Research

13 pages, 1755 KiB

Open AccessArticle

Derivative-Free Iterative Schemes for Multiple Roots of Nonlinear Functions

by Himani Arora, Alicia Cordero, Juan R. Torregrosa, Ramandeep Behl and Sattam Alharbi

Mathematics 2022, 10(9), 1530; https://doi.org/10.3390/math10091530 - 03 May 2022

Cited by 3 | Viewed by 1418

The construction of derivative-free iterative methods for approximating multiple roots of a nonlinear equation is a relatively new line of research. This paper presents a novel family of one-parameter second-order techniques. Our schemes are free from derivatives and have been designed to find [...] Read more.

The construction of derivative-free iterative methods for approximating multiple roots of a nonlinear equation is a relatively new line of research. This paper presents a novel family of one-parameter second-order techniques. Our schemes are free from derivatives and have been designed to find multiple roots (

m \geq 2

). The new techniques involve the weight function approach. The convergence analysis for the new family is presented in the main theorem. In addition, some special cases of the new class are discussed. We also illustrate the applicability of our methods on van der Waals, Planck’s radiation, root clustering, and eigenvalue problems. We also contrast them with the known methods. Finally, the dynamical study of iterative schemes also provides a good overview of their stability. Full article

(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis)

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17 pages, 836 KiB

Open AccessArticle

A Derivative Free Fourth-Order Optimal Scheme for Applied Science Problems

by Ramandeep Behl

Mathematics 2022, 10(9), 1372; https://doi.org/10.3390/math10091372 - 20 Apr 2022

Cited by 3 | Viewed by 1227

Abstract

We suggest a new and cost-effective iterative scheme for nonlinear equations. The main features of the presented scheme are that it does not involve any derivative in the structure, achieves an optimal convergence of fourth-order factors, has more flexibility for obtaining new members, [...] Read more.

We suggest a new and cost-effective iterative scheme for nonlinear equations. The main features of the presented scheme are that it does not involve any derivative in the structure, achieves an optimal convergence of fourth-order factors, has more flexibility for obtaining new members, and is two-point, cost-effective, more stable and yields better numerical results. The derivation of our scheme is based on the weight function technique. The convergence order is studied in three main theorems. We have demonstrated the applicability of our methods on four numerical problems. Out of them, two are real-life cases, while the third one is a root clustering problem and the fourth one is an academic problem. The obtained numerical results illustrate preferable outcomes as compared to the existing ones in terms of absolute residual errors, CPU timing, approximated zeros and absolute error difference between two consecutive iterations. Full article

(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis)

15 pages, 308 KiB

Open AccessArticle

Using Matrix Eigenvalues to Construct an Iterative Method with the Highest Possible Efficiency Index Two

by Malik Zaka Ullah, Vali Torkashvand, Stanford Shateyi and Mir Asma

Mathematics 2022, 10(9), 1370; https://doi.org/10.3390/math10091370 - 20 Apr 2022

Viewed by 1291

Abstract

In this paper, we first derive a family of iterative schemes with fourth order. A weight function is used to maintain its optimality. Then, we transform it into methods with several self-accelerating parameters to reach the highest possible convergence rate 8. For this [...] Read more.

In this paper, we first derive a family of iterative schemes with fourth order. A weight function is used to maintain its optimality. Then, we transform it into methods with several self-accelerating parameters to reach the highest possible convergence rate 8. For this aim, we employ the property of the eigenvalues of the matrices and the technique with memory. Solving several nonlinear test equations shows that the proposed variants have a computational efficiency index of two (maximum amount possible) in practice. Full article

(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis)

18 pages, 330 KiB

Open AccessArticle

Common Attractive Point Results for Two Generalized Nonexpansive Mappings in Uniformly Convex Banach Spaces

by Chadarat Thongphaen, Warunun Inthakon, Suthep Suantai and Narawadee Phudolsitthiphat

Mathematics 2022, 10(8), 1275; https://doi.org/10.3390/math10081275 - 12 Apr 2022

Cited by 1 | Viewed by 1147

Abstract

In this work, we study some basic properties of the set of common attractive points and prove strong convergence results for common attractive points of two generalized nonexpansive mappings in a uniformly convex Banach space. As a consequence, we obtain a common fixed [...] Read more.

In this work, we study some basic properties of the set of common attractive points and prove strong convergence results for common attractive points of two generalized nonexpansive mappings in a uniformly convex Banach space. As a consequence, we obtain a common fixed point result of such mappings and apply it to solving the convex minimization problem. Finally, numerical experiments are given to support our results. Full article

(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis)

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22 pages, 1282 KiB

Open AccessArticle

A Novel n-Point Newton-Type Root-Finding Method of High Computational Efficiency

by Xiaofeng Wang

Mathematics 2022, 10(7), 1144; https://doi.org/10.3390/math10071144 - 02 Apr 2022

Cited by 2 | Viewed by 1029

Abstract

A novel Newton-type n-point iterative method with memory is proposed for solving nonlinear equations, which is constructed by the Hermite interpolation. The proposed iterative method with memory reaches the order [...] Read more.

A novel Newton-type n-point iterative method with memory is proposed for solving nonlinear equations, which is constructed by the Hermite interpolation. The proposed iterative method with memory reaches the order

(2^{n} + 2^{n - 1} - 1 + \sqrt{2^{2 n + 1} + 2^{2 n - 2} + 2^{n} + 1}) / 2

by using n variable parameters. The computational efficiency of the proposed method is higher than that of the existing Newton-type methods with and without memory. To observe the stability of the proposed method, some complex functions are considered under basins of attraction. Basins of attraction show that the proposed method has better stability and requires a lesser number of iterations than various well-known methods. The numerical results support the theoretical results. Full article

(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis)

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23 pages, 1798 KiB

Open AccessArticle

On Numerical Analysis of Bio-Ethanol Production Model with the Effect of Recycling and Death Rates under Fractal Fractional Operators with Three Different Kernels

by Rubayyi T. Alqahtani, Shabir Ahmad and Ali Akgül

Mathematics 2022, 10(7), 1102; https://doi.org/10.3390/math10071102 - 29 Mar 2022

Cited by 5 | Viewed by 1373

Abstract

The main metabolism of yeasts produces bioethanol. Bioethanol, which is produced from biomass and bioenergy crops, has been promoted as one of the most viable alternatives to fossil fuels. The following reaction represents all of the knowledge we have regarding intracellular reactions and [...] Read more.

The main metabolism of yeasts produces bioethanol. Bioethanol, which is produced from biomass and bioenergy crops, has been promoted as one of the most viable alternatives to fossil fuels. The following reaction represents all of the knowledge we have regarding intracellular reactions and their regulatory mechanisms:

b i o m a s s + s u b s t r a t e s \to e t h a n o l + b i o m a s s (m o r e c e l l s) .

Atangana has suggested new operators based on a combination of fractional and fractal calculus. Fractal-fractional operators (FFOs) have frequently been utilized to investigate the dynamics of a physical problem. In this paper, FFOs are used to investigate a nonlinear mathematical model for ethanol production with three different kernels. Famous fixed point results are employed to show the existence and uniqueness of the solution of the FFO ethanol model under the Mittag–Leffler kernel. The concept of non-linear analysis is utilized to demonstrate the model’s Ulam–Hyres stability. The Adams—Bashforth numerical technique, which is based on the Lagrangian interpolation method, is utilized to find the solution of the model under fractal-fractional operators with three different kernels. The numerical results are simulated with MATLAB-17 for several sets of fractional orders and fractal dimensions to show the relationship between components of ethanol production under new operators in various senses. Full article

(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis)

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21 pages, 500 KiB

Open AccessArticle

Dimensionality of Iterative Methods: The Adimensional Scale Invariant Steffensen (ASIS) Method

by Vicente F. Candela

Mathematics 2022, 10(6), 911; https://doi.org/10.3390/math10060911 - 12 Mar 2022

Viewed by 1462

Abstract

The dimensionality of parameters and variables is a fundamental issue in physics but is mostly ignored from a mathematical point of view. Difficulties arising from dimensional inconsistency are overcome by scaling analysis and, often, both concepts, dimensionality and scaling, are confused. In the [...] Read more.

The dimensionality of parameters and variables is a fundamental issue in physics but is mostly ignored from a mathematical point of view. Difficulties arising from dimensional inconsistency are overcome by scaling analysis and, often, both concepts, dimensionality and scaling, are confused. In the particular case of iterative methods for solving nonlinear equations, dimensionality and scaling affect their robustness: While some classical methods, such as Newton’s, are adimensional and scale independent, some other iterations such as Steffensen’s are not; their convergence depends on scaling, and their evaluation needs a dimensional congruence. In this paper, we introduce the concept of an adimensional form of a function in order to study the behavior of iterative methods, thus correcting, if possible, some pathological features. From this adimensional form, we will devise an adimensional and scale invariant method based on Steffensen’s, which we will call the ASIS method. Full article

(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis)

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9 pages, 739 KiB

Open AccessArticle

Revisiting the Copula-Based Trading Method Using the Laplace Marginal Distribution Function

by Tayyebeh Nadaf, Taher Lotfi and Stanford Shateyi

Mathematics 2022, 10(5), 783; https://doi.org/10.3390/math10050783 - 01 Mar 2022

Cited by 3 | Viewed by 2156

Abstract

Pairs trading under the copula approach is revisited in this paper. It is well known that financial returns arising from indices in markets may not follow the features of normal distribution and may exhibit asymmetry or fatter tails, in particular. Due to this, [...] Read more.

Pairs trading under the copula approach is revisited in this paper. It is well known that financial returns arising from indices in markets may not follow the features of normal distribution and may exhibit asymmetry or fatter tails, in particular. Due to this, the Laplace distribution is employed in this work to fit the marginal distribution function, which will then be employed in a copula function. In fact, a multivariate copula function is constructed on two indices (based on the Laplace marginal distribution), enabling us to obtain the associated probabilities required for the process of pairs trade and creating an efficient tool for trading. Full article

(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis)

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27 pages, 736 KiB

Open AccessArticle

Resolution of Initial Value Problems of Ordinary Differential Equations Systems

by Josep Vicent Arnau i Córdoba and Màrius Josep Fullana i Alfonso

Mathematics 2022, 10(4), 593; https://doi.org/10.3390/math10040593 - 14 Feb 2022

Viewed by 1879

Abstract

In this work, we present some techniques applicable to Initial Value Problems when solving a System of Ordinary Differential Equations (ODE). Such techniques should be used when applying adaptive step-size numerical methods. In our case, a Runge-Kutta-Fehlberg algorithm (RKF45) has been employed, but [...] Read more.

In this work, we present some techniques applicable to Initial Value Problems when solving a System of Ordinary Differential Equations (ODE). Such techniques should be used when applying adaptive step-size numerical methods. In our case, a Runge-Kutta-Fehlberg algorithm (RKF45) has been employed, but the procedure presented here can also be applied to other adaptive methods, such as N-body problems, as AP3M or similar ones. By doing so, catastrophic cancellations were eliminated. A mathematical optimization was carried out by introducing the objective function in the ODE System (ODES). Resizing of local errors was also utilised in order to adress the problem. This resize implies the use of certain variables to adjust the integration step while the other variables are used as parameters to determine the coefficients of the ODE system. This resize was executed by using the asymptotic solution of this system. The change of variables is necessary to guarantee the stability of the integration. Therefore, the linearization of the ODES is possible and can be used as a powerful control test. All these tools are applied to a physical problem. The example we present here is the effective numerical resolution of Lemaitre-Tolman-Bondi space-time solutions of Einstein Equations. Full article

(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis)

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13 pages, 659 KiB

Open AccessArticle

Study of Dynamics of a COVID-19 Model for Saudi Arabia with Vaccination Rate, Saturated Treatment Function and Saturated Incidence Rate

by Rubayyi T. Alqahtani and Abdelhamid Ajbar

Mathematics 2021, 9(23), 3134; https://doi.org/10.3390/math9233134 - 05 Dec 2021

Cited by 5 | Viewed by 2645

Abstract

This paper proposes, validates and analyzes the dynamics of the susceptible exposed infectious recovered (SEIR) model for the propagation of COVID-19 in Saudi Arabia, which recorded the largest number of cases in the Arab world. The model incorporates a saturated incidence rate, a [...] Read more.

This paper proposes, validates and analyzes the dynamics of the susceptible exposed infectious recovered (SEIR) model for the propagation of COVID-19 in Saudi Arabia, which recorded the largest number of cases in the Arab world. The model incorporates a saturated incidence rate, a constant vaccination rate and a nonlinear treatment function. The rate of treatment is assumed to be proportional to the number of infected persons when this number is low and reaches a fixed value for large number of infected individuals. The expression of the basic reproduction number is derived, and the model basic stability properties are studied. We show that when the basic reproduction number is less than one the model can predict both a Hopf and backward bifurcations. Simulations are also provided to fit the model to COVID-19 data in Saudi Arabia and to study the effects of the parameters of the treatment function and vaccination rate on disease control. Full article

(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis)

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7 pages, 295 KiB

Open AccessArticle

A New Inversion-Free Iterative Scheme to Compute Maximal and Minimal Solutions of a Nonlinear Matrix Equation

by Malik Zaka Ullah

Mathematics 2021, 9(23), 2994; https://doi.org/10.3390/math9232994 - 23 Nov 2021

Cited by 1 | Viewed by 1297

Abstract

The goal of this article is to investigate a new solver in the form of an iterative method to solve

X + A^{*} X^{- 1} A = I

as an important nonlinear matrix equation (NME), where

A, X, I

[...] Read more.

The goal of this article is to investigate a new solver in the form of an iterative method to solve

X + A^{*} X^{- 1} A = I

as an important nonlinear matrix equation (NME), where

A, X, I

are appropriate matrices. The minimal and maximal solutions of this NME are discussed as Hermitian positive definite (HPD) matrices. The convergence of the scheme is given. Several numerical tests are also provided to support the theoretical discussions. Full article

(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis)

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21 pages, 2492 KiB

Open AccessArticle

Mathematical Analysis of Biodegradation Model under Nonlocal Operator in Caputo Sense

by Rubayyi T. Alqahtani, Shabir Ahmad and Ali Akgül

Mathematics 2021, 9(21), 2787; https://doi.org/10.3390/math9212787 - 03 Nov 2021

Cited by 13 | Viewed by 1309

Abstract

To lower the concentration of organic pollutants in the effluent stream, wastewater must be treated before being discharged into the environment. The question of whether wastewater treatment facilities can successfully reduce the concentration of micropollutants found in their influent streams is becoming increasingly [...] Read more.

To lower the concentration of organic pollutants in the effluent stream, wastewater must be treated before being discharged into the environment. The question of whether wastewater treatment facilities can successfully reduce the concentration of micropollutants found in their influent streams is becoming increasingly pressing. The removal of micropollutants in treatment plants is investigated using a model that incorporates biodegradation and sorption as the key processes of micropollutant removal. This article provides the mathematical analysis of the wastewater model that describes the removal of micropollutant in treatment plants under a non-local operator in Caputo sense. The positivity of the solution is presented for the Caputo fractional model. The steady state’s solution of model and their stability is presented. The fixed point theorems of Leray–Schauder and Banach are used to deduce results regarding the existence of the solution of the model. Ulam–Hyers (UH) types of stabilities are presented via functional analysis. The fractional Euler method is used to find the numerical results of the proposed model. The numerical results are illustrated via graphs to show the effects of recycle ratio and the impact of fractional order on the evolution of the model. Full article

(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis)

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13 pages, 2849 KiB

Open AccessArticle

Memorizing Schröder’s Method as an Efficient Strategy for Estimating Roots of Unknown Multiplicity

by Alicia Cordero, Beny Neta and Juan R. Torregrosa

Mathematics 2021, 9(20), 2570; https://doi.org/10.3390/math9202570 - 13 Oct 2021

Cited by 8 | Viewed by 1424

Abstract

In this paper, we propose, to the best of our knowledge, the first iterative scheme with memory for finding roots whose multiplicity is unknown existing in the literature. It improves the efficiency of a similar procedure without memory due to Schröder and can [...] Read more.

In this paper, we propose, to the best of our knowledge, the first iterative scheme with memory for finding roots whose multiplicity is unknown existing in the literature. It improves the efficiency of a similar procedure without memory due to Schröder and can be considered as a seed to generate higher order methods with similar characteristics. Once its order of convergence is studied, its stability is analyzed showing its good properties, and it is compared numerically in terms of their basins of attraction with similar schemes without memory for finding multiple roots. Full article

(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis)

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15 pages, 2922 KiB

Open AccessArticle

Computational Geometry of Period-3 Hyperbolic Components in the Mandelbrot Set

by Young-Hee Geum and Young-Ik Kim

Mathematics 2021, 9(19), 2519; https://doi.org/10.3390/math9192519 - 07 Oct 2021

Viewed by 1538

Abstract

A parametric theoretical boundary equation of a period-3 hyperbolic component in the Mandelbrot set is established from a perspective of Euclidean plane geometry. We not only calculate the interior area, perimeter and curvature of the boundary line but also derive some relevant geometrical [...] Read more.

A parametric theoretical boundary equation of a period-3 hyperbolic component in the Mandelbrot set is established from a perspective of Euclidean plane geometry. We not only calculate the interior area, perimeter and curvature of the boundary line but also derive some relevant geometrical properties. The budding point of the period-

3 k

component, which is born on the boundary of the period-3 component, and its relevant period-

3 k

points are theoretically obtained by means of Cardano’s formula for the cubic equation. In addition, computational results are presented in tables and figures to support the theoretical background of this paper. Full article

(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis)

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22 pages, 4681 KiB

Open AccessFeature PaperArticle

A Graphic Method for Detecting Multiple Roots Based on Self-Maps of the Hopf Fibration and Nullity Tolerances

by José Ignacio Extreminana-Aldana, José Manuel Gutiérrez-Jiménez, Luis Javier Hernández-Paricio and María Teresa Rivas-Rodríguéz

Mathematics 2021, 9(16), 1914; https://doi.org/10.3390/math9161914 - 11 Aug 2021

Cited by 1 | Viewed by 1302

Abstract

The aim of this paper is to study, from a topological and geometrical point of view, the iteration map obtained by the application of iterative methods (Newton or relaxed Newton’s method) to a polynomial equation. In fact, we present a collection of algorithms [...] Read more.

The aim of this paper is to study, from a topological and geometrical point of view, the iteration map obtained by the application of iterative methods (Newton or relaxed Newton’s method) to a polynomial equation. In fact, we present a collection of algorithms that avoid the problem of overflows caused by denominators close to zero and the problem of indetermination which appears when simultaneously the numerator and denominator are equal to zero. This is solved by working with homogeneous coordinates and the iteration of self-maps of the Hopf fibration. As an application, our algorithms can be used to check the existence of multiple roots for polynomial equations as well as to give a graphical representation of the union of the basins of attraction of simple roots and the union of the basins of multiple roots. Finally, we would like to highlight that all the algorithms developed in this work have been implemented in Julia, a programming language with increasing use in the mathematical community. Full article

(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis)

23 pages, 672 KiB

Open AccessArticle

MHD Laminar Boundary Layer Flow of a Jeffrey Fluid Past a Vertical Plate Influenced by Viscous Dissipation and a Heat Source/Sink

by Hillary Muzara and Stanford Shateyi

Mathematics 2021, 9(16), 1896; https://doi.org/10.3390/math9161896 - 09 Aug 2021

Cited by 7 | Viewed by 1925

Abstract

This study investigates the effects of viscous dissipation and a heat source or sink on the magneto-hydrodynamic laminar boundary layer flow of a Jeffrey fluid past a vertical plate. The governing boundary layer non-linear partial differential equations are reduced to non-linear ordinary differential [...] Read more.

This study investigates the effects of viscous dissipation and a heat source or sink on the magneto-hydrodynamic laminar boundary layer flow of a Jeffrey fluid past a vertical plate. The governing boundary layer non-linear partial differential equations are reduced to non-linear ordinary differential equations using suitable similarity transformations. The resulting system of dimensionless differential equations is then solved numerically using the bivariate spectral quasi-linearisation method. The effects of some physical parameters that include the Schmidt number, Eckert number, radiation parameter, magnetic field parameter, heat generation parameter, and the ratio of relaxation to retardation times on the velocity, temperature, and concentration profiles are presented graphically. Additionally, the influence of some physical parameters on the skin friction coefficient, local Nusselt number, and the local Sherwood number are displayed in tabular form. Full article

(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis)

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14 pages, 396 KiB

Open AccessArticle

Derivative-Free King’s Scheme for Multiple Zeros of Nonlinear Functions

by Ramandeep Behl, Sonia Bhalla, Eulalia Martínez and Majed Aali Alsulami

Mathematics 2021, 9(11), 1242; https://doi.org/10.3390/math9111242 - 28 May 2021

Cited by 10 | Viewed by 1330

Abstract

There is no doubt that the fourth-order King’s family is one of the important ones among its counterparts. However, it has two major problems: the first one is the calculation of the first-order derivative; secondly, it has a linear order of convergence in [...] Read more.

There is no doubt that the fourth-order King’s family is one of the important ones among its counterparts. However, it has two major problems: the first one is the calculation of the first-order derivative; secondly, it has a linear order of convergence in the case of multiple roots. In order to improve these complications, we suggested a new King’s family of iterative methods. The main features of our scheme are the optimal convergence order, being free from derivatives, and working for multiple roots

(m \geq 2)

. In addition, we proposed a main theorem that illustrated the fourth order of convergence. It also satisfied the optimal Kung–Traub conjecture of iterative methods without memory. We compared our scheme with the latest iterative methods of the same order of convergence on several real-life problems. In accordance with the computational results, we concluded that our method showed superior behavior compared to the existing methods. Full article

(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis)

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