• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.21 no.1
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• pp.1-16
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• 2017

The Allen-Cahn equation is solved numerically by operator splitting Fourier spectral methods. The basic idea of the operator splitting method is to decompose the original problem into sub-equations and compose the approximate solution of the original equation using the solutions of the subproblems. The purpose of this paper is to characterize higher order operator splitting schemes and propose several higher order methods. Unlike the first and the second order methods, each of the heat and the free-energy evolution operators has at least one backward evaluation in higher order methods. We investigate the effect of negative time steps on a general form of third order schemes and suggest three third order methods for better stability and accuracy. Two fourth order methods are also presented. The traveling wave solution and a spinodal decomposition problem are used to demonstrate numerical properties and the order of convergence of the proposed methods.

• Journal of electromagnetic engineering and science
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• v.10 no.4
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• pp.296-302
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• 2010

This paper proposes an approximate equation for broad bandwidth conditions in an antenna feeding probe design with a cylindrical magneto material structure (CMMS). The bandwidth calculation has been conducted according to the relation between the distance ($r_m$) between the magneto material and feeding probe, and the magneto material thickness ($t_m$) for a given ${\mu}_r$. The bandwidth of a proposed antenna with CMM feeding structure is improved about 182 %, when ${\mu}_r=20+j0.001$, in comparison with the bandwidth of an antenna without CMMS. The maximum error extent between the bandwidth calculated by the approximation equation and by the numerical calculation of the proposed antenna is about $\pm$3.2 % for ${\mu}_r=10+j0.001$. The approximation equation proposed in this study can solve the conventional problem of the complex process and the long time required for reiterative calculation, and allow simple and precise design with prediction. The accuracy of an approximated equation is compared with the results calculated by a commercial tool and verified by reasonable agreement between them.

• Journal of applied mathematics & informatics
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• v.20 no.1_2
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• pp.133-147
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• 2006

In this paper, by using the Riccati transformation technique we establish some new oscillation criteria for second-order nonlinear perturbed dynamic equation on time scales. An example illustrating our main results is also given.

• Journal of applied mathematics & informatics
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• v.28 no.3_4
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• pp.945-952
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• 2010

In this paper, existence criteria of one solution to a nonlinear first-order periodic boundary value problem of impulsive dynamic equation on time scales are obtained by using the well-known Schaefer fixed-point theorem.

• Journal of Korean Association for Spatial Structures
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• v.24 no.1
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• pp.65-72
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• 2024

This paper aims to advance our understanding of extensible beams with multiple cracks by presenting a crack energy and motion equation, and mathematically justifying the energy functions of axial and bending deformations caused by cracks. Utilizing an extended form of Hamilton's principle, we derive a normalized governing equation for the motion of the extensible beam, taking into account crack energy. To achieve a closed-form solution of the beam equation, we employ a simple approach that incorporates the crack's patching condition into the eigenvalue problem associated with the linear part of the governing equation. This methodology not only yields a valuable eigenmode function but also significantly enhances our understanding of the dynamics of cracked extensible beams. Furthermore, we derive a governing equation that is an ordinary differential equation concerning time, based on orthogonal eigenmodes. This research lays the foundation for further studies, including experimental validations, applications, and the study of damage estimation and detection in the presence of cracks.

• Nonlinear Functional Analysis and Applications
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• v.29 no.3
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• pp.803-823
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• 2024

In this paper, we study the existence and uniqueness of solutions for the fractional time-varying delay integrodifferential equation with multi-point multi-term nonlocal and fractional integral boundary conditions by using fixed point theorems. The fractional derivative considered here is in the Caputo sense. Examples are provided to illustrate the results.

• Kyungpook Mathematical Journal
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• v.47 no.3
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• pp.425-438
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• 2007

In this paper, we establish some new oscillation criteria for a second-order delay dynamic equation $$u^{{\Delta}{\Delta}}(t)+p(t)u(\tau(t))=0$$ on a time scale $\mathbb{T}$. The results can be applied on differential equations when $\mathbb{T}=\mathbb{R}$, delay difference equations when $\mathbb{T}=\mathbb{N}$ and for delay $q$-difference equations when $\mathbb{T}=q^{\mathbb{N}}$ for q > 1.

• The Journal of the Acoustical Society of Korea
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• v.17 no.4E
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• pp.3-10
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• 1998

This paper concerns the dynamical behavior, in probabilistic sense, of a feedforward neural network performing auto association for novelty. Networks of retinotopic topology having a one-to-one correspondence between and output units can be readily trained using back-propagation algorithm, to perform autoassociative mappings. A novelty filter is obtained by subtracting the network output from the input vector. Then the presentation of a "familiar" pattern tends to evoke a null response ; but any anomalous component is enhanced. Such a behavior exhibits a promising feature for enhancement of weak signals in additive noise. As an analysis of the novelty filtering, this paper shows that the probability density function of the weigh converges to Gaussian when the input time series is statistically characterized by nonsymmetrical probability density functions. After output units are locally linearized, the recursive relation for updating the weight of the neural network is converted into a first-order random differential equation. Based on this equation it is shown that the probability density function of the weight satisfies the Fokker-Planck equation. By solving the Fokker-Planck equation, it is found that the weight is Gaussian distributed with time dependent mean and variance.

• Transactions of the Korean Society of Mechanical Engineers B
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• v.22 no.1
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• pp.1-11
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• 1998

For the analysis of phonon heat transfer within short time and spatial scales, conventional macroscopic heat conduction equations with jump boundary conditions are tried and the results are compared to those of equation of phonon radiative transport(EPRT), which is one of microscopic transport equation. In transient state the macroscopic temperatures show far different behavior from EPRT. In steady state the hyperbolic temperatures with temperature jump at the wall from time relaxation model agrees well with EPRT temperatures. Since EPRT is also an approximate form of microscopic transport equation and there are no experimental results to verify the proposed model in this study, we can not conclude whether the approaching method from this study is valid or not. To the authors' knowledge, there are no experimental results available which can be used to test the validity of these models. Such an experiment, while difficult to conduct, would be invaluable.

• Transactions of the Korean Society for Noise and Vibration Engineering
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• v.21 no.4
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• pp.315-324
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• 2011

This paper presents a historical study on the derivation of the differential equation of motion for the single-degree-of-freedom m-k system with the harmonic excitation. It was Euler for the first time in the history of vibration theory who tackled the equation of motion for that system analytically, then gave the solution of the free vibration and described the resonance phenomena of the forced vibration in his famous paper E126 of 1739. As a result of the chronological progress in mechanics like pendulum condition from Galileo to Euler, the author asserts two conjectures that Euler could apply to obtain the equation of motion at that time.