Research

8 pages, 1315 KiB

Open AccessArticle

Numerical Simulation of Pattern Formation on Surfaces Using an Efficient Linear Second-Order Method

by Hyun Geun Lee

Symmetry 2019, 11(8), 1010; https://doi.org/10.3390/sym11081010 - 05 Aug 2019

Cited by 7 | Viewed by 2691

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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15 pages, 883 KiB

Open AccessArticle

A Numerical Solution of Fredholm Integral Equations of the Second Kind Based on Tight Framelets Generated by the Oblique Extension Principle

by Mutaz Mohammad

Symmetry 2019, 11(7), 854; https://doi.org/10.3390/sym11070854 - 02 Jul 2019

Cited by 22 | Viewed by 2557

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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8 pages, 720 KiB

Open AccessArticle

Qualitative Behavior of Solutions of Second Order Differential Equations

by Clemente Cesarano and Omar Bazighifan

Symmetry 2019, 11(6), 777; https://doi.org/10.3390/sym11060777 - 11 Jun 2019

Cited by 36 | Viewed by 2246

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)

13 pages, 284 KiB

Open AccessArticle

Approximation to Logarithmic-Cauchy Type Singular Integrals with Highly Oscillatory Kernels

by SAIRA and Shuhuang Xiang

Symmetry 2019, 11(6), 728; https://doi.org/10.3390/sym11060728 - 28 May 2019

Cited by 2 | Viewed by 2051

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, [...] 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} \frac{f (x) \log (x - α) e^{i k x}}{x - t} d x

,

t \notin (- 1, 1), α \in [- 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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12 pages, 472 KiB

Open AccessFeature PaperArticle

Model for the Evaluation of an Angular Slot’s Coupling Impedance

by Dario Assante and Luigi Verolino

Symmetry 2019, 11(5), 700; https://doi.org/10.3390/sym11050700 - 22 May 2019

Cited by 1 | Viewed by 1933

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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10 pages, 265 KiB

Open AccessFeature PaperArticle

Asymptotic Properties of Solutions of Fourth-Order Delay Differential Equations

by Clemente Cesarano, Sandra Pinelas, Faisal Al-Showaikh and Omar Bazighifan

Symmetry 2019, 11(5), 628; https://doi.org/10.3390/sym11050628 - 03 May 2019

Cited by 32 | Viewed by 2418

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)

15 pages, 939 KiB

Open AccessArticle

Numerical Analysis of Boundary Layer Flow Adjacent to a Thin Needle in Nanofluid with the Presence of Heat Source and Chemical Reaction

by Siti Nur Alwani Salleh, Norfifah Bachok, Norihan Md Arifin and Fadzilah Md Ali

Symmetry 2019, 11(4), 543; https://doi.org/10.3390/sym11040543 - 15 Apr 2019

Cited by 26 | Viewed by 2707

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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11 pages, 2555 KiB

Open AccessArticle

The Third and Fourth Kind Pseudo-Chebyshev Polynomials of Half-Integer Degree

by Clemente Cesarano, Sandra Pinelas and Paolo Emilio Ricci

Symmetry 2019, 11(2), 274; https://doi.org/10.3390/sym11020274 - 20 Feb 2019

Cited by 10 | Viewed by 3620

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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11 pages, 294 KiB

Open AccessArticle

k-Hypergeometric Series Solutions to One Type of Non-Homogeneous k-Hypergeometric Equations

by Shengfeng Li and Yi Dong

Symmetry 2019, 11(2), 262; https://doi.org/10.3390/sym11020262 - 19 Feb 2019

Cited by 13 | Viewed by 3250

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)

30 pages, 1230 KiB

Open AccessArticle

Explicit Integrator of Runge-Kutta Type for Direct Solution of u⁽⁴⁾ = f(x, u, u′, u″)

by Nizam Ghawadri, Norazak Senu, Firas Adel Fawzi, Fudziah Ismail and Zarina Bibi Ibrahim

Symmetry 2019, 11(2), 246; https://doi.org/10.3390/sym11020246 - 16 Feb 2019

Cited by 7 | Viewed by 2841

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 [...] 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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14 pages, 357 KiB

Open AccessArticle

Symplectic Effective Order Numerical Methods for Separable Hamiltonian Systems

by Junaid Ahmad, Yousaf Habib, Shafiq ur Rehman, Azqa Arif, Saba Shafiq and Muhammad Younas

Symmetry 2019, 11(2), 142; https://doi.org/10.3390/sym11020142 - 28 Jan 2019

Cited by 1 | Viewed by 2823

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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16 pages, 644 KiB

Open AccessArticle

A Complex Lie-Symmetry Approach to Calculate First Integrals and Their Numerical Preservation

by Wajeeha Irshad, Yousaf Habib and Muhammad Umar Farooq

Symmetry 2019, 11(1), 11; https://doi.org/10.3390/sym11010011 - 24 Dec 2018

Cited by 2 | Viewed by 2427

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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14 pages, 3608 KiB

Open AccessArticle

Improvement of Risk Assessment Using Numerical Analysis for an Offshore Plant Dipole Antenna

by Yun-Jeong Cho, Kichang Im, Dongkoo Shon, Daehoon Park and Jong-Myon Kim

Symmetry 2018, 10(12), 681; https://doi.org/10.3390/sym10120681 - 01 Dec 2018

Cited by 1 | Viewed by 2591

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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