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Mathematics
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Numerical Analysis in Computational Mathematics
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Numerical Analysis in Computational Mathematics

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

Deadline for manuscript submissions: 15 February 2025 | Viewed by 1128

Special Issue Editors

Dr. Thái Anh Nhan

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Guest Editor
Department of Mathematics and Computer Science, Santa Clara University, Santa Clara, CA 95053-0513, USA
Interests: numerical analysis; applied math; scientific computing
Prof. Dr. Istvan Farago

E-Mail Website
Guest Editor
Department of Applied Analysis, Eötvös Loránd University, Pázmány Péter sétány 1/C, H-1117 Budapest, Hungary
Interests: numerical analysis; differential equations; mathematical modeling of real-life problems; epidemic propagation

Special Issue Information

Dear Colleagues,

The Special Issue on Numerical Analysis in Computational Methods provides a high-quality venue for recent advances in the design and development of numerical and computational algorithms for all aspects in Applied Mathematics. We invite researchers to contribute high-quality original research articles as well as review articles on recent advances including, but not limited to, theoretical and applied numerical algorithms, analysis, and methods, applied and computational methods, scientific computing, computer simulations, optimization techniques, and advances in other branches of applied science with a computational mathematics component.

Dr. Thái Anh Nhan
Prof. Dr. Istvan Farago
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

  • numerical algorithms, analysis, and methods
  • applications of numerical algorithms, analysis, and methods
  • applied and computational mathematics
  • scientific computing
  • computer simulations
  • optimization techniques

Published Papers (2 papers)

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Research

23 pages, 1168 KiB  
Article
Numerical Analysis for Sturm–Liouville Problems with Nonlocal Generalized Boundary Conditions
by Chein-Shan Liu, Chih-Wen Chang and Chung-Lun Kuo
Mathematics 2024, 12(8), 1265; https://doi.org/10.3390/math12081265 - 22 Apr 2024
Viewed by 351
Abstract
For the generalized Sturm–Liouville problem (GSLP), a new formulation is undertaken to reduce the number of unknowns from two to one in the target equation for the determination of eigenvalue. The eigenparameter-dependent shape functions are derived for using in a variable transformation, such [...] Read more.
For the generalized Sturm–Liouville problem (GSLP), a new formulation is undertaken to reduce the number of unknowns from two to one in the target equation for the determination of eigenvalue. The eigenparameter-dependent shape functions are derived for using in a variable transformation, such that the GSLP becomes an initial value problem for a new variable. For the uniqueness of eigenfunction an extra condition is imposed, which renders the right-end value of the new variable available; a derived implicit nonlinear equation is solved by an iterative method without using the differential; we can achieve highly precise eigenvalues. For the nonlocal Sturm–Liouville problem (NSLP), we consider two types of integral boundary conditions on the right end. For the first type of NSLP we can prove sufficient conditions for the positiveness of the eigenvalue. Negative eigenvalues and multiple solutions may exist for the second type of NSLP. We propose a boundary shape function method, a two-dimensional fixed-quasi-Newton method and a combination of them to solve the NSLP with fast convergence and high accuracy. From the aspect of numerical analysis the initial value problem of ordinary differential equations and scalar nonlinear equations are more easily treated than the original GSLP and NSLP, which is the main novelty of the paper to provide the mathematically equivalent and simpler mediums to determine the eigenvalues and eigenfunctions. Full article
(This article belongs to the Special Issue Numerical Analysis in Computational Mathematics)
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22 pages, 349 KiB  
Article
New Memory-Updating Methods in Two-Step Newton’s Variants for Solving Nonlinear Equations with High Efficiency Index
by Chein-Shan Liu and Chih-Wen Chang
Mathematics 2024, 12(4), 581; https://doi.org/10.3390/math12040581 - 15 Feb 2024
Cited by 1 | Viewed by 478
Abstract
In the paper, we iteratively solve a scalar nonlinear equation f(x)=0, where f∈C(I,R),x∈I⊂R, and I includes at least one real root r. [...] Read more.
In the paper, we iteratively solve a scalar nonlinear equation f(x)=0, where f∈C(I,R),x∈I⊂R, and I includes at least one real root r. Three novel two-step iterative schemes equipped with memory updating methods are developed; they are variants of the fixed-point Newton method. A triple data interpolation is carried out by the two-degree Newton polynomial, which is used to update the values of f′(r) and f″(r). The relaxation factor in the supplementary variable is accelerated by imposing an extra condition on the interpolant. The new memory method (NMM) can raise the efficiency index (E.I.) significantly. We apply the NMM to five existing fourth-order iterative methods, and the computed order of convergence (COC) and E.I. are evaluated by numerical tests. When the relaxation factor acceleration technique is combined with the modified Dzˇunic´’s memory method, the value of E.I. is much larger than that predicted by the paper [Kung, H.T.; Traub, J.F. J. Assoc. Comput. Machinery 1974, 21]. for the iterative method without memory. Full article
(This article belongs to the Special Issue Numerical Analysis in Computational Mathematics)
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