Applications of Iterative Methods in Solving Nonlinear Equations

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "General Mathematics, Analysis".

Deadline for manuscript submissions: closed (30 November 2023) | Viewed by 17180

Special Issue Editors


E-Mail Website
Guest Editor
Institute Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera, s/n, 46022 Valencia, Spain
Interests: numerical analysis; mathematical modelling; numerical modeling
Special Issues, Collections and Topics in MDPI journals

E-Mail Website
Guest Editor
Instituto Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera, s/n, 46022 Valencia, Spain
Interests: iterative methods; memory schemes; nonlinear equations and systems; dynamical analysis
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Numerical analysis is a research area of applied mathematics that has experienced an important boom in recent decades. A common problem in science, engineering and economics disciplines lies in the requirement of the solution to a nonlinear equation or system of equations. We resort to approximate solutions where analytical solutions do not reach. One of the strategies consists of the use of iterative methods for solving equations and systems of nonlinear equations.

The design and analysis of iterative methods for solving nonlinear problems is the subject of this Special Issue, as are their potential applications. In this sense, research on memoryless and memory methods, methods to find multiple roots, methods to simultaneously obtain all the solutions of a problem or methods using fractional derivatives, among others, are welcome.

Dr. Francisco I. Chicharro
Dr. Neus Garrido
Dr. Paula Triguero-Navarro
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Fractal and Fractional is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2700 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • iterative methods
  • stability theory
  • methods with memory
  • simultaneous roots
  • multiple roots
  • fractional derivatives
  • fractal dimension
  • nonlinear dynamics
  • mathematical modelling

Related Special Issue

Published Papers (12 papers)

Order results
Result details
Select all
Export citation of selected articles as:

Research

14 pages, 1502 KiB  
Article
Extension of the Optimal Auxiliary Function Method to Solve the System of a Fractional-Order Whitham–Broer–Kaup Equation
by Zainab Alsheekhhussain, Khaled Moaddy, Rasool Shah, Saleh Alshammari, Mohammad Alshammari, M. Mossa Al-Sawalha and Aisha Abdullah Alderremy
Fractal Fract. 2024, 8(1), 1; https://doi.org/10.3390/fractalfract8010001 - 19 Dec 2023
Cited by 4 | Viewed by 1006
Abstract
In this paper, we introduce and implement the optimal auxiliary function method to solve a system of fractional-order Whitham–Broer–Kaup equations, a class of nonlinear partial differential equations with broad applications in mathematical physics. This method provides a systematic and efficient approach to finding [...] Read more.
In this paper, we introduce and implement the optimal auxiliary function method to solve a system of fractional-order Whitham–Broer–Kaup equations, a class of nonlinear partial differential equations with broad applications in mathematical physics. This method provides a systematic and efficient approach to finding accurate solutions for complex systems of fractional-order equations. We give a full analysis using tables and figures to demonstrate the reliability and accuracy of our approach. We confirm the effectiveness of our suggested method in solving the considered equations using numerical simulations and comparisons, emphasizing its potential for applications in a variety of scientific and engineering areas. Full article
(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)
Show Figures

Figure 1

19 pages, 2269 KiB  
Article
A Comparative Analysis of Laplace Residual Power Series and a New Iteration Method for Fitzhugh-Nagumo Equation in the Caputo Operator Framework
by Azzh Saad Alshehry
Fractal Fract. 2023, 7(12), 867; https://doi.org/10.3390/fractalfract7120867 - 06 Dec 2023
Viewed by 853
Abstract
In this paper, I utilize the Laplace residual power series method (LRPSM) along with a novel iteration technique to investigate the Fitzhugh-Nagumo equation within the framework of the Caputo operator. The Fitzhugh-Nagumo equation is a fundamental model for describing excitable systems, playing a [...] Read more.
In this paper, I utilize the Laplace residual power series method (LRPSM) along with a novel iteration technique to investigate the Fitzhugh-Nagumo equation within the framework of the Caputo operator. The Fitzhugh-Nagumo equation is a fundamental model for describing excitable systems, playing a crucial role in understanding various physiological and biological phenomena. The Caputo operator extends the conventional derivative to handle non-local and non-integer-order differential equations, making it a potent tool for modeling complex processes. Our study involves transforming the Fitzhugh-Nagumo equation into its Laplace domain representation, applying the LRPSM to derive a series solution. We then introduce a novel iteration technique to enhance the solution’s convergence properties, enabling more accurate and efficient computations. This approach offers a systematic methodology for solving the Fitzhugh-Nagumo equation with the Caputo operator, providing deeper insights into excitable system dynamics. Numerical examples and comparisons with existing methods demonstrate the accuracy and efficiency of the LRPSM with the new iteration technique, showcasing its potential for solving diverse differential equations involving the Caputo operator and advancing mathematical modeling in various scientific and engineering domains. Full article
(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)
Show Figures

Figure 1

17 pages, 2267 KiB  
Article
Fractal Complexity of a New Biparametric Family of Fourth Optimal Order Based on the Ermakov–Kalitkin Scheme
by Alicia Cordero, Renso V. Rojas-Hiciano, Juan R. Torregrosa and Maria P. Vassileva
Fractal Fract. 2023, 7(6), 459; https://doi.org/10.3390/fractalfract7060459 - 03 Jun 2023
Viewed by 834
Abstract
In this paper, we generalize the scheme proposed by Ermakov and Kalitkin and present a class of two-parameter fourth-order optimal methods, which we call Ermakov’s Hyperfamily. It is a substantial improvement of the classical Newton’s method because it optimizes one that extends the [...] Read more.
In this paper, we generalize the scheme proposed by Ermakov and Kalitkin and present a class of two-parameter fourth-order optimal methods, which we call Ermakov’s Hyperfamily. It is a substantial improvement of the classical Newton’s method because it optimizes one that extends the regions of convergence and is very stable. Another novelty is that it is a class containing as particular cases some classical methods, such as King’s family. From this class, we generate a new uniparametric family, which we call the KLAM, containing the classical Ostrowski and Chun, whose efficiency, stability, and optimality has been proven but also new methods that in many cases outperform these mentioned, as we prove. We demonstrate that it is of a fourth order of convergence, as well as being computationally efficienct. A dynamical study is performed allowing us to choose methods with good stability properties and to avoid chaotic behavior, implicit in the fractal structure defined by the Julia set in the related dynamic planes. Some numerical tests are presented to confirm the theoretical results and to compare the proposed methods with other known methods. Full article
(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)
Show Figures

Figure 1

16 pages, 777 KiB  
Article
A Seventh Order Family of Jarratt Type Iterative Method for Electrical Power Systems
by Saima Yaseen, Fiza Zafar and Francisco I. Chicharro
Fractal Fract. 2023, 7(4), 317; https://doi.org/10.3390/fractalfract7040317 - 06 Apr 2023
Cited by 3 | Viewed by 1386
Abstract
A load flow study referred to as a power flow study is a numerical analysis of the electricity that flows through any electrical power system. For instance, if a transmission line needs to be taken out of service for maintenance, load flow studies [...] Read more.
A load flow study referred to as a power flow study is a numerical analysis of the electricity that flows through any electrical power system. For instance, if a transmission line needs to be taken out of service for maintenance, load flow studies allow us to assess whether the remaining line can carry the load without exceeding its rated capacity. So, we need to understand about the voltage level and voltage phase angle on each bus under steady-state conditions to keep the bus voltage within a specific range. In this paper, our goal is to present a higher order efficient iterative method to carry out a power flow study to determine the voltages (magnitude and angle) for a specific load, generation and network conditions. We introduce a new seventh-order three-step iterative scheme for obtaining approximate solution of nonlinear systems of equations. We attain the seventh-order convergence by using four function evaluations which makes it worthy of interest. Moreover, we show its applicability to the electrical power system for calculating voltages and phase angles. By calculating the bus angle and voltage level, we conclude that the performance of the power system is assessed in a more efficient manner using the new scheme. In addition, dynamical planes of the methods applied on nonlinear systems of equations show global convergence. Full article
(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)
Show Figures

Figure 1

21 pages, 877 KiB  
Article
Convergence Analysis of a New Implicit Iterative Scheme and Its Application to Delay Caputo Fractional Differential Equations
by Austine Efut Ofem, Mfon Okon Udo, Oboyi Joseph, Reny George and Chukwuka Fernando Chikwe
Fractal Fract. 2023, 7(3), 212; https://doi.org/10.3390/fractalfract7030212 - 24 Feb 2023
Cited by 4 | Viewed by 1010
Abstract
This article presents a new three-step implicit iterative method. The proposed method is used to approximate the fixed points of a certain class of pseudocontractive-type operators. Additionally, the strong convergence results of the new iterative procedure are derived. Some examples are constructed to [...] Read more.
This article presents a new three-step implicit iterative method. The proposed method is used to approximate the fixed points of a certain class of pseudocontractive-type operators. Additionally, the strong convergence results of the new iterative procedure are derived. Some examples are constructed to authenticate the assumptions in our main result. At the end, we use our new method to solve a fractional delay differential equation in the sense of Caputo. Our main results improve and generalize the results of many prominent authors in the existing literature. Full article
(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)
Show Figures

Figure 1

12 pages, 516 KiB  
Article
Structured Doubling Algorithm for a Class of Large-Scale Discrete-Time Algebraic Riccati Equations with High-Ranked Constant Term
by Bo Yu, Chengxu Jiang and Ning Dong
Fractal Fract. 2023, 7(2), 193; https://doi.org/10.3390/fractalfract7020193 - 14 Feb 2023
Viewed by 1017
Abstract
Consider the computation of the solution for a class of discrete-time algebraic Riccati equations (DAREs) with the low-ranked coefficient matrix G and the high-ranked constant matrix H. A structured doubling algorithm is proposed for large-scale problems when A is of lowrank. Compared [...] Read more.
Consider the computation of the solution for a class of discrete-time algebraic Riccati equations (DAREs) with the low-ranked coefficient matrix G and the high-ranked constant matrix H. A structured doubling algorithm is proposed for large-scale problems when A is of lowrank. Compared to the existing doubling algorithm of O(2kn) flops at the k-th iteration, the newly developed version merely needs O(n) flops for preprocessing and O((k+1)3m3) flopsfor iterations and is more proper for large-scale computations when mn. The convergence and complexity of the algorithm are subsequently analyzed. Illustrative numerical experiments indicate that the presented algorithm, which consists of a dominant time-consuming preprocessing step and a trivially iterative step, is capable of computing the solution efficiently for large-scale DAREs. Full article
(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)
Show Figures

Figure 1

22 pages, 685 KiB  
Article
Order of Convergence, Extensions of Newton–Simpson Method for Solving Nonlinear Equations and Their Dynamics
by Santhosh George, Ajil Kunnarath, Ramya Sadananda, Jidesh Padikkal and Ioannis K. Argyros
Fractal Fract. 2023, 7(2), 163; https://doi.org/10.3390/fractalfract7020163 - 06 Feb 2023
Cited by 1 | Viewed by 1036
Abstract
Local convergence of order three has been established for the Newton–Simpson method (NS), provided that derivatives up to order four exist. However, these derivatives may not exist and the NS can converge. For this reason, we recover the convergence order based only on [...] Read more.
Local convergence of order three has been established for the Newton–Simpson method (NS), provided that derivatives up to order four exist. However, these derivatives may not exist and the NS can converge. For this reason, we recover the convergence order based only on the first two derivatives. Moreover, the semilocal convergence of NS and some of its extensions not given before is developed. Furthermore, the dynamics are explored for these methods with many illustrations. The study contains examples verifying the theoretical conditions. Full article
(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)
Show Figures

Figure 1

15 pages, 452 KiB  
Article
Spectral Collocation Approach via Normalized Shifted Jacobi Polynomials for the Nonlinear Lane-Emden Equation with Fractal-Fractional Derivative
by Youssri Hassan Youssri and Ahmed Gamal Atta
Fractal Fract. 2023, 7(2), 133; https://doi.org/10.3390/fractalfract7020133 - 31 Jan 2023
Cited by 29 | Viewed by 1796
Abstract
Herein, we adduce, analyze, and come up with spectral collocation procedures to iron out a specific class of nonlinear singular Lane–Emden (LE) equations with generalized Caputo derivatives that appear in the study of astronomical objects. The offered solution is approximated as a truncated [...] Read more.
Herein, we adduce, analyze, and come up with spectral collocation procedures to iron out a specific class of nonlinear singular Lane–Emden (LE) equations with generalized Caputo derivatives that appear in the study of astronomical objects. The offered solution is approximated as a truncated series of the normalized shifted Jacobi polynomials under the assumption that the exact solution is an element in L2. The spectral collocation method is used as a solver to obtain the unknown expansion coefficients. The Jacobi roots are used as collocation nodes. Our solutions can easily be a generalization of the solutions of the classical LE equation, by obtaining a numerical solution based on new parameters, by fixing these parameters to the classical case, we obtain the solution of the classical equation. We provide a meticulous convergence analysis and demonstrate rapid convergence of the truncation error concerning the number of retained modes. Numerical examples show the effectiveness and applicability of the method. The primary benefits of the suggested approach are that we significantly reduce the complexity of the underlying differential equation by solving a nonlinear system of algebraic equations that can be done quickly and accurately using Newton’s method and vanishing initial guesses. Full article
(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)
Show Figures

Figure 1

20 pages, 394 KiB  
Article
Stability and Bifurcation Analysis of Fifth-Order Nonlinear Fractional Difference Equation
by Abdul Khaliq, Irfan Mustafa, Tarek F. Ibrahim, Waleed M. Osman, Bushra R. Al-Sinan, Arafa Abdalrhim Dawood and Manal Yagoub Juma
Fractal Fract. 2023, 7(2), 113; https://doi.org/10.3390/fractalfract7020113 - 23 Jan 2023
Cited by 6 | Viewed by 1783
Abstract
In this paper, a rational difference equation with positive parameters and non-negative conditions is used to determine the presence and direction of the Neimark–Sacker bifurcation. The neimark–Sacker bifurcation of the system is first studied using the characteristic equation. In addition, we study bifurcation [...] Read more.
In this paper, a rational difference equation with positive parameters and non-negative conditions is used to determine the presence and direction of the Neimark–Sacker bifurcation. The neimark–Sacker bifurcation of the system is first studied using the characteristic equation. In addition, we study bifurcation invariant curves from the perspective of normal form theory. A computer simulation is used to illustrate the analytical results. Full article
(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)
Show Figures

Figure 1

12 pages, 494 KiB  
Article
Necessary and Sufficient Conditions for Existence and Uniqueness of Solutions to Nabla Fractional Systems
by Jikai Yang, Hongli Li and Long Zhang
Fractal Fract. 2022, 6(12), 723; https://doi.org/10.3390/fractalfract6120723 - 07 Dec 2022
Viewed by 967
Abstract
In this paper, we study the existence and uniqueness of solutions for nabla fractional systems. By using the properties of bijective functions, we obtain a necessary and sufficient condition ensuring the existence and uniqueness of solutions for a class of fractional discrete systems. [...] Read more.
In this paper, we study the existence and uniqueness of solutions for nabla fractional systems. By using the properties of bijective functions, we obtain a necessary and sufficient condition ensuring the existence and uniqueness of solutions for a class of fractional discrete systems. Furthermore, we derive two sufficient conditions guaranteeing the existence of solutions by means of a nonlinear functional analysis method. In addition, the above conclusions are extended to high-dimensional delayed systems. Finally, two examples are given to illustrate the validity of our results. Full article
(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)
Show Figures

Figure 1

21 pages, 959 KiB  
Article
Extended Comparison between Two Derivative-Free Methods of Order Six for Equations under the Same Conditions
by Samundra Regmi, Ioannis K. Argyros, Christopher I. Argyros and Debasis Sharma
Fractal Fract. 2022, 6(11), 634; https://doi.org/10.3390/fractalfract6110634 - 30 Oct 2022
Viewed by 923
Abstract
Under the same conditions, we propose the extended comparison between two derivative free schemes of order six for addressing equations. The existing convergence technique used the standard Taylor series approach, which requires derivatives up to order seven. In contrast to previous researchers, our [...] Read more.
Under the same conditions, we propose the extended comparison between two derivative free schemes of order six for addressing equations. The existing convergence technique used the standard Taylor series approach, which requires derivatives up to order seven. In contrast to previous researchers, our convergence theorems only demand the first derivative. In addition, formulas for determining the region of uniqueness for the solution, convergence radii, and error estimations are suggested. As a consequence, we broaden the utility of these productive schemes. Moreover, we present a comparison of attraction basins for these schemes to obtain roots of complex polynomial equations. The confirmation of our convergence findings on application problems brings this research to a close. Full article
(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)
Show Figures

Figure 1

15 pages, 1460 KiB  
Article
Parametric Family of Root-Finding Iterative Methods: Fractals of the Basins of Attraction
by José J. Padilla, Francisco I. Chicharro, Alicia Cordero and Juan R. Torregrosa
Fractal Fract. 2022, 6(10), 572; https://doi.org/10.3390/fractalfract6100572 - 08 Oct 2022
Cited by 4 | Viewed by 1650
Abstract
Research interest in iterative multipoint schemes to solve nonlinear problems has increased recently because of the drawbacks of point-to-point methods, which need high-order derivatives to increase the order of convergence. However, this order is not the only key element to classify the iterative [...] Read more.
Research interest in iterative multipoint schemes to solve nonlinear problems has increased recently because of the drawbacks of point-to-point methods, which need high-order derivatives to increase the order of convergence. However, this order is not the only key element to classify the iterative schemes. We aim to design new multipoint fixed point classes without memory, that improve or bring together the existing ones in different areas such as computational efficiency, stability and also convergence order. In this manuscript, we present a family of parametric iterative methods, whose order of convergence is four, that has been designed by using composition and weight function techniques. A qualitative analysis is made, based on complex discrete dynamics, to select those elements of the class with best stability properties on low-degree polynomials. This stable behavior is directly related with the simplicity of the fractals defined by the basins of attraction. In the opposite, particular methods with unstable performance present high-complexity in the fractals of their basins. The stable members are demonstrated also be the best ones in terms of numerical performance of non-polynomial functions, with special emphasis on Colebrook-White equation, with wide applications in Engineering. Full article
(This article belongs to the Special Issue Applications of Iterative Methods in Solving Nonlinear Equations)
Show Figures

Figure 1

Back to TopTop