Mathematics

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15 pages, 1156 KiB

Open AccessFeature PaperArticle

Chebyshev’s Method for Multiple Zeros of Analytic Functions: Convergence, Dynamics and Real-World Applications

by Stoyanka G. Kostadinova and Stoil I. Ivanov

Mathematics 2024, 12(19), 3043; https://doi.org/10.3390/math12193043 - 28 Sep 2024

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This paper deals with the convergence and dynamics of Chebyshev’s method for simple and multiple zeros of analytic functions. We establish a local convergence theorem that provides error estimates and exact domains of initial approximations to guarantee the Q-cubic convergence of the [...] Read more.

This paper deals with the convergence and dynamics of Chebyshev’s method for simple and multiple zeros of analytic functions. We establish a local convergence theorem that provides error estimates and exact domains of initial approximations to guarantee the Q-cubic convergence of the method right from the first iteration. Applications to some real-world problems as well as theoretical and numerical comparison with the famous Halley’s method are also provided. Full article

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

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21 pages, 3083 KiB

Open AccessFeature PaperArticle

The Assessment of the Overall Lifetime Performance Index of Chen Products with Multiple Components

by Shu-Fei Wu and Yu-Lun Huang

Mathematics 2024, 12(13), 2140; https://doi.org/10.3390/math12132140 - 8 Jul 2024

Viewed by 567

Abstract

Process capability indices are widely utilized to evaluate process performance and drive continuous improvements in quality and productivity. Among these indices, the the-larger-the-better lifetime performance index is particularly noteworthy. For products with multiple components, an overall lifetime performance index is used, since it [...] Read more.

Process capability indices are widely utilized to evaluate process performance and drive continuous improvements in quality and productivity. Among these indices, the the-larger-the-better lifetime performance index is particularly noteworthy. For products with multiple components, an overall lifetime performance index is used, since it is a monotonically increasing function of the overall conforming rate and the relationship with each individual lifetime performance index can be determined. For products with the lifetime of the ith component following the Chen distribution, we investigate the maximum likelihood estimator for the overall lifetime performance index and the individual lifetime performance index based on the progressive type I interval censoring sample. Their asymptotic distributions for all lifetime performance indices are also derived. Once the target level for the overall lifetime performance index is specified, the desired level of individual lifetime performance index can be specified. By using the maximum likelihood estimator as the test statistic, a testing procedure to test whether the overall lifetime performance index has reached the target level is developed. The power analysis of the testing procedure is shown with figures, and some findings are summarized. At last, we use one practical example with two components to demonstrate how to implement this testing algorithmic procedure to test if the overall production process has reached the pre-assigned target level. Full article

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

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23 pages, 555 KiB

Open AccessArticle

Partition Entropy as a Measure of Regularity of Music Scales

by Rafael Cubarsi

Mathematics 2024, 12(11), 1658; https://doi.org/10.3390/math12111658 - 25 May 2024

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Abstract

The entropy of the partition generated by an n-tone music scale is proposed to quantify its regularity. The normalized entropy relative to a regular partition and its complementary, here referred to as the bias, allow us to analyze various conditions of similarity [...] Read more.

The entropy of the partition generated by an n-tone music scale is proposed to quantify its regularity. The normalized entropy relative to a regular partition and its complementary, here referred to as the bias, allow us to analyze various conditions of similarity between an arbitrary scale and a regular scale. Interesting particular cases are scales with limited bias because their tones are distributed along specific interval fractions of a regular partition. The most typical case in music concerns partitions associated with well-formed scales generated by a single tone h. These scales are maximal even sets that combine two elementary intervals. Then, the normalized entropy depends on each number of intervals as well as their relative size. When well-formed scales are refined, several nested families stand out with increasing regularity. It is proven that a scale of minimal bias, i.e., with less bias than those with fewer tones, is always a best rational approximation of

l o g_{2} h

. Full article

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

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13 pages, 1882 KiB

Open AccessArticle

Efficient Fourth-Order Weights in Kernel-Type Methods without Increasing the Stencil Size with an Application in a Time-Dependent Fractional PDE Problem

by Tao Liu and Stanford Shateyi

Mathematics 2024, 12(7), 1121; https://doi.org/10.3390/math12071121 - 8 Apr 2024

Cited by 1 | Viewed by 1080

Abstract

An effective strategy to enhance the convergence order of nodal approximations in interpolation or PDE problems is to increase the size of the stencil, albeit at the cost of increased computational burden. In this study, our goal is to improve the convergence orders [...] Read more.

An effective strategy to enhance the convergence order of nodal approximations in interpolation or PDE problems is to increase the size of the stencil, albeit at the cost of increased computational burden. In this study, our goal is to improve the convergence orders for approximating the first and second derivatives of sufficiently differentiable functions using the radial basis function-generated Hermite finite-difference (RBF-HFD) scheme. By utilizing only three equally spaced points in 1D, we are able to boost the convergence rate to four. Extensive tests have been conducted to demonstrate the effectiveness of the proposed theoretical weighting coefficients in solving interpolation and PDE problems. Full article

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

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34 pages, 808 KiB

Open AccessArticle

Explicit P₁ Finite Element Solution of the Maxwell-Wave Equation Coupling Problem with Absorbing b. c.

by Larisa Beilina and Vitoriano Ruas

Mathematics 2024, 12(7), 936; https://doi.org/10.3390/math12070936 - 22 Mar 2024

Viewed by 821

Abstract

In this paper, we address the approximation of the coupling problem for the wave equation and Maxwell’s equations of electromagnetism in the time domain in terms of electric field by means of a nodal linear finite element discretization in space, combined with a [...] Read more.

In this paper, we address the approximation of the coupling problem for the wave equation and Maxwell’s equations of electromagnetism in the time domain in terms of electric field by means of a nodal linear finite element discretization in space, combined with a classical explicit finite difference scheme for time discretization. Our study applies to a particular case where the dielectric permittivity has a constant value outside a subdomain, whose closure does not intersect the boundary of the domain where the problem is defined. Inside this subdomain, Maxwell’s equations hold. Outside this subdomain, the wave equation holds, which may correspond to Maxwell’s equations with a constant permittivity under certain conditions. We consider as a model the case of first-order absorbing boundary conditions. First-order error estimates are proven in the sense of two norms involving first-order time and space derivatives under reasonable assumptions, among which lies a CFL condition for hyperbolic equations. The theoretical estimates are validated by numerical computations, which also show that the scheme is globally of the second order in the maximum norm in time and in the least-squares norm in space. Full article

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

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16 pages, 312 KiB

Open AccessArticle

Collocation Technique Based on Chebyshev Polynomials to Solve Emden–Fowler-Type Singular Boundary Value Problems with Derivative Dependence

by Shabanam Kumari, Arvind Kumar Singh and Utsav Gupta

Mathematics 2024, 12(4), 592; https://doi.org/10.3390/math12040592 - 17 Feb 2024

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Abstract

In this work, an innovative technique is presented to solve Emden–Fowler-type singular boundary value problems (SBVPs) with derivative dependence. These types of problems have significant applications in applied mathematics and astrophysics. Initially, the differential equation is transformed into a Fredholm integral equation, which [...] Read more.

In this work, an innovative technique is presented to solve Emden–Fowler-type singular boundary value problems (SBVPs) with derivative dependence. These types of problems have significant applications in applied mathematics and astrophysics. Initially, the differential equation is transformed into a Fredholm integral equation, which is then converted into a system of nonlinear equations using the collocation technique based on Chebyshev polynomials. Subsequently, an iterative numerical approach, such as Newton’s method, is employed on the system of nonlinear equations to obtain an approximate solution. Error analysis is included to assess the accuracy of the obtained solutions and provide insights into the reliability of the numerical results. Furthermore, we graphically compare the residual errors of the current method to the previously established method for various examples. Full article

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

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19 pages, 1832 KiB

Open AccessArticle

The Computational Testing Procedure for the Comprehensive Lifetime Performance Index of Burr XII Products in Multiple Production Lines

by Shu-Fei Wu, Pei-Hsuan Kuo and Wen-Shuenn Deng

Mathematics 2024, 12(4), 584; https://doi.org/10.3390/math12040584 - 16 Feb 2024

Cited by 1 | Viewed by 732

Abstract

Extending from a single production line to multiple production lines, a comprehensive life performance index is proposed for evaluating the quality of lifetime products. The connection between the comprehensive lifetime performance index and the individual lifetime performance index is explored. For products with [...] Read more.

Extending from a single production line to multiple production lines, a comprehensive life performance index is proposed for evaluating the quality of lifetime products. The connection between the comprehensive lifetime performance index and the individual lifetime performance index is explored. For products with a lifetime following Burr XII distribution for the ith production line, the maximum likelihood estimation method and the corresponding asymptotic distribution for all lifetime performance indices are derived. Checking whether the comprehensive lifetime performance index has achieved the target value is essentially the same as testing whether each individual lifetime performance index has reached its corresponding target value. A testing procedure is proposed for a given significance level using the maximum likelihood estimator as the test statistic, and the power analysis is presented through graphical representations. For the power analysis, the impacts of sample size, the number of inspection intervals, the removal probability, the level of significance, and the number of production lines on the test power are analyzed, and the results show that there is a monotonic relationship between the test power and the above five impact factors. To illustrate how to apply the proposed testing procedure, we give one practical example with two production lines to test whether the comprehensive production process is capable. Full article

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

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14 pages, 339 KiB

Open AccessArticle

Convergence of the Euler Method for Impulsive Neutral Delay Differential Equations

by Yang Sun, Gui-Lai Zhang, Zhi-Wei Wang and Tao Liu

Mathematics 2023, 11(22), 4684; https://doi.org/10.3390/math11224684 - 17 Nov 2023

Cited by 1 | Viewed by 961

Abstract

In this paper, we are concerned with a fixed stepsize Euler method for a class of linear impulsive neutral delay differential equations. By taking the partition nodes for the Euler scheme and employing the linear interpolation, we strictly prove the method is convergent [...] Read more.

In this paper, we are concerned with a fixed stepsize Euler method for a class of linear impulsive neutral delay differential equations. By taking the partition nodes for the Euler scheme and employing the linear interpolation, we strictly prove the method is convergent of order one. Two examples illustrating the efficiency results are also presented. Full article

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

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13 pages, 819 KiB

Open AccessArticle

Explicit Symplectic Runge–Kutta–Nyström Methods Based on Roots of Shifted Legendre Polynomial

by Jun Zhang, Jingjing Zhang and Shangyou Zhang

Mathematics 2023, 11(20), 4291; https://doi.org/10.3390/math11204291 - 15 Oct 2023

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Abstract

To date, all explicit symplectic Runge–Kutta–Nyström methods of order five or above are derived by numerical solutions of order condition equations and symplectic condition. In this paper, we derive 124 sets of seven-stage fifth-order explicit symplectic Runge–Kutta–Nyström methods with closed-form coefficients in the [...] Read more.

To date, all explicit symplectic Runge–Kutta–Nyström methods of order five or above are derived by numerical solutions of order condition equations and symplectic condition. In this paper, we derive 124 sets of seven-stage fifth-order explicit symplectic Runge–Kutta–Nyström methods with closed-form coefficients in the Butcher tableau using the roots of a degree-3 shifted Legendre polynomial. One method is analyzed and its P-stable interval is derived. Numerical tests on the two newly discovered methods are performed, showing their long-time stability and large step size stability over some existing methods. Full article

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

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18 pages, 753 KiB

Open AccessArticle

Enhancing the Convergence Order from p to p + 3 in Iterative Methods for Solving Nonlinear Systems of Equations without the Use of Jacobian Matrices

by Alicia Cordero, Miguel A. Leonardo-Sepúlveda, Juan R. Torregrosa and María P. Vassileva

Mathematics 2023, 11(20), 4238; https://doi.org/10.3390/math11204238 - 10 Oct 2023

Viewed by 1260

Abstract

In this paper, we present an innovative technique that improves the convergence order of iterative schemes that do not require the evaluation of Jacobian matrices. As far as we know, this is the first technique that allows us the achievement of an increase, [...] Read more.

In this paper, we present an innovative technique that improves the convergence order of iterative schemes that do not require the evaluation of Jacobian matrices. As far as we know, this is the first technique that allows us the achievement of an increase, from p to

p + 3

units, in the order of convergence. This is constructed from any Jacobian-free scheme of order p. We conduct comprehensive numerical tests first in academical examples to validate the theoretical results, showing the efficiency and effectiveness of the new Jacobian-free schemes. Then, we apply them on the non-differentiable partial differential equations that models the nutrient diffusion in a biological substrate. Full article

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

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13 pages, 318 KiB

Open AccessArticle

A New R-Function to Estimate the PDF of the Product of Two Uncorrelated Normal Variables

by Antonio Seijas-Macias, Amílcar Oliveira and Teresa A. Oliveira

Mathematics 2023, 11(16), 3515; https://doi.org/10.3390/math11163515 - 14 Aug 2023

Viewed by 1168

Abstract

This paper analyses the implementation of a procedure using the software R to calculate the Probability Density Function (PDF) of the product of two uncorrelated Normally Distributed Random Variables. The problem of estimating the distribution of the product of two random variables has [...] Read more.

This paper analyses the implementation of a procedure using the software R to calculate the Probability Density Function (PDF) of the product of two uncorrelated Normally Distributed Random Variables. The problem of estimating the distribution of the product of two random variables has been solved for some particular cases, but there is no unique expression for all possible situations. In our study, we chose Rohatgi’s theorem as a basis for approximating the product of two uncorrelated Normally Distributed Random Variables. The numerical approximation of the product PDF was calculated using a function that we implemented in R. Several numerical examples show that the approximations obtained in R fit the theoretical values of the product distributions. The results obtained with our R function are very positive when we compare them with the Monte Carlo Simulation of the product of the two variables. Full article

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

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