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Numerical Analysis and Modeling
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Numerical Analysis and Modeling

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

Deadline for manuscript submissions: 31 July 2024 | Viewed by 14290

Special Issue Editors

Prof. Dr. Juan R. Torregrosa

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Guest Editor
Department of Applied Mathematics and Institute for Multidisciplinary Mathematics (im2), Universitat Politècnica de València, 46022 Valencia, 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, “Numerical Analysis and Modeling”, of Mathematics (MDPI) 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 of, and the use of 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 interests of numerical schemes include approximation, simulation, and estimation, and they are used in virtually every scientific field.

On the other hand, differential equations have been demonstrated to play a key role in modeling the dynamics of many phenomena belonging to different realms, including physics, chemistry, finance, social sciences, etc. Since their classical formulation, via ordinary derivatives, a number of other classes of differential equations have been proposed, such as delay, fractional, functional, or integro-differential equations. The mathematical and numerical analysis of all of these types of differential equations is still a hot topic in mathematics.

Some possible topics of interest include numerical stability, interpolation, approximation, quadrature methods, numerical linear algebra, ordinary differential equations, initial and boundary conditions, partial differential equations, numerical fractional analyses, 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 R. Torregrosa
Prof. Dr. Alicia Cordero Barbero
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
  • numerical methods for differential equations
  • random and stochastic differential equations
  • mathematical modeling using differential equations
  • optimization
  • discretization
  • iterative methods
  • stability theory

Published Papers (10 papers)

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Research

18 pages, 321 KiB  
Article
A One-Parameter Family of Methods with a Higher Order of Convergence for Equations in a Banach Space
by Ramandeep Behl, Ioannis K. Argyros and Sattam Alharbi
Mathematics 2024, 12(9), 1278; https://doi.org/10.3390/math12091278 - 23 Apr 2024
Viewed by 306
Abstract
The conventional approach of the local convergence analysis of an iterative method on Rm, with m a natural number, depends on Taylor series expansion. This technique often requires the calculation of high-order derivatives. However, those derivatives may not be part of [...] Read more.
The conventional approach of the local convergence analysis of an iterative method on Rm, with m a natural number, depends on Taylor series expansion. This technique often requires the calculation of high-order derivatives. However, those derivatives may not be part of the proposed method(s). In this way, the method(s) can face several limitations, particularly the use of higher-order derivatives and a lack of information about a priori computable error bounds on the solution distance or uniqueness. In this paper, we address these drawbacks by conducting the local convergence analysis within the broader framework of a Banach space. We have selected an important family of high convergence order methods to demonstrate our technique as an example. However, due to its generality, our technique can be used on any other iterative method using inverses of linear operators along the same line. Our analysis not only extends in Rm spaces but also provides convergence conditions based on the operators used in the method, which offer the applicability of the method in a broader area. Additionally, we introduce a novel semilocal convergence analysis not presented before in such studies. Both forms of convergence analysis depend on the concept of generalized continuity and provide a deeper understanding of convergence properties. Our methodology not only enhances the applicability of the suggested method(s) but also provides suitability for applied science problems. The computational results also support the theoretical aspects. Full article
(This article belongs to the Special Issue Numerical Analysis and Modeling)
16 pages, 414 KiB  
Article
A Class of Efficient Sixth-Order Iterative Methods for Solving the Nonlinear Shear Model of a Reinforced Concrete Beam
by José J. Padilla, Francisco I. Chicharro, Alicia Cordero, Alejandro M. Hernández-Díaz and Juan R. Torregrosa
Mathematics 2024, 12(3), 499; https://doi.org/10.3390/math12030499 - 05 Feb 2024
Viewed by 546
Abstract
In this paper, we present a three-step sixth-order class of iterative schemes to estimate the solutions of a nonlinear system of equations. This procedure is designed by means of a weight function technique. We apply this procedure for predicting the shear strength of [...] Read more.
In this paper, we present a three-step sixth-order class of iterative schemes to estimate the solutions of a nonlinear system of equations. This procedure is designed by means of a weight function technique. We apply this procedure for predicting the shear strength of a reinforced concrete beam. The values for the parameters of the nonlinear system describing this problem were randomly selected inside the prescribed ranges by technical standards for structural concrete. Moreover, some of these parameters were fixed taking into consideration the solvability region of the adopted steel constitutive model. The effectiveness of the new class is also compared with other current schemes in terms of the computational efficiency and numerical performance, with very good results. The advantages of this new class come from the low computational cost, due to the existence of an only inverse operator. Full article
(This article belongs to the Special Issue Numerical Analysis and Modeling)
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17 pages, 1822 KiB  
Article
A Modified Structured Spectral HS Method for Nonlinear Least Squares Problems and Applications in Robot Arm Control
by Rabiu Bashir Yunus, Nooraini Zainuddin, Hanita Daud, Ramani Kannan, Samsul Ariffin Abdul Karim and Mahmoud Muhammad Yahaya
Mathematics 2023, 11(14), 3215; https://doi.org/10.3390/math11143215 - 21 Jul 2023
Viewed by 892
Abstract
This paper proposes a modification to the Hestenes-Stiefel (HS) method by devising a spectral parameter using a modified secant relation to solve nonlinear least-squares problems. Notably, in the implementation, the proposed method differs from existing approaches, in that it does not require a [...] Read more.
This paper proposes a modification to the Hestenes-Stiefel (HS) method by devising a spectral parameter using a modified secant relation to solve nonlinear least-squares problems. Notably, in the implementation, the proposed method differs from existing approaches, in that it does not require a safeguarding strategy and its Hessian matrix is positive and definite throughout the iteration process. Numerical experiments are conducted on a range of test problems, with 120 instances to demonstrate the efficacy of the proposed algorithm by comparing it with existing techniques in the literature. However, the results obtained validate the effectiveness of the proposed method in terms of the standard metrics of comparison. Additionally, the proposed approach is applied to address motion-control problems in a robotic model, resulting in favorable outcomes in terms of the robot’s motion characteristics. Full article
(This article belongs to the Special Issue Numerical Analysis and Modeling)
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12 pages, 636 KiB  
Article
Investigation of Higher Order Localized Approximations for a Fractional Pricing Model in Finance
by Malik Zaka Ullah, Abdullah Khamis Alzahrani, Hashim Mohammed Alshehri and Stanford Shateyi
Mathematics 2023, 11(12), 2641; https://doi.org/10.3390/math11122641 - 09 Jun 2023
Viewed by 642
Abstract
In this work, by considering spatial uniform meshes and stencils having five adjacent discretization nodes, we furnish a numerical scheme to solve the time-fractional Black–Scholes (partial differential equation) PDE to price financial options under the generalized multiquadric radial basis function (RBF). The time-fractional [...] Read more.
In this work, by considering spatial uniform meshes and stencils having five adjacent discretization nodes, we furnish a numerical scheme to solve the time-fractional Black–Scholes (partial differential equation) PDE to price financial options under the generalized multiquadric radial basis function (RBF). The time-fractional derivative is estimated by an L1-scheme but the spatial variable is discretized using fourth-order RBF-FD methodology. As a matter of fact, the PDE problem is transformed in the form of a linear set of algebraic equations. To support analytical discussions, numerical tests are furnished and reveal the efficacy of the presented solver. Full article
(This article belongs to the Special Issue Numerical Analysis and Modeling)
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17 pages, 338 KiB  
Article
The Existence Problems of Solutions for a Class of Differential Variational–Hemivariational Inequality Problems
by Shih-Sen Chang, Salahuddin, A. A. H. Ahmadini, Lin Wang and Gang Wang
Mathematics 2023, 11(9), 2066; https://doi.org/10.3390/math11092066 - 26 Apr 2023
Viewed by 1024
Abstract
In this work, we used reflexive Banach spaces to study the differential variational—hemivariational inequality problems with constraints. We established a sequence of perturbed differential variational–hemivariational inequality problems with perturbed constraints and penalty coefficients. Then, for each perturbed inequality, we proved the unique solvability [...] Read more.
In this work, we used reflexive Banach spaces to study the differential variational—hemivariational inequality problems with constraints. We established a sequence of perturbed differential variational–hemivariational inequality problems with perturbed constraints and penalty coefficients. Then, for each perturbed inequality, we proved the unique solvability and convergence of the solutions to the problems. Following that, we proposed a mathematical model for a viscoelastic rod in unilateral contact equilibrium, where the unknowns were the displacement field and the history of the deformation. We used the abstract penalty method in the analysis of this inequality and provided the corresponding mechanical interpretations. Full article
(This article belongs to the Special Issue Numerical Analysis and Modeling)
15 pages, 727 KiB  
Article
Performance of a New Sixth-Order Class of Iterative Schemes for Solving Non-Linear Systems of Equations
by Marlon Moscoso-Martínez, Francisco I. Chicharro, Alicia Cordero and Juan R. Torregrosa
Mathematics 2023, 11(6), 1374; https://doi.org/10.3390/math11061374 - 12 Mar 2023
Cited by 1 | Viewed by 1117
Abstract
This manuscript is focused on a new parametric class of multi-step iterative procedures to find the solutions of systems of nonlinear equations. Starting from Ostrowski’s scheme, the class is constructed by adding a Newton step with a Jacobian matrix taken from the previous [...] Read more.
This manuscript is focused on a new parametric class of multi-step iterative procedures to find the solutions of systems of nonlinear equations. Starting from Ostrowski’s scheme, the class is constructed by adding a Newton step with a Jacobian matrix taken from the previous step and employing a divided difference operator, resulting in a triparametric scheme with a convergence order of four. The convergence order of the family can be accelerated to six by setting two parameters, resulting in a uniparametric family. We performed dynamic and numerical development to analyze the stability of the sixth-order family. Previous studies for scalar functions allow us to isolate those elements of the family with stable performance for solving practical problems. In this regard, we present dynamical planes showing the complexity of the family. In addition, the numerical properties of the class are analyzed with several test problems. Full article
(This article belongs to the Special Issue Numerical Analysis and Modeling)
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9 pages, 2511 KiB  
Article
Comparative Study of Numerical Methods for Solving the Fresnel Integral in Aperiodic Diffractive Lenses
by Adrián Garmendía-Martínez, Francisco M. Muñoz-Pérez, Walter D. Furlan, Fernando Giménez, Juan C. Castro-Palacio, Juan A. Monsoriu and Vicente Ferrando
Mathematics 2023, 11(4), 946; https://doi.org/10.3390/math11040946 - 13 Feb 2023
Cited by 2 | Viewed by 1613
Abstract
In this work, we present a comparative analysis of different numerical methods to obtain the focusing properties of the zone plates based on Fibonacci and Cantor sequences. The Fresnel approximation was solved numerically in order to obtain the axial irradiance provided by these [...] Read more.
In this work, we present a comparative analysis of different numerical methods to obtain the focusing properties of the zone plates based on Fibonacci and Cantor sequences. The Fresnel approximation was solved numerically in order to obtain the axial irradiance provided by these diffractive lenses. Two different methods were applied. The first one is based on numerical integration, specifically the Simpson integration method and the two-dimensional Gaussian quadrature. The second consisted in the implementation of the Fast Fourier Transform in both one and two dimensions. The axial irradiance of the lenses, the relative error with respect to the analytical solution, and the calculation time required by each method are analyzed and compared. From this analysis it was concluded that the Gauss method presents the best balance between accuracy and computation time. This analysis could be useful to decide the most convenient numerical method to be used for the study of more complex diffractive structures. Full article
(This article belongs to the Special Issue Numerical Analysis and Modeling)
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18 pages, 338 KiB  
Article
Extending the Applicability of Highly Efficient Iterative Methods for Nonlinear Equations and Their Applications
by Ramandeep Behl, Ioannis K. Argyros, Fouad Othman Mallawi and Samaher Khalaf Alharbi
Mathematics 2023, 11(1), 125; https://doi.org/10.3390/math11010125 - 27 Dec 2022
Viewed by 1019
Abstract
Numerous three-step methods of high convergence order have been developed to produce sequences approximating solutions of equations usually defined on the Euclidean space with a finite dimension. The local convergence order is determined by Taylor expansions requiring the existence of derivatives that are [...] Read more.
Numerous three-step methods of high convergence order have been developed to produce sequences approximating solutions of equations usually defined on the Euclidean space with a finite dimension. The local convergence order is determined by Taylor expansions requiring the existence of derivatives that are not present on the methods. The more interesting semi-local convergence analysis for these methods has not been considered before. The semi-local is also provided based on generalized ω-continuity conditions on the derivative of the operator involved and the majorizing sequences, thus limiting their usage to only solving equations with operators that are many times differentiable. However, these methods may convergence to a solution of the equation even if these high-order derivatives do not exist. That is why a methodology is utilized on two sixth convergence order methods and in the more general setting of a Banach space. This time, the convergence depends only on the operators and the first derivative on the method. Therefore, by this methodology the applicability of the methods is in the extended area. Although this methodology is demonstrated on two competing and efficient methods, it can also be utilized for the same reasons on other methods involving inverses of operators that are linear. This is the motivation and novelty of the paper. The numerical applications further validate the theoretical results both in the local as well as the semi-local convergence case. Full article
(This article belongs to the Special Issue Numerical Analysis and Modeling)
17 pages, 331 KiB  
Article
A Polynomial Fitting Problem: The Orthogonal Distances Method
by Luis Alberto Cantera-Cantera, Cristóbal Vargas-Jarillo, Sergio Isaí Palomino-Reséndiz, Yair Lozano-Hernández and Carlos Manuel Montelongo-Vázquez
Mathematics 2022, 10(23), 4596; https://doi.org/10.3390/math10234596 - 04 Dec 2022
Cited by 1 | Viewed by 4025
Abstract
The classical curve-fitting problem to relate two variables, x and y, deals with polynomials. Generally, this problem is solved by the least squares method (LS), where the minimization function considers the vertical errors from the data points to the fitting curve. Another [...] Read more.
The classical curve-fitting problem to relate two variables, x and y, deals with polynomials. Generally, this problem is solved by the least squares method (LS), where the minimization function considers the vertical errors from the data points to the fitting curve. Another curve-fitting method is total least squares (TLS), which takes into account errors in both x and y variables. A further method is the orthogonal distances method (OD), which minimizes the sum of the squares of orthogonal distances from the data points to the fitting curve. In this work, we develop the OD method for the polynomial fitting of degree n and compare the TLS and OD methods. The results show that TLS and OD methods are not equivalent in general; however, both methods get the same estimates when a polynomial of degree 1 without an independent coefficient is considered. As examples, we consider the calibration curve-fitting problem of a R-type thermocouple by polynomials of degrees 1 to 4, with and without an independent coefficient, using the LS, TLS and OD methods. Full article
(This article belongs to the Special Issue Numerical Analysis and Modeling)
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13 pages, 1008 KiB  
Article
On the Conditional Value at Risk Based on the Laplace Distribution with Application in GARCH Model
by Malik Zaka Ullah, Fouad Othman Mallawi, Mir Asma and Stanford Shateyi
Mathematics 2022, 10(16), 3018; https://doi.org/10.3390/math10163018 - 22 Aug 2022
Cited by 4 | Viewed by 1756
Abstract
In this article, the Laplace distribution is employed in lieu of the well-known normal distribution for finding better scalar values of risk. Explicit formulas for value-at-risk (VaR) and conditional value-at-risk (CVaR) are studied and used to manage the risk involved in a stock [...] Read more.
In this article, the Laplace distribution is employed in lieu of the well-known normal distribution for finding better scalar values of risk. Explicit formulas for value-at-risk (VaR) and conditional value-at-risk (CVaR) are studied and used to manage the risk involved in a stock movement by using the GARCH model. Numerical simulations are given for a variety of stocks in equity markets to uphold the findings. Full article
(This article belongs to the Special Issue Numerical Analysis and Modeling)
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