Numerical Methods II

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 March 2022) | Viewed by 5432

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Department of Mathematics, Technical University of Cluj-Napoca, 400641 Cluj-Napoca, Romania
Interests: wavelet analysis
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Special Issue Information

Dear Colleagues,

This Special Issue, “Numerical Methods II”, is open for submissions and welcomes papers from a broad interdisciplinary area, since ‘numerical methods’ are a specific form of mathematics that involves the creating and use of algorithms to map out the mathematical core of a practical problem.

Numerical methods naturally find application in all fields of engineering, physical sciences, life sciences, social sciences, medicine, business, and even arts. The common uses of numerical methods include approximation, simulation, and estimation, and there is almost no scientific field in which numerical methods do not find use.

Some subjects included in ‘numerical methods’ are IEEE arithmetic, root finding, systems of equations, least-squares estimation, maximum likelihood estimation, interpolation, numeric integration and differentiation—the list may go on and on. MSC 2010 subject classification for numerical methods includes 30C30 (in conformal mapping theory), 31C20 (in connection with discrete potential theory), and 60H35 (computational methods for stochastic equations), and most of the subjects in 37Mxx (approximation methods and numerical treatment of dynamical systems), 49Mxx (numerical methods), 65XX (numerical analysis), 74Sxx (numerical methods for deformable solids), 76Mxx (basic methods in fluid mechanics), 78Mxx (basic methods for optics and electromagnetic theory), 80Mxx (basic methods for classical thermodynamics and heat transfer), 82Bxx (equilibrium statistical mechanics), 82Cxx (time-dependent statistical mechanics), and 91Gxx (mathematical finance). Topics of interest include (but are not limited to) the numerical methods for approximation, simulation, and estimation.

Prof. Dr. Lorentz Jäntschi
Prof. Daniela Roșca
Guest Editors

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Keywords

  • Asymptotic stability
  • Boundary element method
  • Diffusion
  • Elasticity
  • Errors
  • Finite element method
  • Flow of fluids
  • Hydrodynamics
  • Image processing
  • Integration
  • Markov processes
  • Monte Carlo methods
  • Numerical algorithms
  • Numerical methods
  • Optimization
  • Partial differential equations
  • Robustness
  • Simulation
  • Stress analysis
  • System stability
  • Turbulence
  • Uncertainty analysis
  • Vibrations
  • Wavelets

Published Papers (3 papers)

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Research

15 pages, 303 KiB  
Article
The Mathematical Simulation for the Photocatalytic Fatigue of Polymer Nanocomposites Using the Monte Carlo Methods
by Andrey V. Orekhov, Yurii M. Artemev and Galina V. Pavilaynen
Mathematics 2022, 10(9), 1613; https://doi.org/10.3390/math10091613 - 09 May 2022
Cited by 1 | Viewed by 1331
Abstract
We consider an approach to mathematical modeling of photodegradation of polymer nanocomposites with photoactive additives using the Monte Carlo methods. We principally pay attention to the strength decrease of these materials under solar light action. We propose a new term, “photocatalytic fatigue”, which [...] Read more.
We consider an approach to mathematical modeling of photodegradation of polymer nanocomposites with photoactive additives using the Monte Carlo methods. We principally pay attention to the strength decrease of these materials under solar light action. We propose a new term, “photocatalytic fatigue”, which we apply to the particular case when the mechanical strength decreases only owing to the presence of photocatalytically active components in polymeric nanocomposite material. The propriety of the term is based on a relative similarity of photostimulated mechanical destructive processes in nanocomposites with photoactive additives and mechanical destructive processes typical for metal high-cycle fatigue. Formation of the stress concentrations is one of the major causes of fatigue cracks generation in metals. Photocatalytic active nanoparticles of semiconductors initiate a generation of the stress concentrations under sunlight irradiation. The proposed mathematical model is a Wöhler curve analog for the metal high-cycle fatigue. We assume that equations for high-cycle fatigue curves of samples with stress concentrations could be used in mathematical modeling of polymer nanocomposites photodegradation. In this way, we replace the number of loading cycles with the exposition time in the equations. In the case of polypropylene and polyester samples with photoactive titanium dioxide, the experimental parameters of phenomenological equations for “photocatalytic fatigue” are calculated using one of the Monte Carlo methods based on the random search algorithm. The calculating scheme includes a solution of the extreme task of finding of the minimum of nonnegative transcendent multivariable function, which is a relative average quadratic deviation of calculated values of polymeric nanocomposite stress in comparison with corresponding experimental values. The applicability of the “photocatalytic fatigue” model for polymer nanocomposites with photoactive nanoparticles is confirmed by the example of polypropylene and polyester samples. The approximation error of the experimental strength values for them did not exceed 2%. Full article
(This article belongs to the Special Issue Numerical Methods II)
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13 pages, 343 KiB  
Article
Approximate Solutions for a Class of Nonlinear Fredholm and Volterra Integro-Differential Equations Using the Polynomial Least Squares Method
by Bogdan Căruntu and Mădălina Sofia Paşca
Mathematics 2021, 9(21), 2692; https://doi.org/10.3390/math9212692 - 23 Oct 2021
Cited by 3 | Viewed by 1500
Abstract
We apply the polynomial least squares method to obtain approximate analytical solutions for a very general class of nonlinear Fredholm and Volterra integro-differential equations. The method is a relatively simple and straightforward one, but its precision for this type of equations is very [...] Read more.
We apply the polynomial least squares method to obtain approximate analytical solutions for a very general class of nonlinear Fredholm and Volterra integro-differential equations. The method is a relatively simple and straightforward one, but its precision for this type of equations is very high, a fact that is illustrated by the numerical examples presented. The comparison with previous approximations computed for the included test problems emphasizes the method’s simplicity and accuracy. Full article
(This article belongs to the Special Issue Numerical Methods II)
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17 pages, 1520 KiB  
Article
ABS-Based Direct Method for Solving Complex Systems of Linear Equations
by József Abaffy and Szabina Fodor
Mathematics 2021, 9(19), 2527; https://doi.org/10.3390/math9192527 - 08 Oct 2021
Cited by 1 | Viewed by 1593
Abstract
Efficient solution of linear systems of equations is one of the central topics of numerical computation. Linear systems with complex coefficients arise from various physics and quantum chemistry problems. In this paper, we propose a novel ABS-based algorithm, which is able to solve [...] Read more.
Efficient solution of linear systems of equations is one of the central topics of numerical computation. Linear systems with complex coefficients arise from various physics and quantum chemistry problems. In this paper, we propose a novel ABS-based algorithm, which is able to solve complex systems of linear equations. Theoretical analysis is given to highlight the basic features of our new algorithm. Four variants of our algorithm were also implemented and intensively tested on randomly generated full and sparse matrices and real-life problems. The results of numerical experiments reveal that our ABS-based algorithm is able to compute the solution with high accuracy. The performance of our algorithm was compared with a commercially available software, Matlab’s mldivide (\) algorithm. Our algorithm outperformed the Matlab algorithm in most cases in terms of computational accuracy. These results expand the practical usefulness of our algorithm. Full article
(This article belongs to the Special Issue Numerical Methods II)
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