Lanyin Sun, Siya Wen and Ziwei Dong

Axioms 2025, 14(3), 168; https://doi.org/10.3390/axioms14030168 - 25 Feb 2025

Viewed by 344

In this paper, we employ interpolation and projection methodologies to establish a superconvergence outcome for the Stokes problem, as approximated by the mixed finite element method (FEM) utilizing Bernstein polynomial basis functions. It is widely recognized that the convergence rate of the FEM [...] Read more.

In this paper, we employ interpolation and projection methodologies to establish a superconvergence outcome for the Stokes problem, as approximated by the mixed finite element method (FEM) utilizing Bernstein polynomial basis functions. It is widely recognized that the convergence rate of the FEM in the L²-norm is

O (h^{m + 2})

. However, this paper presents an innovative superconvergence result: specifically, in terms of the L²-norm, the error convergence rate between the mixed finite element approximate solution and the local projection is

O (h^{m + 2})

, with m denoting the order of the Bernstein polynomial basis function. Full article

(This article belongs to the Special Issue Numerical Analysis and Applied Mathematics)

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12 pages, 453 KiB

Open AccessArticle

On the Derivation of a Fast Solver for Nonlinear Systems of Equations Utilizing Frozen Substeps with Applications

by Mingming Liu and Stanford Shateyi

Axioms 2025, 14(2), 77; https://doi.org/10.3390/axioms14020077 - 21 Jan 2025

Viewed by 657

Abstract

In this manuscript, we propose a multi-step framework for solving nonlinear systems of algebraic equations. To improve the solver’s efficiency, the Jacobian matrix is held constant during the second sub-step, while a specialized strategy is applied in the third sub-step to maximize convergence [...] Read more.

In this manuscript, we propose a multi-step framework for solving nonlinear systems of algebraic equations. To improve the solver’s efficiency, the Jacobian matrix is held constant during the second sub-step, while a specialized strategy is applied in the third sub-step to maximize convergence speed without necessitating additional Jacobian evaluations. The proposed method achieves fifth-order convergence for simple roots, with its theoretical convergence established. Finally, computational experiments are conducted to illustrate the performance of the proposed solver in addressing nonlinear equation systems. Full article

(This article belongs to the Special Issue Numerical Analysis and Applied Mathematics)

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18 pages, 332 KiB

Open AccessArticle

Cantelli’s Bounds for Generalized Tail Inequalities

by Nicola Apollonio

Axioms 2025, 14(1), 43; https://doi.org/10.3390/axioms14010043 - 6 Jan 2025

Viewed by 633

Abstract

Let X be a centered random vector in a finite-dimensional real inner product space

E

. For a subset C of the ambient vector space V of

E

and

x, y \in V

, write

x ⪯_{C} y

if [...] Read more.

Let X be a centered random vector in a finite-dimensional real inner product space

E

. For a subset C of the ambient vector space V of

E

and

x, y \in V

, write

x ⪯_{C} y

if

y - x \in C

. If C is a closed convex cone in

E

, then

⪯_{C}

is a preorder on V, whereas if C is a proper cone in

E

, then

⪯_{C}

is actually a partial order on V. In this paper, we give sharp Cantelli-type inequalities for generalized tail probabilities such as

\Pr \{X ⪰_{C} b\}

for

b \in V

. These inequalities are obtained by “scalarizing”

X ⪰_{C} b

via cone duality and then by minimizing the classical univariate Cantelli’s bound over the scalarized inequalities. Three diverse applications to random matrices, tails of linear images of random vectors, and network homophily are also given. Full article

(This article belongs to the Special Issue Numerical Analysis and Applied Mathematics)

15 pages, 475 KiB

Open AccessArticle

On the Construction of a Two-Step Sixth-Order Scheme to Find the Drazin Generalized Inverse

by Keyang Zhang, Fazlollah Soleymani and Stanford Shateyi

Axioms 2025, 14(1), 22; https://doi.org/10.3390/axioms14010022 - 30 Dec 2024

Viewed by 552

Abstract

This study introduces a numerically efficient iterative solver for computing the Drazin generalized inverse, addressing a critical need for high-performance methods in matrix computations. The proposed two-step scheme achieves sixth-order convergence, distinguishing it as a higher-order method that outperforms several existing approaches. A [...] Read more.

This study introduces a numerically efficient iterative solver for computing the Drazin generalized inverse, addressing a critical need for high-performance methods in matrix computations. The proposed two-step scheme achieves sixth-order convergence, distinguishing it as a higher-order method that outperforms several existing approaches. A rigorous convergence analysis is provided, highlighting the importance of selecting an appropriate initial value to ensure robustness. Extensive numerical experiments validate the analytical findings, showcasing the method’s superior speed and efficiency, making it an advancement in iterative solvers for generalized inverses. Full article

(This article belongs to the Special Issue Numerical Analysis and Applied Mathematics)

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12 pages, 2013 KiB

Open AccessArticle

Kink Wave Phenomena in the Nonlinear Partial Differential Equation Representing the Transmission Line Model of Microtubules for Nanoionic Currents

by Safyan Mukhtar, Weaam Alhejaili, Mohammad Alqudah, Ali M. Mahnashi, Rasool Shah and Samir A. El-Tantawy

Axioms 2024, 13(10), 686; https://doi.org/10.3390/axioms13100686 - 2 Oct 2024

Cited by 1 | Viewed by 1143

Abstract

This paper provides several new traveling wave solutions for a nonlinear partial differential equation (PDE) by applying symbolic computation and a new approach, the Riccati–Bernoulli sub-ODE method, in a computer algebra system. Herein, employing the Bäcklund transformation, we solve a nonlinear PDE associated [...] Read more.

This paper provides several new traveling wave solutions for a nonlinear partial differential equation (PDE) by applying symbolic computation and a new approach, the Riccati–Bernoulli sub-ODE method, in a computer algebra system. Herein, employing the Bäcklund transformation, we solve a nonlinear PDE associated with nanobiosciences and biophysics based on the transmission line model of microtubules for nanoionic currents. The equation introduced here in this form is suitable for critical nanoscience concerns like cell signaling and might continue to explain some of the basic cognitive functions in neurons. We employ advanced procedures to replicate the previously detected solitary waves. We offer our solutions in graphical forms, such as 3D and contour plots, using Mathematica. We can generalize the elementary method to other nonlinear equations in physics, requiring only a few steps. Full article

(This article belongs to the Special Issue Numerical Analysis and Applied Mathematics)

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24 pages, 613 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Round-Off Error Suppression by Statistical Averaging

by Andrej Liptaj

Axioms 2024, 13(9), 615; https://doi.org/10.3390/axioms13090615 - 11 Sep 2024

Viewed by 1046

Abstract

Regarding round-off errors as random is often a necessary simplification to describe their behavior. Assuming, in addition, the symmetry of their distributions, we show that one can, in unstable (ill-conditioned) computer calculations, suppress their effect by statistical averaging. For this, one slightly perturbs [...] Read more.

Regarding round-off errors as random is often a necessary simplification to describe their behavior. Assuming, in addition, the symmetry of their distributions, we show that one can, in unstable (ill-conditioned) computer calculations, suppress their effect by statistical averaging. For this, one slightly perturbs the argument of

f (x_{0})

many times and averages the resulting function values. In this text, we forward arguments to support the assumed properties of round-off errors and critically evaluate the validity of the averaging approach in several numerical experiments. Full article

(This article belongs to the Special Issue Numerical Analysis and Applied Mathematics)

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15 pages, 295 KiB

Open AccessArticle

High-Order, Accurate Finite Difference Schemes for Fourth-Order Differential Equations

by Allaberen Ashyralyev and Ibrahim Mohammed Ibrahım

Axioms 2024, 13(2), 90; https://doi.org/10.3390/axioms13020090 - 30 Jan 2024

Cited by 2 | Viewed by 2349

Abstract

This article is devoted to the study of high-order, accurate difference schemes’ numerical solutions of local and non-local problems for ordinary differential equations of the fourth order. Local and non-local problems for ordinary differential equations with constant coefficients can be solved by classical [...] Read more.

This article is devoted to the study of high-order, accurate difference schemes’ numerical solutions of local and non-local problems for ordinary differential equations of the fourth order. Local and non-local problems for ordinary differential equations with constant coefficients can be solved by classical integral transform methods. However, these classical methods can be used simply in the case when the differential equation has constant coefficients. We study fourth-order differential equations with dependent coefficients and their corresponding boundary value problems. Novel compact numerical solutions of high-order, accurate finite difference schemes generated by Taylor’s decomposition on five points have been studied in these problems. Numerical experiments support the theoretical statements for the solution of these difference schemes. Full article

(This article belongs to the Special Issue Numerical Analysis and Applied Mathematics)

24 pages, 2582 KiB

Open AccessArticle

Stability Analysis of a New Fourth-Order Optimal Iterative Scheme for Nonlinear Equations

by Alicia Cordero, José A. Reyes, Juan R. Torregrosa and María P. Vassileva

Axioms 2024, 13(1), 34; https://doi.org/10.3390/axioms13010034 - 31 Dec 2023

Cited by 5 | Viewed by 1907

Abstract

In this paper, a new parametric class of optimal fourth-order iterative methods to estimate the solutions of nonlinear equations is presented. After the convergence analysis, a study of the stability of this class is made using the tools of complex discrete dynamics, allowing [...] Read more.

In this paper, a new parametric class of optimal fourth-order iterative methods to estimate the solutions of nonlinear equations is presented. After the convergence analysis, a study of the stability of this class is made using the tools of complex discrete dynamics, allowing those elements of the class with lower dependence on initial estimations to be selected in order to find a very stable subfamily. Numerical tests indicate that the stable members perform better on quadratic polynomials than the unstable ones when applied to other non-polynomial functions. Moreover, the performance of the best elements of the family are compared with known methods, showing robust and stable behaviour. Full article

(This article belongs to the Special Issue Numerical Analysis and Applied Mathematics)

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16 pages, 3260 KiB

Open AccessArticle

Two Dynamic Remarks on the Chebyshev–Halley Family of Iterative Methods for Solving Nonlinear Equations

by José M. Gutiérrez and Víctor Galilea

Axioms 2023, 12(12), 1114; https://doi.org/10.3390/axioms12121114 - 12 Dec 2023

Cited by 1 | Viewed by 1716

Abstract

The aim of this paper is to delve into the dynamic study of the well-known Chebyshev–Halley family of iterative methods for solving nonlinear equations. Our objectives are twofold: On the one hand, we are interested in characterizing the existence of extraneous attracting fixed [...] Read more.

The aim of this paper is to delve into the dynamic study of the well-known Chebyshev–Halley family of iterative methods for solving nonlinear equations. Our objectives are twofold: On the one hand, we are interested in characterizing the existence of extraneous attracting fixed points when the methods in the family are applied to polynomial equations. On the other hand, we are also interested in studying the free critical points of the methods in the family, as a previous step to determine the existence of attracting cycles. In both cases, we want to identify situations where the methods in the family have bad behavior from the root-finding point of view. Finally, and joining these two studies, we look for polynomials for which there are methods in the family where these two situations happen simultaneously. The rational map obtained by applying a method in the Chebyshev–Halley family to a polynomial has both super-attracting extraneous fixed points and super-attracting cycles different from the roots of the polynomial. Full article

(This article belongs to the Special Issue Numerical Analysis and Applied Mathematics)

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