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New Trends and Developments in Numerical Analysis: 2nd Edition
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New Trends and Developments in Numerical Analysis: 2nd Edition

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: 20 January 2025 | Viewed by 2845

Special Issue Editors

Prof. Dr. Juan Ramón Torregrosa Sánchez

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Guest Editor
Institute for Multidisciplinary Mathematics, Universitat Politècnica de València, 46022 València, Spain
Interests: iterative processes; matrix analysis; numerical analysis
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Alicia Cordero

E-Mail Website
Guest Editor
School of Telecommunications Engineering, Universitat Politècnica de València, 46022 Valencia, Spain
Interests: numerical analysis; iterative methods; nonlinear problems; discrete dynamics, real and complex
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Francisco I. Chicharro

E-Mail Website
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

Special Issue Information

Dear Colleagues,

This Special Issue Mathematics invites both original and survey manuscripts that bring together new mathematical tools and numerical methods for computational problems. Numerical analysis is a specific area of mathematics that involves designing, proving the convergence and using algorithms to map out the mathematical core of a practical problem and find its solution.

Numerical methods are applied in all fields of engineering, physical sciences, life sciences, social sciences, medicine, business, etc. The main aspects of interest for numerical schemes include approximation, simulation and estimation, used in virtually every scientific field.

The possible topics of interest include the following: numerical stability, interpolation, approximation, quadrature methods, numerical linear algebra, ordinary differential equations, with initial and boundary conditions, partial differential equations, numerical fractional analysis, optimization, integral equations, iterative methods for solving nonlinear equations and systems, etc., and their applications for solving real problems in sciences and engineering.

Prof. Dr. Juan Ramón Torregrosa Sánchez
Prof. Dr. Alicia Cordero
Prof. Dr. Francisco I. Chicharro
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. Mathematics is an international peer-reviewed open access semimonthly 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 2600 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

  • interpolation
  • approximation
  • numerical integration
  • optimization
  • stability
  • convergence
  • eigenvalues
  • discretization
  • iterative methods

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Related Special Issue

Published Papers (4 papers)

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Research

15 pages, 558 KiB  
Article
Enhanced Ninth-Order Memory-Based Iterative Technique for Efficiently Solving Nonlinear Equations
by Shubham Kumar Mittal, Sunil Panday and Lorentz Jäntschi
Mathematics 2024, 12(22), 3490; https://doi.org/10.3390/math12223490 - 8 Nov 2024
Viewed by 349
Abstract
In this article, we present a novel three-step with-memory iterative method for solving nonlinear equations. We have improved the convergence order of a well-known optimal eighth-order iterative method by converting it into a with-memory version. The Hermite interpolating polynomial is utilized to compute [...] Read more.
In this article, we present a novel three-step with-memory iterative method for solving nonlinear equations. We have improved the convergence order of a well-known optimal eighth-order iterative method by converting it into a with-memory version. The Hermite interpolating polynomial is utilized to compute a self-accelerating parameter that improves the convergence order. The proposed uni-parametric with-memory iterative method improves its R-order of convergence from 8 to 8.8989. Additionally, no more function evaluations are required to achieve this improvement in convergence order. Furthermore, the efficiency index has increased from 1.6818 to 1.7272. The proposed method is shown to be more effective than some well-known existing methods, as shown by extensive numerical testing on a variety of problems. Full article
(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis: 2nd Edition)
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20 pages, 358 KiB  
Article
Two-Step Fifth-Order Efficient Jacobian-Free Iterative Method for Solving Nonlinear Systems
by Alicia Cordero, Javier G. Maimó, Antmel Rodríguez-Cabral and Juan R. Torregrosa
Mathematics 2024, 12(21), 3341; https://doi.org/10.3390/math12213341 - 24 Oct 2024
Viewed by 965
Abstract
This article introduces a novel two-step fifth-order Jacobian-free iterative method aimed at efficiently solving systems of nonlinear equations. The method leverages the benefits of Jacobian-free approaches, utilizing divided differences to circumvent the computationally intensive calculation of Jacobian matrices. This adaptation significantly reduces computational [...] Read more.
This article introduces a novel two-step fifth-order Jacobian-free iterative method aimed at efficiently solving systems of nonlinear equations. The method leverages the benefits of Jacobian-free approaches, utilizing divided differences to circumvent the computationally intensive calculation of Jacobian matrices. This adaptation significantly reduces computational overhead and simplifies the implementation process while maintaining high convergence rates. We demonstrate that this method achieves fifth-order convergence under specific parameter settings, with broad applicability across various types of nonlinear systems. The effectiveness of the proposed method is validated through a series of numerical experiments that confirm its superior performance in terms of accuracy and computational efficiency compared to existing methods. Full article
(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis: 2nd Edition)
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9 pages, 669 KiB  
Article
Influence of Fractional Order on the Behavior of a Normalized Time-Fractional SIR Model
by Junseok Kim
Mathematics 2024, 12(19), 3081; https://doi.org/10.3390/math12193081 - 1 Oct 2024
Viewed by 623
Abstract
In this paper, we propose a novel normalized time-fractional susceptible–infected–removed (SIR) model that incorporates memory effects into epidemiological dynamics. The proposed model is based on a newly developed normalized time-fractional derivative, which is similar to the well-known Caputo fractional derivative but is characterized [...] Read more.
In this paper, we propose a novel normalized time-fractional susceptible–infected–removed (SIR) model that incorporates memory effects into epidemiological dynamics. The proposed model is based on a newly developed normalized time-fractional derivative, which is similar to the well-known Caputo fractional derivative but is characterized by the property that the sum of its weight function equals one. This unity property is crucial because it helps with evaluating how the fractional order influences the behavior of time-fractional differential equations over time. The normalized time-fractional derivative, with its unity property, provides an intuitive understanding of how fractional orders influence the SIR model’s dynamics and enables systematic exploration of how changes in the fractional order affect the model’s behavior. We numerically investigate how these variations impact the epidemiological dynamics of our normalized time-fractional SIR model and highlight the role of fractional order in improving the accuracy of infectious disease predictions. The appendix provides the program code for the model. Full article
(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis: 2nd Edition)
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22 pages, 357 KiB  
Article
Accelerating the Speed of Convergence for High-Order Methods to Solve Equations
by Ramandeep Behl, Ioannis K. Argyros and Sattam Alharbi
Mathematics 2024, 12(17), 2785; https://doi.org/10.3390/math12172785 - 9 Sep 2024
Viewed by 460
Abstract
This article introduces a multistep method for developing sequences that solve Banach space-valued equations. It provides error estimates, a radius of convergence, and uniqueness results. Our approach improves the applicability of the recommended method and addresses challenges in applied science. The theoretical advancements [...] Read more.
This article introduces a multistep method for developing sequences that solve Banach space-valued equations. It provides error estimates, a radius of convergence, and uniqueness results. Our approach improves the applicability of the recommended method and addresses challenges in applied science. The theoretical advancements are supported by comprehensive computational results, demonstrating the practical applicability and robustness of the earlier method. We ensure more reliable and precise solutions to Banach space-valued equations by providing computable error estimates and a clear radius of convergence for the considered method. We conclude that our work significantly improves the practical utility of multistep methods, offering a rigorous and computable approach to solving complex equations in Banach spaces, with strong theoretical and computational results. Full article
(This article belongs to the Special Issue New Trends and Developments in Numerical Analysis: 2nd Edition)
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