Special Issue "Numerical Analysis or Numerical Method in Symmetry"

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: closed (31 May 2019).

Special Issue Editor

Guest Editor
Prof. Clemente Cesarano Website E-Mail
Section of Mathematics – International Telematic University UNINETTUNO
Phone: +39.0669207675
Interests: special functions; orthogonal polynomials; fractional calculus; numerical methods; ODE and PDE

Special Issue Information

Dear Colleagues,

Numerical methods and, in particular, numerical analysis represent an important field of investigation in modern mathematical research. In recent years, numerical analysis has undertaken a various lines of application in different areas of applied mathematics and, moreover, in applied sciences, such as biology, physics, engineering, and so on. However, part of the research on the topic of numerical analysis cannot exclude the fundamental role played by approximation theory and some of the tools used to develop this research. In this Special Issue, we want to draw attention to mathematical methods used in numerical analysis, such as special functions, orthogonal polynomials and their theoretical instruments, such as Lie algebra, to investigate the concepts and properties of some special and adavanced methods that are useful in the description of solutions of linear and non-linear differential equations. A further field of investigation is devoted to the theory and related properties of fractional calculus with its suitable application to numerical methods.

Prof. Clemente Cesarano
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 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.

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1400 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

  • Special Functions
  • Orthogonal Polynomials
  • Fractional Calculus
  • Numerical Methods
  • ODE and PDE

Published Papers (13 papers)

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Research

Open AccessArticle
Numerical Simulation of Pattern Formation on Surfaces Using an Efficient Linear Second-Order Method
Symmetry 2019, 11(8), 1010; https://doi.org/10.3390/sym11081010 - 05 Aug 2019
Abstract
We present an efficient linear second-order method for a Swift–Hohenberg (SH) type of a partial differential equation having quadratic-cubic nonlinearity on surfaces to simulate pattern formation on surfaces numerically. The equation is symmetric under a change of sign of the density field if [...] Read more.
We present an efficient linear second-order method for a Swift–Hohenberg (SH) type of a partial differential equation having quadratic-cubic nonlinearity on surfaces to simulate pattern formation on surfaces numerically. The equation is symmetric under a change of sign of the density field if there is no quadratic nonlinearity. We introduce a narrow band neighborhood of a surface and extend the equation on the surface to the narrow band domain. By applying a pseudo-Neumann boundary condition through the closest point, the Laplace–Beltrami operator can be replaced by the standard Laplacian operator. The equation on the narrow band domain is split into one linear and two nonlinear subequations, where the nonlinear subequations are independent of spatial derivatives and thus are ordinary differential equations and have closed-form solutions. Therefore, we only solve the linear subequation on the narrow band domain using the Crank–Nicolson method. Numerical experiments on various surfaces are given verifying the accuracy and efficiency of the proposed method. Full article
(This article belongs to the Special Issue Numerical Analysis or Numerical Method in Symmetry)
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Open AccessArticle
A Numerical Solution of Fredholm Integral Equations of the Second Kind Based on Tight Framelets Generated by the Oblique Extension Principle
Symmetry 2019, 11(7), 854; https://doi.org/10.3390/sym11070854 - 02 Jul 2019
Abstract
In this paper, we present a new computational method for solving linear Fredholm integral equations of the second kind, which is based on the use of B-spline quasi-affine tight framelet systems generated by the unitary and oblique extension principles. We convert the [...] Read more.
In this paper, we present a new computational method for solving linear Fredholm integral equations of the second kind, which is based on the use of B-spline quasi-affine tight framelet systems generated by the unitary and oblique extension principles. We convert the integral equation to a system of linear equations. We provide an example of the construction of quasi-affine tight framelet systems. We also give some numerical evidence to illustrate our method. The numerical results confirm that the method is efficient, very effective and accurate. Full article
(This article belongs to the Special Issue Numerical Analysis or Numerical Method in Symmetry)
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Open AccessArticle
Qualitative Behavior of Solutions of Second Order Differential Equations
Symmetry 2019, 11(6), 777; https://doi.org/10.3390/sym11060777 - 11 Jun 2019
Cited by 2
Abstract
In this work, we study the oscillation of second-order delay differential equations, by employing a refinement of the generalized Riccati substitution. We establish a new oscillation criterion. Symmetry ideas are often invisible in these studies, but they help us decide the right way [...] Read more.
In this work, we study the oscillation of second-order delay differential equations, by employing a refinement of the generalized Riccati substitution. We establish a new oscillation criterion. Symmetry ideas are often invisible in these studies, but they help us decide the right way to study them, and to show us the correct direction for future developments. We illustrate the results with some examples. Full article
(This article belongs to the Special Issue Numerical Analysis or Numerical Method in Symmetry)
Open AccessArticle
Approximation to Logarithmic-Cauchy Type Singular Integrals with Highly Oscillatory Kernels
Symmetry 2019, 11(6), 728; https://doi.org/10.3390/sym11060728 - 28 May 2019
Abstract
In this paper, a fast and accurate numerical Clenshaw-Curtis quadrature is proposed for the approximation of highly oscillatory integrals with Cauchy and logarithmic singularities, 1 1 f ( x ) log ( x α ) e i k x x [...] Read more.
In this paper, a fast and accurate numerical Clenshaw-Curtis quadrature is proposed for the approximation of highly oscillatory integrals with Cauchy and logarithmic singularities, 1 1 f ( x ) log ( x α ) e i k x x t d x , t ( 1 , 1 ) , α [ 1 , 1 ] for a smooth function f ( x ) . This method consists of evaluation of the modified moments by stable recurrence relation and Cauchy kernel is solved by steepest descent method that transforms the oscillatory integral into the sum of line integrals. Later theoretical analysis and high accuracy of the method is illustrated by some examples. Full article
(This article belongs to the Special Issue Numerical Analysis or Numerical Method in Symmetry)
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Open AccessFeature PaperArticle
Model for the Evaluation of an Angular Slot’s Coupling Impedance
Symmetry 2019, 11(5), 700; https://doi.org/10.3390/sym11050700 - 22 May 2019
Abstract
In high energy particle accelerators, a careful modeling of the electromagnetic interaction between the particle beam and the structure is essential to ensure the performance of the experiments. Particular interest arises in the presence of angular discontinuities of the structure, due to the [...] Read more.
In high energy particle accelerators, a careful modeling of the electromagnetic interaction between the particle beam and the structure is essential to ensure the performance of the experiments. Particular interest arises in the presence of angular discontinuities of the structure, due to the asymmetrical behavior. In this case, semi-analytical models allow one to reduce the computational effort and to better understand the physics of the phenomena, with respect to purely numerical models. In the paper, a model for analyzing the electromagnetic interaction between a traveling charge particle and a perfectly conducting angular slot of a negligible thickness is discussed. The particle travels at a constant velocity along a straight line parallel to the axis of symmetry of the strip. The longitudinal and transverse coupling impedances are therefore evaluated for a wide range of parameters. Full article
(This article belongs to the Special Issue Numerical Analysis or Numerical Method in Symmetry)
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Open AccessFeature PaperArticle
Asymptotic Properties of Solutions of Fourth-Order Delay Differential Equations
Symmetry 2019, 11(5), 628; https://doi.org/10.3390/sym11050628 - 03 May 2019
Cited by 5
Abstract
In the paper, we study the oscillation of fourth-order delay differential equations, the present authors used a Riccati transformation and the comparison technique for the fourth order delay differential equation, and that was compared with the oscillation of the certain second order differential [...] Read more.
In the paper, we study the oscillation of fourth-order delay differential equations, the present authors used a Riccati transformation and the comparison technique for the fourth order delay differential equation, and that was compared with the oscillation of the certain second order differential equation. Our results extend and improve many well-known results for oscillation of solutions to a class of fourth-order delay differential equations. Some examples are also presented to test the strength and applicability of the results obtained. Full article
(This article belongs to the Special Issue Numerical Analysis or Numerical Method in Symmetry)
Open AccessArticle
Numerical Analysis of Boundary Layer Flow Adjacent to a Thin Needle in Nanofluid with the Presence of Heat Source and Chemical Reaction
Symmetry 2019, 11(4), 543; https://doi.org/10.3390/sym11040543 - 15 Apr 2019
Cited by 1
Abstract
The steady boundary layer flow of a nanofluid past a thin needle under the influences of heat generation and chemical reaction is analyzed in the present work. The mathematical model has been formulated by using Buongiornos’s nanofluid model which incorporates the effect of [...] Read more.
The steady boundary layer flow of a nanofluid past a thin needle under the influences of heat generation and chemical reaction is analyzed in the present work. The mathematical model has been formulated by using Buongiornos’s nanofluid model which incorporates the effect of the Brownian motion and thermophoretic diffusion. The governing coupled partial differential equations are transformed into a set of nonlinear ordinary differential equations by using appropriate similarity transformations. These equations are then computed numerically through MATLAB software using the implemented package called bvp4c. The influences of various parameters such as Brownian motion, thermophoresis, velocity ratio, needle thickness, heat generation and chemical reaction parameters on the flow, heat and mass characteristics are investigated. The physical characteristics which include the skin friction, heat and mass transfers, velocity, temperature and concentration are further elaborated with the variation of governing parameters and presented through graphs. It is observed that the multiple (dual) solutions are likely to exist when the needle moves against the direction of the fluid flow. It is also noticed that the reduction in needle thickness contributes to the enlargement of the region of the dual solutions. The determination of the stable solution has been done using a stability analysis. The results indicate that the upper branch solutions are linearly stable, while the lower branch solutions are linearly unstable. The study also revealed that the rate of heat transfer is a decreasing function of heat generation parameter, while the rate of mass transfer is an increasing function of heat generation and chemical reaction parameters. Full article
(This article belongs to the Special Issue Numerical Analysis or Numerical Method in Symmetry)
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Open AccessArticle
The Third and Fourth Kind Pseudo-Chebyshev Polynomials of Half-Integer Degree
Symmetry 2019, 11(2), 274; https://doi.org/10.3390/sym11020274 - 20 Feb 2019
Abstract
New sets of orthogonal functions, which correspond to the first, second, third, and fourth kind Chebyshev polynomials with half-integer indexes, have been recently introduced. In this article, links of these new sets of irrational functions to the third and fourth kind Chebyshev polynomials [...] Read more.
New sets of orthogonal functions, which correspond to the first, second, third, and fourth kind Chebyshev polynomials with half-integer indexes, have been recently introduced. In this article, links of these new sets of irrational functions to the third and fourth kind Chebyshev polynomials are highlighted and their connections with the classical Chebyshev polynomials are shown. Full article
(This article belongs to the Special Issue Numerical Analysis or Numerical Method in Symmetry)
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Open AccessArticle
k-Hypergeometric Series Solutions to One Type of Non-Homogeneous k-Hypergeometric Equations
Symmetry 2019, 11(2), 262; https://doi.org/10.3390/sym11020262 - 19 Feb 2019
Cited by 1
Abstract
In this paper, we expound on the hypergeometric series solutions for the second-order non-homogeneous k-hypergeometric differential equation with the polynomial term. The general solutions of this equation are obtained in the form of k-hypergeometric series based on the Frobenius method. Lastly, [...] Read more.
In this paper, we expound on the hypergeometric series solutions for the second-order non-homogeneous k-hypergeometric differential equation with the polynomial term. The general solutions of this equation are obtained in the form of k-hypergeometric series based on the Frobenius method. Lastly, we employ the result of the theorem to find the solutions of several non-homogeneous k-hypergeometric differential equations. Full article
(This article belongs to the Special Issue Numerical Analysis or Numerical Method in Symmetry)
Open AccessArticle
Explicit Integrator of Runge-Kutta Type for Direct Solution of u(4) = f(x, u, u′, u″)
Symmetry 2019, 11(2), 246; https://doi.org/10.3390/sym11020246 - 16 Feb 2019
Abstract
The primary contribution of this work is to develop direct processes of explicit Runge-Kutta type (RKT) as solutions for any fourth-order ordinary differential equation (ODEs) of the structure u ( 4 ) = f ( x , u , u , u [...] Read more.
The primary contribution of this work is to develop direct processes of explicit Runge-Kutta type (RKT) as solutions for any fourth-order ordinary differential equation (ODEs) of the structure u ( 4 ) = f ( x , u , u , u ) and denoted as RKTF method. We presented the associated B-series and quad-colored tree theory with the aim of deriving the prerequisites of the said order. Depending on the order conditions, the method with algebraic order four with a three-stage and order five with a four-stage denoted as RKTF4 and RKTF5 are discussed, respectively. Numerical outcomes are offered to interpret the accuracy and efficacy of the new techniques via comparisons with various currently available RK techniques after converting the problems into a system of first-order ODE systems. Application of the new methods in real-life problems in ship dynamics is discussed. Full article
(This article belongs to the Special Issue Numerical Analysis or Numerical Method in Symmetry)
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Open AccessArticle
Symplectic Effective Order Numerical Methods for Separable Hamiltonian Systems
Symmetry 2019, 11(2), 142; https://doi.org/10.3390/sym11020142 - 28 Jan 2019
Cited by 1
Abstract
A family of explicit symplectic partitioned Runge-Kutta methods are derived with effective order 3 for the numerical integration of separable Hamiltonian systems. The proposed explicit methods are more efficient than existing symplectic implicit Runge-Kutta methods. A selection of numerical experiments on separable Hamiltonian [...] Read more.
A family of explicit symplectic partitioned Runge-Kutta methods are derived with effective order 3 for the numerical integration of separable Hamiltonian systems. The proposed explicit methods are more efficient than existing symplectic implicit Runge-Kutta methods. A selection of numerical experiments on separable Hamiltonian system confirming the efficiency of the approach is also provided with good energy conservation. Full article
(This article belongs to the Special Issue Numerical Analysis or Numerical Method in Symmetry)
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Open AccessArticle
A Complex Lie-Symmetry Approach to Calculate First Integrals and Their Numerical Preservation
Symmetry 2019, 11(1), 11; https://doi.org/10.3390/sym11010011 - 24 Dec 2018
Abstract
We calculated Noether-like operators and first integrals of a scalar second-order ordinary differential equation using the complex Lie-symmetry method. We numerically integrated the equations using a symplectic Runge–Kutta method. It was seen that these structure-preserving numerical methods provide qualitatively correct numerical results, and [...] Read more.
We calculated Noether-like operators and first integrals of a scalar second-order ordinary differential equation using the complex Lie-symmetry method. We numerically integrated the equations using a symplectic Runge–Kutta method. It was seen that these structure-preserving numerical methods provide qualitatively correct numerical results, and good preservation of first integrals is obtained. Full article
(This article belongs to the Special Issue Numerical Analysis or Numerical Method in Symmetry)
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Open AccessArticle
Improvement of Risk Assessment Using Numerical Analysis for an Offshore Plant Dipole Antenna
Symmetry 2018, 10(12), 681; https://doi.org/10.3390/sym10120681 - 01 Dec 2018
Abstract
This paper proposes a numerical analysis method for improving risk assessment of radio frequency (RF) hazards. To compare the results of conventional code analysis, the values required for dipole antenna risk assessment, which is widely used in offshore plants based on the British [...] Read more.
This paper proposes a numerical analysis method for improving risk assessment of radio frequency (RF) hazards. To compare the results of conventional code analysis, the values required for dipole antenna risk assessment, which is widely used in offshore plants based on the British standards (BS) guide, are calculated using the proposed numerical analysis. Based on the BS (published document CENELEC technical report (PD CLC/TR) 50427:2004 and international electrotechnical commission (IEC) 60079 for an offshore plant dipole antenna, an initial assessment, a full assessment, and on-site test procedures are performed to determine if there is a potential risk of high-frequency ignition. Alternatively, numerical analysis is performed using the Ansys high frequency structure simulator (HFSS) tool to compare results based on the BS guide. The proposed method computes the effective field strength and power for the antenna without any special consideration of the structure to simplify the calculation. Experimental results show that the proposed numerical analysis outperforms the risk assessment based on the BS guide in accuracy of the evaluation. Full article
(This article belongs to the Special Issue Numerical Analysis or Numerical Method in Symmetry)
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