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Symmetry
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Numerical Analysis, Approximation Theory, Differential Equations
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Numerical Analysis, Approximation Theory, Differential Equations

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 July 2023) | Viewed by 15369

Special Issue Editor

Dr. Teodora Cătinaș

E-Mail Website
Guest Editor
Faculty of Mathematics and Computer Science, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania
Interests: approximation theory; numerical analysis

Special Issue Information

Dear Colleagues,

This Special Issue is mainly devoted to recent research results in numerical analysis, approximation theory (including multivariate interpolation), scientific computing, differential equations, and symmetries in the applied mathematics field. Numerical analysis and approximation theory are essential parts of the field of applied mathematics. They constitute fields of active research and continual development with applications to real life problems. For example, the results in domains such as wavelets, multivariate spline functions, radial functions, etc., have practical applications in the fields of computer aided design, geometric modelling, geodesy, image analysis, etc. The problem of the interpolation of arbitrarily spaced data is encountered in such areas as geology, cartography, earth sciences, etc. Symmetry methods also play an essential role in real world applications. Differential equations are essential tools for modeling different processes appearing in science. They give rise to important questions such as the existence and uniqueness of the solution, stability, numerical methods of approximation, symmetry methods of evolution equations, etc. We welcome papers on topics including, but not limited to, the following:

  • numerical analysis;
  • scientific computing and algorithms;
  • approximation theory;
  • differential equations;
  • symmetries;
  • multivariate interpolation;
  • interpolation of scattered data;
  • symmetries;
  • numerical linear algebra;
  • numerical integral equations;
  • special polynomial sequences in approximation theory;
  • boundary value problems;
  • quadrature formulas;
  • orthogonal polynomials;
  • numerical methods for nonlinear equations;
  • numerical methods for differential equations.

Dr. Teodora Cătinaș
Guest Editor

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. Symmetry 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 2400 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.

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Published Papers (9 papers)

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Research

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16 pages, 2704 KiB  
Article
Numerical Investigation of Fractional-Order Fornberg–Whitham Equations in the Framework of Aboodh Transformation
by Saima Noor, Ma’mon Abu Hammad, Rasool Shah, Albandari W. Alrowaily and Samir A. El-Tantawy
Symmetry 2023, 15(7), 1353; https://doi.org/10.3390/sym15071353 - 3 Jul 2023
Cited by 7 | Viewed by 1308
Abstract
In this investigation, the fractional Fornberg–Whitham equation (FFWE) is solved and analyzed via the variational iteration method (VIM) and Adomian decomposition method (ADM) with the help of the Aboodh transformation (AT). The FFWE is an important model for describing several nonlinear wave propagations [...] Read more.
In this investigation, the fractional Fornberg–Whitham equation (FFWE) is solved and analyzed via the variational iteration method (VIM) and Adomian decomposition method (ADM) with the help of the Aboodh transformation (AT). The FFWE is an important model for describing several nonlinear wave propagations in various fields of science and plasma physics. The AT provides a powerful tool for transforming fractional-order differential equations (DEs) into integer-order ones, making them more amenable to analytical solutions. Accordingly, the main objective of this investigation is to demonstrate the effectiveness and accuracy of ADM and VIM in deriving some approximations for the FFWE. Furthermore, we highlight the advantages and potential applications of these methods in solving other fractional-order nonlinear problems in several scientific fields, especially in plasma physics and some engineering problems. Full article
(This article belongs to the Special Issue Numerical Analysis, Approximation Theory, Differential Equations)
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18 pages, 348 KiB  
Article
An Improved Symmetric Numerical Approach for Systems of Second-Order Two-Point BVPs
by Busyra Latif, Md Yushalify Misro, Samsul Ariffin Abdul Karim and Ishak Hashim
Symmetry 2023, 15(6), 1166; https://doi.org/10.3390/sym15061166 - 29 May 2023
Cited by 1 | Viewed by 1481
Abstract
This study deals with the numerical solution of a class of linear systems of second-order boundary value problems (BVPs) using a new symmetric cubic B-spline method (NCBM). This is a typical cubic B-spline collocation method powered by new approximations for second-order derivatives. The [...] Read more.
This study deals with the numerical solution of a class of linear systems of second-order boundary value problems (BVPs) using a new symmetric cubic B-spline method (NCBM). This is a typical cubic B-spline collocation method powered by new approximations for second-order derivatives. The flexibility and high order precision of B-spline functions allow them to approximate the answers. These functions have a symmetrical property. The new second-order approximation plays an important role in producing more accurate results up to a fifth-order accuracy. To verify the proposed method’s accuracy, it is tested on three linear systems of ordinary differential equations with multiple step sizes. The numerical findings by the present method are quite similar to the exact solutions available in the literature. We discovered that when the step size decreased, the computational errors decreased, resulting in better precision. In addition, details of maximum errors are investigated. Moreover, simple implementation and straightforward computations are the main advantages of the offered method. This method yields improved results, even if it does not require using free parameters. Thus, it can be concluded that the offered scheme is reliable and efficient. Full article
(This article belongs to the Special Issue Numerical Analysis, Approximation Theory, Differential Equations)
9 pages, 261 KiB  
Article
Quasi-Monomiality Principle and Certain Properties of Degenerate Hybrid Special Polynomials
by Rabab Alyusof
Symmetry 2023, 15(2), 407; https://doi.org/10.3390/sym15020407 - 3 Feb 2023
Cited by 3 | Viewed by 1196
Abstract
This article aims to introduce degenerate hybrid type Appell polynomials HQm(u,v,w;η) and establishes their quasi-monomial characteristics. Additionally, a number of features of these polynomials are established, including symmetric identities, implicit summation formulae, differential equations, series definition and operational formalism. Full article
(This article belongs to the Special Issue Numerical Analysis, Approximation Theory, Differential Equations)
22 pages, 8116 KiB  
Article
Some New Time and Cost Efficient Quadrature Formulas to Compute Integrals Using Derivatives with Error Analysis
by Sara Mahesar, Muhammad Mujtaba Shaikh, Muhammad Saleem Chandio and Abdul Wasim Shaikh
Symmetry 2022, 14(12), 2611; https://doi.org/10.3390/sym14122611 - 9 Dec 2022
Cited by 2 | Viewed by 1359
Abstract
In this research, some new and efficient quadrature rules are proposed involving the combination of function and its first derivative evaluations at equally spaced data points with the main focus on their computational efficiency in terms of cost and time usage. The methods [...] Read more.
In this research, some new and efficient quadrature rules are proposed involving the combination of function and its first derivative evaluations at equally spaced data points with the main focus on their computational efficiency in terms of cost and time usage. The methods are theoretically derived, and theorems on the order of accuracy, degree of precision and error terms are proved. The proposed methods are semi-open-type rules with derivatives. The order of accuracy and degree of precision of the proposed methods are higher than the classical rules for which a systematic and symmetrical ascendancy has been proved. Various numerical tests are performed to compare the performance of the proposed methods with the existing methods in terms of accuracy, precision, leading local and global truncation errors, numerical convergence rates and computational cost with average CPU usage. In addition to the classical semi-open rules, the proposed methods have also been compared with some Gauss–Legendre methods for performance evaluation on various integrals involving some oscillatory, periodic and integrals with derivative singularities. The analysis of the results proves that the devised techniques are more efficient than the classical semi-open Newton–Cotes rules from theoretical and numerical perspectives because of promisingly reduced functional cost and lesser execution times. The proposed methods compete well with the spectral Gauss–Legendre rules, and in some cases outperform. Symmetric error distributions have been observed in regular cases of integrands, whereas asymmetrical behavior is evidenced in oscillatory and highly nonlinear cases. Full article
(This article belongs to the Special Issue Numerical Analysis, Approximation Theory, Differential Equations)
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13 pages, 408 KiB  
Article
A New Numerical Approach for Variable-Order Time-Fractional Modified Subdiffusion Equation via Riemann–Liouville Fractional Derivative
by Dowlath Fathima, Muhammad Naeem, Umair Ali, Abdul Hamid Ganie and Farah Aini Abdullah
Symmetry 2022, 14(11), 2462; https://doi.org/10.3390/sym14112462 - 21 Nov 2022
Cited by 2 | Viewed by 1995
Abstract
Fractional differential equations describe nature adequately because of the symmetry properties that describe physical and biological processes. In this paper, a new approximation is found for the variable-order (VO) Riemann–Liouville fractional derivative (RLFD) operator; on that basis, an efficient numerical approach is formulated [...] Read more.
Fractional differential equations describe nature adequately because of the symmetry properties that describe physical and biological processes. In this paper, a new approximation is found for the variable-order (VO) Riemann–Liouville fractional derivative (RLFD) operator; on that basis, an efficient numerical approach is formulated for VO time-fractional modified subdiffusion equations (TFMSDE). Complete theoretical analysis is performed, such as stability by the Fourier series, consistency, and convergence, and the feasibility of the proposed approach is also discussed. A numerical example illustrates that the proposed scheme demonstrates high accuracy, and that the obtained results are more feasible and accurate. Full article
(This article belongs to the Special Issue Numerical Analysis, Approximation Theory, Differential Equations)
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11 pages, 1867 KiB  
Article
Cheney–Sharma Type Operators on a Triangle with Straight Sides
by Teodora Cătinaş
Symmetry 2022, 14(11), 2446; https://doi.org/10.3390/sym14112446 - 18 Nov 2022
Cited by 2 | Viewed by 1233
Abstract
We consider two types of Cheney–Sharma operators for functions defined on a triangle with all straight sides. We construct their product and Boolean sum, we study their interpolation properties and the orders of accuracy and we give different expressions of the corresponding remainders, [...] Read more.
We consider two types of Cheney–Sharma operators for functions defined on a triangle with all straight sides. We construct their product and Boolean sum, we study their interpolation properties and the orders of accuracy and we give different expressions of the corresponding remainders, highlighting the symmetry between the methods. We also give some illustrative numerical examples. Full article
(This article belongs to the Special Issue Numerical Analysis, Approximation Theory, Differential Equations)
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23 pages, 742 KiB  
Article
Highly Accurate Compact Finite Difference Schemes for Two-Point Boundary Value Problems with Robin Boundary Conditions
by James Malele, Phumlani Dlamini and Simphiwe Simelane
Symmetry 2022, 14(8), 1720; https://doi.org/10.3390/sym14081720 - 18 Aug 2022
Cited by 7 | Viewed by 2439
Abstract
In this study, a high-order compact finite difference method is used to solve boundary value problems with Robin boundary conditions. The norm is to use a first-order finite difference scheme to approximate Neumann and Robin boundary conditions, but that compromises the accuracy of [...] Read more.
In this study, a high-order compact finite difference method is used to solve boundary value problems with Robin boundary conditions. The norm is to use a first-order finite difference scheme to approximate Neumann and Robin boundary conditions, but that compromises the accuracy of the entire scheme. As a result, new higher-order finite difference schemes for approximating Robin boundary conditions are developed in this work. Six examples for testing the applicability and performance of the method are considered. Convergence analysis is provided, and it is consistent with the numerical results. The results are compared with the exact solutions and published results from other methods. The method produces highly accurate results, which are displayed in tables and graphs. Full article
(This article belongs to the Special Issue Numerical Analysis, Approximation Theory, Differential Equations)
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7 pages, 578 KiB  
Article
A Constrained Shepard Type Operator for Modeling and Visualization of Scattered Data
by Teodora Cătinaş
Symmetry 2022, 14(2), 240; https://doi.org/10.3390/sym14020240 - 26 Jan 2022
Cited by 1 | Viewed by 1844
Abstract
For solving the problem of modeling and visualization of scattered data that should preserve some constraints, we use a modified Shepard type operator that is required to fulfill some special conditions, highlighting the symmetry with other methods. We illustrate the properties of the [...] Read more.
For solving the problem of modeling and visualization of scattered data that should preserve some constraints, we use a modified Shepard type operator that is required to fulfill some special conditions, highlighting the symmetry with other methods. We illustrate the properties of the obtained operators by some numerical examples. Full article
(This article belongs to the Special Issue Numerical Analysis, Approximation Theory, Differential Equations)
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Review

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18 pages, 905 KiB  
Review
A Review on Some Linear Positive Operators Defined on Triangles
by Teodora Cătinaş
Symmetry 2022, 14(9), 1880; https://doi.org/10.3390/sym14091880 - 8 Sep 2022
Viewed by 1331
Abstract
We consider results regarding Bernstein and Cheney–Sharma-type operators that interpolate functions defined on triangles with straight and curved sides and we introduce a new Cheney–Sharma-type operator for the triangle with one curved side, highlighting the symmetry between the methods. We present some properties [...] Read more.
We consider results regarding Bernstein and Cheney–Sharma-type operators that interpolate functions defined on triangles with straight and curved sides and we introduce a new Cheney–Sharma-type operator for the triangle with one curved side, highlighting the symmetry between the methods. We present some properties of the operators, their products and Boolean sums and some results regarding the remainders of the corresponding approximation formulas, using modulus of continuity and Peano’s theorem. Additionally, we consider some numerical examples to show the approximation properties of the given operators. Full article
(This article belongs to the Special Issue Numerical Analysis, Approximation Theory, Differential Equations)
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