Special Issue "Numerical Methods"

A special issue of Mathematics (ISSN 2227-7390).

Deadline for manuscript submissions: 30 June 2020.

Special Issue Editors

Prof. Lorentz Jäntschi
Website
Guest Editor
Department of Physics and Chemistry, Technical University of Cluj-Napoca, Romania
Interests: applied mathematics; applied informatics
Special Issues and Collections in MDPI journals
Prof. Daniela Roșca
Website
Guest Editor
Department of Mathematics, Technical University of Cluj-Napoca, Romania
Interests: wavelet analysis

Special Issue Information

Dear Colleagues,

This Special Issue, “Numerical Methods” is open for submissions and welcomes papers from a broad interdisciplinary area, since ‘numerical methods’ are a specific form of mathematics that involves 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 a 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. Lorentz Jäntschi
Prof. Daniela Daniela Roșca
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 papers will be 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 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 1200 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

  • 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 (6 papers)

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Research

Open AccessArticle
On q-Quasi-Newton’s Method for Unconstrained Multiobjective Optimization Problems
Mathematics 2020, 8(4), 616; https://doi.org/10.3390/math8040616 - 17 Apr 2020
Abstract
A parameter-free optimization technique is applied in Quasi-Newton’s method for solving unconstrained multiobjective optimization problems. The components of the Hessian matrix are constructed using q-derivative, which is positive definite at every iteration. The step-length is computed by an Armijo-like rule which is [...] Read more.
A parameter-free optimization technique is applied in Quasi-Newton’s method for solving unconstrained multiobjective optimization problems. The components of the Hessian matrix are constructed using q-derivative, which is positive definite at every iteration. The step-length is computed by an Armijo-like rule which is responsible to escape the point from local minimum to global minimum at every iteration due to q-derivative. Further, the rate of convergence is proved as a superlinear in a local neighborhood of a minimum point based on q-derivative. Finally, the numerical experiments show better performance. Full article
(This article belongs to the Special Issue Numerical Methods)
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Open AccessArticle
Detecting Extreme Values with Order Statistics in Samples from Continuous Distributions
Mathematics 2020, 8(2), 216; https://doi.org/10.3390/math8020216 - 08 Feb 2020
Cited by 1
Abstract
In the subject of statistics for engineering, physics, computer science, chemistry, and earth sciences, one of the sampling challenges is the accuracy, or, in other words, how representative the sample is of the population from which it was drawn. A series of statistics [...] Read more.
In the subject of statistics for engineering, physics, computer science, chemistry, and earth sciences, one of the sampling challenges is the accuracy, or, in other words, how representative the sample is of the population from which it was drawn. A series of statistics were developed to measure the departure between the population (theoretical) and the sample (observed) distributions. Another connected issue is the presence of extreme values—possible observations that may have been wrongly collected—which do not belong to the population selected for study. By subjecting those two issues to study, we hereby propose a new statistic for assessing the quality of sampling intended to be used for any continuous distribution. Depending on the sample size, the proposed statistic is operational for known distributions (with a known probability density function) and provides the risk of being in error while assuming that a certain sample has been drawn from a population. A strategy for sample analysis, by analyzing the information about quality of the sampling provided by the order statistics in use, is proposed. A case study was conducted assessing the quality of sampling for ten cases, the latter being used to provide a pattern analysis of the statistics. Full article
(This article belongs to the Special Issue Numerical Methods)
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Open AccessArticle
Exact Solutions to the Maxmin Problem max‖Ax‖ Subject to ‖Bx‖≤1
Mathematics 2020, 8(1), 85; https://doi.org/10.3390/math8010085 - 04 Jan 2020
Cited by 1
Abstract
In this manuscript we provide an exact solution to the maxmin problem maxAx subject to Bx1, where A and B are real matrices. This problem comes from a remodeling of maxA [...] Read more.
In this manuscript we provide an exact solution to the maxmin problem max A x subject to B x 1 , where A and B are real matrices. This problem comes from a remodeling of max A x subject to min B x , because the latter problem has no solution. Our mathematical method comes from the Abstract Operator Theory, whose strong machinery allows us to reduce the first problem to max C x subject to x 1 , which can be solved exactly by relying on supporting vectors. Finally, as appendices, we provide two applications of our solution: first, we construct a truly optimal minimum stored-energy Transcranian Magnetic Stimulation (TMS) coil, and second, we find an optimal geolocation involving statistical variables. Full article
(This article belongs to the Special Issue Numerical Methods)
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Open AccessArticle
Finite Integration Method with Shifted Chebyshev Polynomials for Solving Time-Fractional Burgers’ Equations
Mathematics 2019, 7(12), 1201; https://doi.org/10.3390/math7121201 - 07 Dec 2019
Abstract
The Burgers’ equation is one of the nonlinear partial differential equations that has been studied by many researchers, especially, in terms of the fractional derivatives. In this article, the numerical algorithms are invented to obtain the approximate solutions of time-fractional Burgers’ equations both [...] Read more.
The Burgers’ equation is one of the nonlinear partial differential equations that has been studied by many researchers, especially, in terms of the fractional derivatives. In this article, the numerical algorithms are invented to obtain the approximate solutions of time-fractional Burgers’ equations both in one and two dimensions as well as time-fractional coupled Burgers’ equations which their fractional derivatives are described in the Caputo sense. These proposed algorithms are constructed by applying the finite integration method combined with the shifted Chebyshev polynomials to deal the spatial discretizations and further using the forward difference quotient to handle the temporal discretizations. Moreover, numerical examples demonstrate the ability of the proposed method to produce the decent approximate solutions in terms of accuracy. The rate of convergence and computational cost for each example are also presented. Full article
(This article belongs to the Special Issue Numerical Methods)
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Open AccessArticle
Convergence Analysis and Complex Geometry of an Efficient Derivative-Free Iterative Method
Mathematics 2019, 7(10), 919; https://doi.org/10.3390/math7100919 - 02 Oct 2019
Abstract
To locate a locally-unique solution of a nonlinear equation, the local convergence analysis of a derivative-free fifth order method is studied in Banach space. This approach provides radius of convergence and error bounds under the hypotheses based on the first Fréchet-derivative only. Such [...] Read more.
To locate a locally-unique solution of a nonlinear equation, the local convergence analysis of a derivative-free fifth order method is studied in Banach space. This approach provides radius of convergence and error bounds under the hypotheses based on the first Fréchet-derivative only. Such estimates are not introduced in the earlier procedures employing Taylor’s expansion of higher derivatives that may not exist or may be expensive to compute. The convergence domain of the method is also shown by a visual approach, namely basins of attraction. Theoretical results are endorsed via numerical experiments that show the cases where earlier results cannot be applicable. Full article
(This article belongs to the Special Issue Numerical Methods)
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Open AccessArticle
Numerical Solution of the Cauchy-Type Singular Integral Equation with a Highly Oscillatory Kernel Function
Mathematics 2019, 7(10), 872; https://doi.org/10.3390/math7100872 - 20 Sep 2019
Cited by 2
Abstract
This paper aims to present a Clenshaw–Curtis–Filon quadrature to approximate the
solution of various cases of Cauchy-type singular integral equations (CSIEs) of the second kind with
a highly oscillatory kernel function. We adduce that the zero case oscillation (k = 0) proposed [...] Read more.
This paper aims to present a Clenshaw–Curtis–Filon quadrature to approximate the
solution of various cases of Cauchy-type singular integral equations (CSIEs) of the second kind with
a highly oscillatory kernel function. We adduce that the zero case oscillation (k = 0) proposed method
gives more accurate results than the scheme introduced in Dezhbord at el. (2016) and Eshkuvatov
at el. (2009) for small values of N. Finally, this paper illustrates some error analyses and numerical
results for CSIEs. Full article
(This article belongs to the Special Issue Numerical Methods)
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