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Fractal Fract
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Iterative Methods in Solving Nonlinear Equations Based on Fractal and...
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Iterative Methods in Solving Nonlinear Equations Based on Fractal and Fractional Perspective

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

Deadline for manuscript submissions: 31 July 2025 | Viewed by 1196

Special Issue Editors

Prof. Dr. Neus Garrido

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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
Prof. Dr. Francisco I. Chicharro

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Guest Editor
Institute Matemática Multidisciplinar, Universitat Politècnica de València, Camino de Vera, S/N, 46022 Valencia, Spain
Interests: iterative methods; nonlinear equations and systems; complex dynamics
Special Issues, Collections and Topics in MDPI journals
Dr. Paula Triguero-Navarro

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 a significant boom in recent decades. A common problem in science, engineering and economics disciplines lies in the requirement of the solution for a nonlinear equation or system of equations. We resort to approximate solutions in cases where analytical solutions are not adequate. One of these 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.

Prof. Dr. Neus Garrido
Prof. Dr. Francisco I. Chicharro
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

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Related Special Issue

Published Papers (2 papers)

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Research

26 pages, 5138 KiB  
Article
On Traub–Steffensen-Type Iteration Schemes With and Without Memory: Fractal Analysis Using Basins of Attraction
by Moin-ud-Din Junjua, Shahid Abdullah, Munish Kansal and Shabbir Ahmad
Fractal Fract. 2024, 8(12), 698; https://doi.org/10.3390/fractalfract8120698 (registering DOI) - 26 Nov 2024
Abstract
This paper investigates the design and stability of Traub–Steffensen-type iteration schemes with and without memory for solving nonlinear equations. Steffensen’s method overcomes the drawback of the derivative evaluation of Newton’s scheme, but it has, in general, smaller sets of initial guesses that converge [...] Read more.
This paper investigates the design and stability of Traub–Steffensen-type iteration schemes with and without memory for solving nonlinear equations. Steffensen’s method overcomes the drawback of the derivative evaluation of Newton’s scheme, but it has, in general, smaller sets of initial guesses that converge to the desired root. Despite this drawback of Steffensen’s method, several researchers have developed higher-order iterative methods based on Steffensen’s scheme. Traub introduced a free parameter in Steffensen’s scheme to obtain the first parametric iteration method, which provides larger basins of attraction for specific values of the parameter. In this paper, we introduce a two-step derivative free fourth-order optimal iteration scheme based on Traub’s method by employing three free parameters and a weight function. We further extend it into a two-step eighth-order iteration scheme by means of memory with the help of suitable approximations of the involved parameters using Newton’s interpolation. The convergence analysis demonstrates that the proposed iteration scheme without memory has an order of convergence of 4, while its memory-based extension achieves an order of convergence of at least 7.993, attaining the efficiency index 7.9931/3≈2. Two special cases of the proposed iteration scheme are also presented. Notably, the proposed methods compete with any optimal j-point method without memory. We affirm the superiority of the proposed iteration schemes in terms of efficiency index, absolute error, computational order of convergence, basins of attraction, and CPU time using comparisons with several existing iterative methods of similar kinds across diverse nonlinear equations. In general, for the comparison of iterative schemes, the basins of iteration are investigated on simple polynomials of the form zn−1 in the complex plane. However, we investigate the stability and regions of convergence of the proposed iteration methods in comparison with some existing methods on a variety of nonlinear equations in terms of fractals of basins of attraction. The proposed iteration schemes generate the basins of attraction in less time with simple fractals and wider regions of convergence, confirming their stability and superiority in comparison with the existing methods. Full article
(This article belongs to the Special Issue Iterative Methods in Solving Nonlinear Equations Based on Fractal and Fractional Perspective)
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21 pages, 665 KiB  
Article
A Note on Fractional Third-Order Partial Differential Equations and the Generalized Laplace Transform Decomposition Method
by Hassan Eltayeb and Diaa Eldin Elgezouli
Fractal Fract. 2024, 8(10), 602; https://doi.org/10.3390/fractalfract8100602 - 15 Oct 2024
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Abstract
This paper establishes a unique approach known as the multi-generalized Laplace transform decomposition method (MGLTDM) to solve linear and nonlinear dispersive KdV-type equations. This method combines the multi-generalized Laplace transform (MGLT) with the decomposition method (DM), and offers a strong procedure for handling [...] Read more.
This paper establishes a unique approach known as the multi-generalized Laplace transform decomposition method (MGLTDM) to solve linear and nonlinear dispersive KdV-type equations. This method combines the multi-generalized Laplace transform (MGLT) with the decomposition method (DM), and offers a strong procedure for handling complicated equations. To verify the applicability and validity of this method, some ideal problems of dispersive KDV-type equations are discussed and the outcoming approximate solutions are stated in sequential form. The results show that the MGLTDM is a dependable and powerful technique to deal with physical problems in diverse implementations. Full article
(This article belongs to the Special Issue Iterative Methods in Solving Nonlinear Equations Based on Fractal and Fractional Perspective)
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