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Mathematics
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Theory and Applications of Numerical Analysis
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Theory and Applications of Numerical Analysis

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (29 February 2024) | Viewed by 5988

Special Issue Editor

Prof. Dr. Riccardo Fazio

E-Mail Website
Guest Editor
Department of Mathematics, Computer Science, PhysicalSciences and Earth Sciences, University of Messina, Viale F. StagnoD’Alcontres 31, 98166 Messina, Italy
Interests: numerical analysis; numerical methods for ODEs and PDEs

Special Issue Information

Dear Colleagues,

Mathematical modeling of real-life problems usually results in functional equations, such as ordinary or partial differential equations, integral and integrodifferential equations, and stochastic equations. Ordinary and partial differential equations (PDEs) have become useful tools for describing the natural phenomena of science and engineering models. Therefore, it is becoming increasingly important to be familiar with all traditional and recently developed methods for solving ordinary and partial differential equations and the implementations of these methods.

The aim of this Special Issue is to bring together the leading researchers of Mathematics related to the numerical solution of ordinary and partial differential problems in order for them to share their original research work. Most welcomed will be papers concerning the definition and application of new methods to the numerical solution of the problems of interest. Particularly, recent developments of free, moving or interface problems or problems defined on infinite intervals and domains will be included in the main focus of the issue.

Prof. Dr. Riccardo Fazio
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. 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 methods
  • ordinary differential problems
  • partial differential problems
  • free boundary problems
  • moving boundary problems
  • interface problems
  • problems defined on infinite intervals or domains

Published Papers (5 papers)

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Research

30 pages, 548 KiB  
Article
On Highly Efficient Fractional Numerical Method for Solving Nonlinear Engineering Models
by Mudassir Shams and Bruno Carpentieri
Mathematics 2023, 11(24), 4914; https://doi.org/10.3390/math11244914 - 10 Dec 2023
Cited by 1 | Viewed by 679
Abstract
We proposed and analyzed the fractional simultaneous technique for approximating all the roots of nonlinear equations in this research study. The newly developed fractional Caputo-type simultaneous scheme’s order of convergence is 3ς+5, according to convergence analysis. Engineering-related numerical test [...] Read more.
We proposed and analyzed the fractional simultaneous technique for approximating all the roots of nonlinear equations in this research study. The newly developed fractional Caputo-type simultaneous scheme’s order of convergence is 3ς+5, according to convergence analysis. Engineering-related numerical test problems are taken into consideration to demonstrate the efficiency and stability of fractional numerical schemes when compared to previously published numerical iterative methods. The newly developed fractional simultaneous approach converges on random starting guess values at random times, demonstrating its global convergence behavior. Although the newly developed method shows global convergent behavior when all starting guess values are distinct, the method diverges otherwise. The total computational time, number of iterations, error graphs and maximum residual error all clearly illustrate the stability and consistency of the developed scheme. The rate of convergence increases as the fractional parameter’s value rises from 0.1 to 1.0. Full article
(This article belongs to the Special Issue Theory and Applications of Numerical Analysis)
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14 pages, 629 KiB  
Article
Dynamics and Bifurcations of a Discrete-Time Moran-Ricker Model with a Time Delay
by Bo Li, Zimeng Yuan and Zohreh Eskandari
Mathematics 2023, 11(11), 2446; https://doi.org/10.3390/math11112446 - 25 May 2023
Cited by 2 | Viewed by 1071
Abstract
This study investigates the dynamics of limited homogeneous populations based on the Moran-Ricker model with time delay. The delay in density dependence caused the preceding generation to consume fewer resources, leading to a decrease in the required resources. Multimodality is evident in the [...] Read more.
This study investigates the dynamics of limited homogeneous populations based on the Moran-Ricker model with time delay. The delay in density dependence caused the preceding generation to consume fewer resources, leading to a decrease in the required resources. Multimodality is evident in the model. Some insect species can be described by the Moran–Ricker model with a time delay. Bifurcations associated with flipping, doubling, and Neimark–Sacker for codimension-one (codim-1) model can be analyzed using bifurcation theory and the normal form method. We also investigate codimension-two (codim-2) bifurcations corresponding to 1:2, 1:3, and 1:4 resonances. In addition to demonstrating the accuracy of theoretical results, numerical simulations are obtained using bifurcation diagrams and phase portraits. Full article
(This article belongs to the Special Issue Theory and Applications of Numerical Analysis)
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8 pages, 274 KiB  
Article
Gauss Quadrature Method for System of Absolute Value Equations
by Lei Shi, Javed Iqbal, Faiqa Riaz and Muhammad Arif
Mathematics 2023, 11(9), 2069; https://doi.org/10.3390/math11092069 - 27 Apr 2023
Cited by 1 | Viewed by 1058
Abstract
In this paper, an iterative method was considered for solving the absolute value equation (AVE). We suggest a two-step method in which the well-known Gauss quadrature rule is the corrector step and the generalized Newton method is taken as the predictor step. The [...] Read more.
In this paper, an iterative method was considered for solving the absolute value equation (AVE). We suggest a two-step method in which the well-known Gauss quadrature rule is the corrector step and the generalized Newton method is taken as the predictor step. The convergence of the proposed method is established under some acceptable conditions. Numerical examples prove the consistency and capability of this new method. Full article
(This article belongs to the Special Issue Theory and Applications of Numerical Analysis)
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12 pages, 425 KiB  
Article
Ansatz and Averaging Methods for Modeling the (Un)Conserved Complex Duffing Oscillators
by Weaam Alhejaili, Alvaro H. Salas and Samir A. El-Tantawy
Mathematics 2023, 11(9), 2007; https://doi.org/10.3390/math11092007 - 24 Apr 2023
Cited by 1 | Viewed by 933
Abstract
In this study, both the ansatz and averaging methods are carried out for analyzing the complex Duffing oscillators including the undamped/conserved complex Duffing oscillator (CDO) and the damped/unconserved CDO to obtain some approximate analytical solutions. To analyze the conserved CDO, it is reduced [...] Read more.
In this study, both the ansatz and averaging methods are carried out for analyzing the complex Duffing oscillators including the undamped/conserved complex Duffing oscillator (CDO) and the damped/unconserved CDO to obtain some approximate analytical solutions. To analyze the conserved CDO, it is reduced to two decoupled conserved Duffing oscillators. After that, the exact solution of the conserved Duffing oscillator is employed to derive an approximation of the conserved CDO in terms of the Jacobi elliptic function. To analyze the damped CDO, two methodologies are considered. For the first methodology, the damped CDO is reduced to two decoupled damped Duffing oscillators, and the ansatz method is devoted to analyzing the damped Duffing oscillator. Accordingly, an approximation of the damped CDO in terms of trigonometric functions is obtained. In the second methodology, the averaging method is applied directly to the damped CDO to derive an approximation in terms of trigonometric functions. All the obtained solutions are compared with the fourth-order Runge–Kutta (RK4) numerical approximations. This study may help many researchers interested in the field of plasma physics to interpret their laboratory and observations results. Full article
(This article belongs to the Special Issue Theory and Applications of Numerical Analysis)
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19 pages, 6616 KiB  
Article
An Improved Material Point Method with Aggregated and Smoothed Bernstein Functions
by Zheng Zhu, Tengfei Bao, Xi Zhu, Jian Gong, Yuhan Hu and Jingying Zhang
Mathematics 2023, 11(4), 907; https://doi.org/10.3390/math11040907 - 10 Feb 2023
Viewed by 1563
Abstract
Nodal shape functions and their gradients are vital in transferring physical information within the material point method (MPM). Their continuity is related to numerical stability and accuracy, and their support domain size affects computational efficiency. In this paper, a scheme of aggregated and [...] Read more.
Nodal shape functions and their gradients are vital in transferring physical information within the material point method (MPM). Their continuity is related to numerical stability and accuracy, and their support domain size affects computational efficiency. In this paper, a scheme of aggregated and smoothed Bernstein functions is proposed to improve the MPM. In detail, the Bernstein polynomials are smoothed with a convolution reformation to eliminate the cell crossing error, and an aggregation strategy is implemented to cut down the node amount required for field probing. Hierarchical MPM variants are obtained with choices of original Bernstein polynomials and degrees of smoothing. Numerical examples show that mass, momentum, and energy conservations are all well met, and no cell crossing noise exists. In addition, solution accuracy and numerical stability are significantly improved in large deformation problems. Full article
(This article belongs to the Special Issue Theory and Applications of Numerical Analysis)
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