• East Asian mathematical journal
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• 제27권1호
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• pp.51-65
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• 2011

In the paper, we study questions on classical solvability of nonlocal problems for a third-order linear hyperbolic equation in a rectangular domain. The Riemann method is applied to the Goursat problem and solution is obtained in the integral form. Investigated problems are reduced to the uniquely solvable Volterra-type equation of second kind. Influence effects of coefficients at lowest derivatives on correctness of studied problems are detected.

• Kyungpook Mathematical Journal
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• 제21권1호
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• pp.71-74
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• 1981

In this paper a general third order differential equation encountered in the flow of fluids in general and hopefully elsewhere. Sufficient conditions for existence & uniqueness of its solutions are given.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제21권1호
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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.

• 충청수학회지
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• 제23권3호
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• pp.443-452
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• 2010

This work deals with the existence of uncountably many bounded positive solutions for the third order nonlinear neutral delay differential equation $$\frac{d^3}{dt^3}[x(t)+p(t)x(t-{\tau})]+f(t,x(t-{{\tau}_1}),{\ldots},x(t-{{\tau}_k}))=0,\;t{\geq}t_0$$ where ${\tau}>0$, ${\tau}_i{\in}{\mathbb{R}^+}$ for $i{\in}\{1,2,{\ldots},k\}$, $p{\in}C([t_0,+{\infty}),{\mathbb{R}^+})$ and $f{\in}C([t_0,+{\infty}){\times}{\mathbb{R}^k},{\mathbb{R}})$.

• 한국광학회지
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• 제16권5호
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• pp.423-427
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• 2005

본 연구는, 원리적인 다양한 장점에도 불구하고 현실적인 제약으로 인해 실제 설계과정에서 잘 적용되지 않는, 자이델 3차 수차를 간편하게 다룰 수 있는 방안을 제안한다. 먼저 유한 물체거리를 갖는 2 반사경계에 대해 자이델 3차의 구면수차계수를 유도한다. 여기서, 유도된 구면수차계수는 고차의 비선형 방정식으로 표현되는데, 그 구성은 설계변수(물체거리, 주경 및 부경의 곡률, 주경과 부경 사이의 거리)와 유효초점거리로 이루어진다. 해석적으로 표현된 고차의 비선형 구면수차 방정식은 컴퓨터를 이용한 수치기법에 의해 근사적인 제로조건을 만족하도록 풀려진다. 이렇게 구해진 다양한 수치 해들을 주의 깊게 통찰하면 주경과 부경의 곡률 간에 선형성이 존재함을 파악할 수 있다. 즉, 결과적으로 주경과 부경의 곡률들을 선형맞춤(linear fitting)하면 곡률선형방정식이 얻어지는데, 이의 의미는 약간의 대수적인 계산으로 최적화의 초기 입력 데이터를 손쉽게 얻을 수 있는 가능성을 제시한 것이다. 한편, 응용외의 순수 수차론적인 관점에서 본다면, 본 연구의 특징은 유한 물체거리를 갖는 2 반사경계의 주경 및 부경의 곡률들이 구면수차가 거의 제로가 되는 조건 하에서 상호간에 선형 관계가 존재하였다는 것이다.

• 센서학회지
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• 제21권2호
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• pp.109-113
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• 2012

We propose an improved interpolating equation to express temperature-resistance characteristics for modern industrial platinum resistance thermometers (PRTs). Callendar-van Dusen equation which has been widely used for platinum resistance thermometer fails to fully describe temperature characteristics of high quality PRTs and leaves systematic residual when the calibration point include temperatures above $300^{\circ}C$. Expanding Callendar-van Dusen to higher-order polynomial drastically improves the uncertainty of the fitting even with reduced degrees of freedom of the fitting. We found that in the fourth-order polynomial fitting, the third-order and fourth-order coefficients have a strong correlation. Using the correlation, we suggest an improved interpolating equation in the form of fourth-order polynomial, but with three fitting parameters. Applying this interpolating equation reduced the uncertainty of the fitting to 32 % of that resulted from the traditional Callendar-van Dusen. This improvement was better than that from a simple third-order polynomial despite that the degrees of the freedom of the fitting was the same.

• Journal of applied mathematics & informatics
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• 제26권5_6호
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• pp.1057-1069
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• 2008

We consider singularly perturbed Boundary Value Problems (BVPs) for third and fourth order Ordinary Differential Equations(ODEs) of convection-diffusion type with discontinuous source term and a small positive parameter multiplying the highest derivative. Because of the type of Boundary Conditions(BCs) imposed on these equations these problems can be transformed into weakly coupled systems. In this system, the first equation does not have the small parameter but the second contains it. In this paper a computational method named as 'An asymptotic finite element method' for solving these systems is presented. In this method we first find an zero order asymptotic approximation to the solution and then the system is decoupled by replacing the first component of the solution by this approximation in the second equation. Then the second equation is independently solved by a fitted mesh Finite Element Method (FEM). Numerical experiments support our theoritical results.

• Journal of applied mathematics & informatics
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• 제18권1_2호
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• pp.405-417
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• 2005

In this paper, the oscillation and asymptotic stability behavior of a third order linear impulsive equation are investigated. A lemma is presented to deal with the sign relation of the nonoscillatory solutions and their derived functions. By the lemma explicit sufficient conditions are obtained for all solutions either oscillating or asymptotically tending to zero. Two illustrative examples are proposed to demonstrate the effectiveness of the conditions.

• 한국수자원학회논문집
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• 제41권11호
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• pp.1107-1121
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• 2008

GIUH 유도식 대신 삼각형 가정의 간략식이 많은 연구에서 사용되고 있으나, 다양한 지형매개변수를 가진 유역에서 GIUH 간략식의 적용성에 대해서 알려진 바가 없다. 본 연구에서는 지도의 해상도를 변화시켜 방림, 상안미, 무성, 병천 등 4개 소유역을 각각 3차와 4차 하천유역으로 나타내고, GIUH 유도식과 간략식을 사용하여 각 유역의 순간단위도를 산정하였다. 각각의 순간단위도에 $8{\sim}16$개의 호우사상을 적용하여 유출수문곡선을 계산하였으며, 이를 관측 유량과 비교하였다. 대상유역들을 3차 하천유역으로 가정하였을 경우, 유도식은 첨두유량을 관측값보다 작게, 간략식은 크게 산정하는 경향이 있었다. 대상유역들을 4차 하천유역으로 가정하였을 경우, 유도식은 관측값을 가장 잘 반영하는 것으로 나타났으며, 간략식은 관측값보다 크게 산정하는 것으로 나타났다. 또한 간략식은 유역에 따라 관측값에 대한 정확성의 편차가 크게 나타난 반면, 유도식은 편차가 작고 안정적인 결과를 제시하는 것으로 나타났다. 따라서 미계측 유역에서 순간단위도를 산정할 때는 간략식보다는 유도식을 사용하는 것이 모형의 정확성을 향상시킬 수 있을 것으로 판단된다.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제3권1호
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• pp.47-52
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• 1996

We consider the third order linear homogeneous differential equation L$_3$(y) = y(equation omitted) ＋ P($\chi$)y' ＋ Q($\chi$)y = 0 (E) P($\chi$) $\geq$ 0, Q($\chi$) ＞ 0 and P($\chi$)/Q($\chi$) is nondecreasing on [${\alpha}$, $\infty$) for some real number ${\alpha}$. (1) In this paper we discuss the distribution of zeros of solutions and a condition of oscillatory for equation (E).(omitted)