• Kyungpook Mathematical Journal
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• 제47권2호
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• pp.175-190
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• 2007

By employing the Riccati transformation technique some new oscillation criteria for the second-order neutral delay dynamic equation $$(y(t)+r(t)y({\tau}(t)))^{{\Delta}{\Delta}}+p(t)y(\delta(t))=0$$, on a time scale $\mathbb{T}$ are established. Our results as a special case when $\mathbb{T}=\mathbb{R}$ and $\mathbb{T}=\mathbb{N}$ improve some well known oscillation criteria for second order neutral delay differential and difference equations, and when $\mathbb{T}=q^{\mathbb{N}}$, i.e., for second-order $q$-neutral difference equations our results are essentially new and can be applied on different types of time scales. Some examples are considered to illustrate the main results.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제17권4호
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• pp.289-298
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• 2010

In this paper, we propose a second-order prediction/correction (SPC) domain decomposition method for solving one dimensional linear hyperbolic partial differential equation $u_{tt}+a(x,t)u_t+b(x,t)u=c(x,t)u_{xx}+{\int}(x,t)$. The method can be applied to variable coefficients problems and singular problems. Unconditional stability and error analysis of the method have been carried out. Numerical results support stability and efficiency of the method.

• 호남수학학술지
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• 제37권3호
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• pp.299-315
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• 2015

In this paper, we study the first-order system least-squares (FOSLS) spectral method for parabolic partial differential equations. There were lots of least-squares approaches to solve elliptic partial differential equations using finite element approximation. Also, some approaches using spectral methods have been studied in recent. In order to solve the parabolic partial differential equations in parallel, we consider a parallel numerical method based on a hybrid method of the frequency-domain method and first-order system least-squares method. First, we transform the parabolic problem in the space-time domain to the elliptic problems in the space-frequency domain. Second, we solve each elliptic problem in parallel for some frequencies using the first-order system least-squares method. And then we take the discrete inverse Fourier transforms in order to obtain the approximate solution in the space-time domain. We will introduce such a hybrid method and then present a numerical experiment.

• 대한수학회지
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• 제36권5호
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• pp.959-1008
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• 1999

This paper considers foundational issues related to connections in the tangent bundle of a manifold. The approach makes use of second order tangent vectors, i.e., vectors tangent to the tangent bundle. The resulting second order tangent bundle has certain properties, above and beyond those of a typical tangent bundle. In particular, it has a natural secondary vector bundle structure and a canonical involution that interchanges the two structures. The involution provides a nice way to understand the torsion of a connection. The latter parts of the paper deal with the Levi-Civita connection of a Riemannian manifold. The idea is to get at the connection by first finding its.spary. This is a second order vector field that encodes the second order differential equation for geodesics. The paper also develops some machinery involving lifts of vector fields form a manifold to its tangent bundle and uses a variational approach to produce the Riemannian spray.

• 한국전산구조공학회논문집
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• 제31권6호
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• pp.331-337
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• 2018

본 논문은 MLS(moving least squares) 차분법의 1차 미분 근사함수를 바탕으로 시간에 따른 수치해석이 가능한 해석기법을 제시한다. 오직 1차 미분 근사함수로만 지배방정식을 이산화했으며, 근사함수를 조립하는 형태로 전체 시스템 방정식을 구성하여 차분법으로 이산화된 운동방정식이 유한요소법(finite element method)과 유사한 모습을 갖게 되었다. 운동방정식을 시간적분하기 위해서 중앙차분법(central difference method)을 사용하였다. 유한요소 알고리즘을 통해서 MLS 차분법과 유한요소법의 고유진동 해석을 수행하였으며, 두 해석결과를 비교하였다. 또한, 동적해석 결과를 기존의 2차 미분 근사함수를 활용한 해석결과와 함께 도시함으로써 제안된 수치기법의 정확성을 검증하였다. 1차 미분 근사함수를 조립하는 과정에서 해석결과의 떨림현상이 억제되었으며 상대적으로 균일한 응력분포를 구할 수 있었다.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제25권4호
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• pp.224-261
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• 2021

The interest in implicit-explicit (IMEX) integration methods has emerged as an alternative for dealing in a computationally cost-effective way with stiff ordinary differential equations arising from practical modeling problems. In this paper, we introduce implicit-explicit second derivative linear multi-step methods (IMEX SDLMM) with error control. The proposed IMEX SDLMM is based on second derivative backward differentiation formulas (SDBDF) and recursive SDBDF. The IMEX second derivative schemes are constructed with order p ranging from p = 1 to 8. The methods are numerically validated on well-known stiff equations.

• 융합보안논문지
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• 제5권3호
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• pp.93-97
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• 2005

MacCormack 방법은 hyperbolic 편미분 방정식의 근을 구하는데 많이 쓰이는 방법으로 그 정확도가 2차 오더가 된다. 하지만 이 방법으로 편미분방정식을 풀 경우 불연속인 점에서는 엔트로피를 만족하지 않는 경우가 있어 우리는 임의의 항을 첨가하여 근을 구해야한다. 이 임의의 항을 첨가하지 않고 직접 방정식으로부터 구하는 방법을 생각하는데 있어서 기존의 MacCormack 방법에 새 central scheme의 개념을 이용하면 전형적인 MacCormack 방법의 정확도와 장점을 보존할 수 있다. 이 새로운 방법을 이용하여 1D Burgers' 방정식과 1D Euler gas dynamic 방정식에 활용하여 그 결과를 살펴본다.

• 한국조명전기설비학회지:조명전기설비
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• 제5권4호
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• pp.35-42
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• 1991

Recently, HID lamps have been considered as important in regard to the trend of energy saving, and increasingly and diversely used in various ways. This paper will show the simulating models concerning high-pressure arc discharge system directly applicable for its design and manufacture, and analyze its various characteristics. For warm-up characteristics, the evaporating process of inner atoms is described in terms of second-order differential equation: for the thermal conduction from are axis to discharge wall and outer bulb, its transfer process is introduced according to five first-order differential equations. Under the steady state satisfying LTE, the time-variant characteristics are suggested by means of time-dependent energy balance equation derived from fluid equations, approximation of radiation energy and material functions in the discharge tube. The simulating models concerning these equations are then applied for high-pressure mercury lamp.

• Journal of applied mathematics & informatics
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• 제28권1_2호
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• pp.411-423
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• 2010

In this paper, a numerical method for singular perturbation problems arising in chemical reactor theory for general singularly perturbed two point boundary value problems with boundary layer at one end(left or right) of the underlying interval is presented. The original second order differential equation is replaced by an approximate first order differential equation with a small deviating argument. By using the trapezoidal formula we obtain a three term recurrence relation, which is solved using Thomas Algorithm. To demonstrate the applicability of the method, we have solved four linear (two left and two right end boundary layer) and one nonlinear problems. From the results, it is observed that the present method approximates the exact or the asymptotic expansion solution very well.

• Journal of applied mathematics & informatics
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• 제26권5_6호
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• pp.1273-1287
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• 2008

In this paper, the numerical integration method for general singularly perturbed two point boundary value problems with mixed boundary conditions of both left and right end boundary layer is presented. The original second order differential equation is replaced by an approximate first order differential equation with a small deviating argument. By using the trapezoidal formula we obtain a three term recurrence relation, which is solved using Thomas Algorithm. To demonstrate the applicability of the method, we have solved four linear (two left and two right end boundary layer) and one nonlinear problems. From the results, it is observed that the present method approximates the exact or the asymptotic expansion solution very well.