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Mathematics
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Numerical Methods for Solving Nonlinear Problems in Science and Engineering
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Numerical Methods for Solving Nonlinear Problems in Science and Engineering

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

Deadline for manuscript submissions: 30 March 2026 | Viewed by 565

Special Issue Editors

Prof. Dr. Juan R. Torregrosa

E-Mail Website
Guest Editor
1. Department of Applied Mathematics, Universitat Politècnica de València, 46022 Valencia, Spain
2. Institute for Multidisciplinary Mathematics (IM2), Universitat Politècnica de València, 46022 Valencia, Spain
Interests: numerical analysis; matrix analysis; dynamic 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

Special Issue Information

Dear Colleagues,

The growing complexity of real-world problems, as well as new advances in established computational theories, demand a wide range of innovative approaches to numerical simulations and the discovery of new and efficient calculation algorithms.

This Special Issue of Mathematics invites both original and survey manuscripts that bring together new mathematical tools and numerical methods for computational problems. Its main objective is to present the latest advances at the intersection between classical numerical methods, scientific machine learning, and other numerical approaches.

Numerical analysis is branch of mathematics concerned with designing algorithms, proving their convergence, and applying them to identify and solve the mathematical core of practical problems.

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

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, and iterative methods for solving nonlinear equations, systems, real problems in sciences, and engineering.

Prof. Dr. Juan R. Torregrosa
Prof. Dr. Alicia Cordero
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 250 words) can be sent to the Editorial Office for assessment.

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
  • singular values decomposition
  • discretization
  • iterative methods

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Published Papers (1 paper)

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Research

34 pages, 3328 KB  
Article
A New Perspective on the Convergence of Mean-Based Methods for Nonlinear Equations
by Alicia Cordero, María Emilia Maldonado Machuca and Juan R. Torregrosa
Mathematics 2025, 13(21), 3525; https://doi.org/10.3390/math13213525 - 3 Nov 2025
Viewed by 389
Abstract
Many problems in science, engineering, and economics require solving of nonlinear equations, often arising from attempts to model natural systems and predict their behavior. In this context, iterative methods provide an effective approach to approximate the roots of nonlinear functions. This work introduces [...] Read more.
Many problems in science, engineering, and economics require solving of nonlinear equations, often arising from attempts to model natural systems and predict their behavior. In this context, iterative methods provide an effective approach to approximate the roots of nonlinear functions. This work introduces five new parametric families of multipoint iterative methods specifically designed for solving nonlinear equations. Each family is built upon a two-step scheme: the first step applies the classical Newton method, while the second incorporates a convex mean, a weight function, and a frozen derivative (i.e., the same derivative from the previous step). The careful design of the weight function was essential to ensure fourth-order convergence while allowing arbitrary parameter values. The proposed methods are theoretically analyzed and dynamically characterized using tools such as stability surfaces, parameter planes, and dynamical planes on the Riemann sphere. These analyses reveal regions of stability and divergence, helping identify suitable parameter values that guarantee convergence to the root. Moreover, a general result proves that all the proposed optimal parametric families of iterative methods are topologically equivalent, under conjugation. Numerical experiments confirm the robustness and efficiency of the methods, often surpassing classical approaches in terms of convergence speed and accuracy. Overall, the results demonstrate that convex-mean-based parametric methods offer a flexible and stable framework for the reliable numerical solution of nonlinear equations. Full article
(This article belongs to the Special Issue Numerical Methods for Solving Nonlinear Problems in Science and Engineering)
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