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Mathematics
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Numerical Methods for Approximation of Functions and Data
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Numerical Methods for Approximation of Functions and Data

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

Deadline for manuscript submissions: closed (31 July 2023) | Viewed by 8903

Special Issue Editors

Dr. Pravin Singh

E-Mail Website
Guest Editor
Department of Mathematics, University of Kwazulu-Natal, Durban 4041, South Africa
Interests: numerical analysis; approximation theory; collocation; linear algebra
Dr. Naben Parumasur

E-Mail Website
Guest Editor
Department of Mathematics, University of Kwazulu-Natal, Durban 4041, South Africa
Interests: spectral; pseudo-spectral and collocation methods for PDEs; mathematical modelling and mathematical biology
Prof. Dr. Vijay Kumar Kukreja

E-Mail Website
Guest Editor
Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal 148106, India
Interests: mathematical modelling; numerical analysis; BVPs

Special Issue Information

Dear Colleagues,

The aim of this Special Issue is to publish original research articles covering recent advances in any of the areas included in approximation of functions and data. Particular emphasis is placed on innovative numerical methods for the approximation of continuous functions and data sets, with some applications. Topics include but are not limited to polynomial and rational function approximation,  discrete and continuous least squares application, regression analysis, curve fitting, interpolation and extrapolation, and approximation using orthogonal functions. Applications to real-world problems are encouraged with the presentation of algorithms and comparison with existing techniques. 

Dr. Pravin Singh
Dr. Naben Parumasur
Prof. Dr. Vijay Kumar Kukreja
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.

Published Papers (5 papers)

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Research

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18 pages, 689 KiB  
Article
Selectivity Estimation of Inequality Joins in Databases
by Diogo Repas, Zhicheng Luo, Maxime Schoemans and Mahmoud Sakr
Mathematics 2023, 11(6), 1383; https://doi.org/10.3390/math11061383 - 13 Mar 2023
Viewed by 1763
Abstract
Selectivity estimation refers to the ability of the SQL query optimizer to estimate the size of the results of a predicate in the query. It is the main calculation based on which the optimizer can select the least expensive plan to execute. While [...] Read more.
Selectivity estimation refers to the ability of the SQL query optimizer to estimate the size of the results of a predicate in the query. It is the main calculation based on which the optimizer can select the least expensive plan to execute. While the problem has been known since the mid-1970s, we were surprised that there are no solutions in the literature for the selectivity estimation of inequality joins. By testing four common database systems: Oracle, SQL-Server, PostgreSQL, and MySQL, we found that the open-source systems PostgreSQL and MySQL lack this estimation. Oracle and SQL-Server make fairly accurate estimations, yet their algorithms are secret. This paper, thus, proposes an algorithm for inequality join selectivity estimation. The proposed algorithm was implemented in PostgreSQL and sent as a patch to be included in the next releases. We compared this implementation with the above DBMS for three different data distributions (uniform, normal, and Zipfian) and showed that our algorithm provides extremely accurate estimations (below 0.1% average error), outperforming the other systems by an order of magnitude. Full article
(This article belongs to the Special Issue Numerical Methods for Approximation of Functions and Data)
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7 pages, 764 KiB  
Article
The Improved Stability Analysis of Numerical Method for Stochastic Delay Differential Equations
by Yu Zhang, Enying Zhang and Longsuo Li
Mathematics 2022, 10(18), 3366; https://doi.org/10.3390/math10183366 - 16 Sep 2022
Cited by 2 | Viewed by 940
Abstract
In this paper, the improved split-step θ method, named the split-step composite θ method, is proposed to study the mean-square stability for stochastic differential equations with a fixed time delay. Under the global Lipschitz and linear growth conditions, it is proved that the [...] Read more.
In this paper, the improved split-step θ method, named the split-step composite θ method, is proposed to study the mean-square stability for stochastic differential equations with a fixed time delay. Under the global Lipschitz and linear growth conditions, it is proved that the split-step composite θ method with θ≥0.5 shows mean-square stability. An approach to improving numerical stability is illustrated by choices of parameters of this method. Some numerical examples are presented to show the accordance between the theoretical and numerical results. Full article
(This article belongs to the Special Issue Numerical Methods for Approximation of Functions and Data)
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13 pages, 490 KiB  
Article
On One- and Two-Dimensional α–Stancu–Schurer–Kantorovich Operators and Their Approximation Properties
by Md. Heshamuddin, Nadeem Rao, Bishnu P. Lamichhane, Adem Kiliçman and Mohammad Ayman-Mursaleen
Mathematics 2022, 10(18), 3227; https://doi.org/10.3390/math10183227 - 06 Sep 2022
Cited by 11 | Viewed by 1429
Abstract
The goal of this research article is to introduce a sequence of α–Stancu–Schurer–Kantorovich operators. We calculate moments and central moments and find the order of approximation with the aid of modulus of continuity. A Voronovskaja-type approximation result is also proven. Next, error [...] Read more.
The goal of this research article is to introduce a sequence of α–Stancu–Schurer–Kantorovich operators. We calculate moments and central moments and find the order of approximation with the aid of modulus of continuity. A Voronovskaja-type approximation result is also proven. Next, error analysis and convergence of the operators for certain functions are presented numerically and graphically. Furthermore, two-dimensional α–Stancu–Schurer–Kantorovich operators are constructed and their rate of convergence, graphical representation of approximation and numerical error estimates are presented. Full article
(This article belongs to the Special Issue Numerical Methods for Approximation of Functions and Data)
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Review

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28 pages, 429 KiB  
Review
Survey of Hermite Interpolating Polynomials for the Solution of Differential Equations
by Archna Kumari and Vijay K. Kukreja
Mathematics 2023, 11(14), 3157; https://doi.org/10.3390/math11143157 - 18 Jul 2023
Cited by 2 | Viewed by 2434
Abstract
With progress on both the theoretical and the computational fronts, the use of Hermite interpolation for mathematical modeling has become an established tool in applied science. This article aims to provide an overview of the most widely used Hermite interpolating polynomials and their [...] Read more.
With progress on both the theoretical and the computational fronts, the use of Hermite interpolation for mathematical modeling has become an established tool in applied science. This article aims to provide an overview of the most widely used Hermite interpolating polynomials and their implementation in various algorithms to solve different types of differential equations, which have important applications in different areas of science and engineering. The Hermite interpolating polynomials, their generalization, properties, and applications are provided in this article. Full article
(This article belongs to the Special Issue Numerical Methods for Approximation of Functions and Data)
21 pages, 3982 KiB  
Review
Efficient Solution of Burgers’, Modified Burgers’ and KdV–Burgers’ Equations Using B-Spline Approximation Functions
by Nabendra Parumasur, Rasheed A. Adetona and Pravin Singh
Mathematics 2023, 11(8), 1847; https://doi.org/10.3390/math11081847 - 13 Apr 2023
Cited by 4 | Viewed by 1178
Abstract
This paper discusses the application of the orthogonal collocation on finite elements (OCFE) method using quadratic and cubic B-spline basis functions on partial differential equations. Collocation is performed at Gaussian points to obtain an optimal solution, hence the name orthogonal collocation. The method [...] Read more.
This paper discusses the application of the orthogonal collocation on finite elements (OCFE) method using quadratic and cubic B-spline basis functions on partial differential equations. Collocation is performed at Gaussian points to obtain an optimal solution, hence the name orthogonal collocation. The method is used to solve various cases of Burgers’ equations, including the modified Burgers’ equation. The KdV–Burgers’ equation is considered as a test case for the OCFE method using cubic splines. The results compare favourably with existing results. The stability and convergence of the method are also given consideration. The method is unconditionally stable and second-order accurate in time and space. Full article
(This article belongs to the Special Issue Numerical Methods for Approximation of Functions and Data)
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