• Journal of Educational Research in Mathematics
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• v.14 no.3
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• pp.221-237
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• 2004

This research analyzed mathematical discourse of the students in an RME-based differential equations course at a university in order to investigate the social transformation of the students' conceptual model of differential equations. The analysis focused on the change in the students' use of conceptual metaphor for differential equations and pedagogical factors promoting the change. The analysis shows that discrete and quantitative conceptual model was prevalent in the beginning of the semester However, continuous and qualitative conceptual model emerged through the negotiation of mathematical meaning based on the inquiry of context problems. The participation in the project class has a positive impact on the extension of the students' conceptual model of differential equations and increases the fluency of the students' problem solving in differential equations. Moreover, this paper provides a discussion to identify the pedagogical factors Involved with the transformation of the students' conceptual model. The discussion highlights the sociocultural aspect of teaching and learning of mathematics and provides implications to improve teaching of mathematics in school.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.25 no.3
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• pp.185-194
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• 2012

This paper deals with free vibrations of the tapered Timoshenko beam in which both the rotatory inertia and shear deformation are included. The cross section of the tapered beam is chosen as the rectangular cross section whose depth is constant but breadth is varied with the parabolic function. The fourth order ordinary differential equation with respect the vertical deflection governing free vibrations of such beam is derived based on the Timoshenko beam theory. This governing equation is solved for determining the natural frequencies corresponding with their mode shapes. In the numerical examples, three end constraints of the hinged-hinged, hinged-clamped and clamped-clamped ends are considered. The effects of various beam parameters on natural frequencies are extensively discussed. The mode shapes of both the deflections and stress resultants are presented, in which the composing rates due to bending rotation and shear deformation are determined.

• Geophysics and Geophysical Exploration
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• v.26 no.3
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• pp.114-125
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• 2023

The physics-informed neural network (PINN) has been proposed to overcome the limitations of various numerical methods used to solve partial differential equations (PDEs) and the drawbacks of purely data-driven machine learning. The PINN directly applies PDEs to the construction of the loss function, introducing physical constraints to machine learning training. This technique can also be applied to wave equation modeling. However, to solve the wave equation using the PINN, second-order differentiations with respect to input data must be performed during neural network training, and the resulting wavefields contain complex dynamical phenomena, requiring careful strategies. This tutorial elucidates the fundamental concepts of the PINN and discusses considerations for wave equation modeling using the PINN approach. These considerations include spatial coordinate normalization, the selection of activation functions, and strategies for incorporating physics loss. Our experimental results demonstrated that normalizing the spatial coordinates of the training data leads to a more accurate reflection of initial conditions in neural network training for wave equation modeling. Furthermore, the characteristics of various functions were compared to select an appropriate activation function for wavefield prediction using neural networks. These comparisons focused on their differentiation with respect to input data and their convergence properties. Finally, the results of two scenarios for incorporating physics loss into the loss function during neural network training were compared. Through numerical experiments, a curriculum-based learning strategy, applying physics loss after the initial training steps, was more effective than utilizing physics loss from the early training steps. In addition, the effectiveness of the PINN technique was confirmed by comparing these results with those of training without any use of physics loss.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.20 no.5
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• pp.581-592
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• 2007

This paper deals with the elastica of deflected cantilever column with the constant volume. The columns are subjected to combined loads consisted of an axial compressive load and a couple moment at the free end. Differential equations governing the elastica of such column are derived, in which both the effects of taper type and shear deformation are included. Three kinds of taper types are considered: linear, parabolic and sinusoidal tapers. Differential equations are solved numerically to obtain the elastica of objective columns. The effects of various system parameters on the elastica are investigated extensively. Experimental studies were carried out in order to verify the theoretical results of non-linear behavior of the elasticas.

• Journal of Korean Society of Steel Construction
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• v.9 no.2 s.31
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• pp.221-228
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• 1997

The main purpose of this paper is to determine the dynamic optimal shapes of simple beam-columns with the constant volume. The parabolic function is chosen as the variable equation for the depth of regular polygon cross-section. The ordinary differential equation including the effect of axial load is applied to calculate the natural frequencies. The Runge-Kutta and Regula-Falsi methods are used to integrate the differential equation and compute the frequencies, respectively. Then the dynamic optimal shape whose lowest natural frequency is highest is determined by reading the critical value of the frequency versus section ratio curve plotted by the frequency data. In the numerical examples, the simple beam-columns are analysed and the numerical results of this study are shown in tables and figures.

• Korean Journal of Optics and Photonics
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• v.4 no.4
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• pp.464-473
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• 1993

The dynamics of phase grating formation with visible light in an optical fiber is investigated. Adopting a simple two-photon local bleaching model, it is shown that the grating self-organize into an ideal grating, where the writing frequency is always in the center of the local band gap, as it evolves. The evolution at each point in the fiber is described in terms of a universal parameter that reduces the coupled partial differential equations describing the system to ordinary differential equatior~s. These equations are used to prove that there exists a fixed point of the grating growth process that corresponds to a perfectly phase-mached grating.

• Journal of the Computational Structural Engineering Institute of Korea
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• v.21 no.4
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• pp.357-364
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• 2008

This paper deals with the free vibrations of arches with rectangular hollow section having constant area. The differential equations governing free vibrations of arches are derived in polar coordinates, in which the effect of rotatory inertia is included. Natural frequencies is computed numerically for parabolic arches with clamped-clamped, clamped-hinged and hinged-hinged ends. Comparisons of natural frequencies between this study and reference are made to validate theories and numerical methods developed herein. The lowest four natural frequency parameters are reported, with the rotatory inertia, as functions of three non-dimensional system parameters: the breadth ratio, the thickness ratio and the shape ratio

• The Journal of Korean Institute of Communications and Information Sciences
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• v.15 no.7
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• pp.553-564
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• 1990

The co-efficient matrix of linear second-order partial differential equations in the general form is partitioned with (n-1)x(n-1) submartices and is transformed into the block tridiagonal system. Then the cyclic odd-even reduction technique is applied to this system with the large-grain data granularity and the block cyclic reduction algorithm to solve unknown vectors of this system is created. But this block cyclic reduction technique is not suitable for the parallel processing system because of its parallelism chanigng at every computing stages. So a new algorithm for solving linear second-order partical differential equations is presentes by the block cyclic reduction technique which is modified in order to keep its parallelism constant, and to reduce gteatly its execution time. Both of these algoriths are compared and studied.

• KSCE Journal of Civil and Environmental Engineering Research
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• v.29 no.1A
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• pp.53-60
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• 2009

This paper deals with the geometrical nonlinear analyses of post-buckled columns with variable cross-section. The objective columns having variable cross-section of the width, depth and square tapers are supported by both hinged ends. By using the Bernoulli-Euler beam theory, differential equations governing the elastica of post-buckled column and their boundary conditions are derived. The solution methods of these differential equations which have two unknown parameters are developed. As the numerical results, equilibrium paths, elasticas and stress resultants of the post-buckled columns are presented. Laboratory scaled experiments were conducted for validating the theories developed in this study.

• Proceedings of the Korean Nuclear Society Conference
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• 1997.05a
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• pp.286-291
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• 1997

이상 유동에서의 음파 전달 현상을 비평형, 비균질 이상 유동 방정식에 의하여 이론적으로 유도하였다 개발된 방법은 이상 계면에서의 압력 불연속성을 표면 장력 방정식에 의하여 해결하였으며, 이로 인하여 이상 유동 지배 방정식의 불량 설정된 초기치 문제(Ⅰ11-posed initial value problem)가 완전한 쌍곡형 편 미분 방정식군(Complete hyperbolic partial differential equation system)으로 만들어졌다. 새로이 개발된 방정식의 고유값인 음파의 속도는 실험 결과와 정확히 일치한다.