• Communications of the Korean Mathematical Society
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• v.17 no.1
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• pp.103-120
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• 2002

A second order upwind method for linear hyperbolic systems is studied in this paper. The method approximates solutions as piecewise linear functions, and state variables and slopes of the linear functions for next time step are computed separately. We present a new method for the computation of slopes, derived from an upwinding difference for a derivative. For nonoscillatory solutions, a monotonicity algorithm is also proposed by modifying an existing algorithm. To validate our second order upwind method, numerical results for linear advection equations and linear systems for elastic and acoustic waves are given.

• Journal of the Korean Mathematical Society
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• v.53 no.4
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• pp.737-767
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• 2016

This paper concerns closed hypersurfaces of dimension $n{\geq}2$ in the hyperbolic space ${\mathbb{H}}_{\kappa}^{n+1}$ of constant sectional curvature evolving in direction of its normal vector, where the speed equals a power ${\beta}{\geq}1$ of the mean curvature. The main result is that if the initial closed, weakly h-convex hypersurface satisfies that the ratio of the biggest and smallest principal curvature at everywhere is close enough to 1, depending only on n and ${\beta}$, then under the flow this is maintained, there exists a unique, smooth solution of the flow which converges to a single point in ${\mathbb{H}}_{\kappa}^{n+1}$ in a maximal finite time, and when rescaling appropriately, the evolving hypersurfaces exponential convergence to a unit geodesic sphere of ${\mathbb{H}}_{\kappa}^{n+1}$.

• Bulletin of the Korean Mathematical Society
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• v.46 no.3
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• pp.409-434
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• 2009

In this paper, we consider a generalized Riemann problem of the first order hyperbolic conservation laws. For the case that excludes the centered wave, we prove that the generalized Riemann problem admits a unique piecewise smooth solution u = u(t, x), and this solution has a structure similar to the similarity solution u = $U{(\frac{x}{t})}$ of the correspondin Riemann problem in the neighborhood of the origin provided that the coefficients of the system and the initial conditions are sufficiently smooth.

• Journal of the Korean Mathematical Society
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• v.48 no.1
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• pp.63-82
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• 2011

On the hyperbolic space $D^n$, codimension-one totally geodesic foliations of class $C^k$ are classified. Except for the unique parabolic homogeneous foliation, the set of all such foliations is in one-one correspondence (up to isometry) with the set of all functions z : [0, $\pi$] $\rightarrow$ $S^{n-1}$ of class $C^{k-1}$ with z(0) = $e_1$ = z($\pi$) satisfying |z'(r)| ${\leq}1$ for all r, modulo an isometric action by O(n-1) ${\times}\mathbb{R}{\times}\mathbb{Z}_2$. Since 1-dimensional metric foliations on $D^n$ are always either homogeneous or flat (that is, their orthogonal distributions are integrable), this classifies all 1-dimensional metric foliations as well. Equations of leaves for a non-trivial family of metric foliations on $D^2$ (called "fifth-line") are found.

• Honam Mathematical Journal
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• v.29 no.4
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• pp.723-732
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• 2007

We investigate the existence of solutions u(x, t) for a perturbation b[$(\xi+\eta+1)^+-1$] of the hyperbolic system with Dirichlet boundary condition (0.1) = $L\xi-{\mu}[(\xi+\eta+1)^+-1]+f$ in $(-\frac{\pi}{2},\frac{\pi}{2}\;{\times})\;\mathbb{R}$, $L\eta={\nu}[(\xi+\eta+1)^+-1]+f$ in $(-\frac{\pi}{2},\frac{\pi}{2}\;{\times})\;\mathbb{R}$ where $u^+$ = max{u,0}, ${\mu},\nu$ are nonzero constants. Here $\xi,\eta$ are periodic functions.

• Smart Structures and Systems
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• v.15 no.6
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• pp.1569-1582
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• 2015

Two hyperbolic displacement models are used for the bending response of simply-supported orthotropic laminated composite plates resting on two-parameter elastic foundations under mechanical loading. The models contain hyperbolic expressions to account for the parabolic distributions of transverse shear stresses and to satisfy the zero shear-stress conditions at the top and bottom surfaces of the plates. The present theory takes into account not only the transverse shear strains, but also their parabolic variation across the plate thickness and requires no shear correction coefficients in computing the shear stresses. The governing equations are derived and their closed-form solutions are obtained. The accuracy of the models presented is demonstrated by comparing the results obtained with solutions of other theories models given in the literature. It is found that the theories proposed can predict the bending analysis of cross-ply laminated composite plates resting on elastic foundations rather accurately. The effects of Winkler and Pasternak foundation parameters, transverse shear deformations, plate aspect ratio, and side-to-thickness ratio on deflections and stresses are investigated.

• Proceedings of the Korean Geotechical Society Conference
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• 2004.03b
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• pp.765-774
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• 2004

This paper presents a damage assesment method for buried pipelines subjected to Deep Excavation-induced ground movements. Ground deformation characteristics resulting from 3D finite element analysis was represented mathematically by a hyperbolic tangential function. A parametric study was performed on excavation depth and burial position of pipeline. The result of the parametric study indicate that length of hyperbolic tangential function affects the results of damage assessment. Using numerical studies for buried pipeline response to ground movements by relative flexibility of the pipe-soil system. The result of numerical studies are presented in forms of design charts which can be readily used for various condition encountered in practices.

• Journal of Ocean Engineering and Technology
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• v.5 no.1
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• pp.35-44
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• 1991

This study was carried out for the purpose of pre-estimating long-term settlement under condition of actual field soil's property, in case of building up industrial sites on the marine deposit silty clay located at West Coast in Korea. This study analyzed Hyperbolic Method, Square Root Time Method and Exponential Function Method with utilization of measured survey values of settlement in In-Cheon Namdong Industrial Sites. In the future, for the continuos utilization, it seemed to be needed that further the survey values of fields should be accurartely measured for the analysis of more accurate pre-estimate about long-term settlement. Among the prediction methods of settlement Hyperbolic Method seemed to be the best fitting method for measured data. The settlement equations were derived from above three methods, for long-term settlements.

• Proceedings of the KSR Conference
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• 2001.10a
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• pp.552-557
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• 2001

Copper slag is the by-producted material on the proceeding of refining the copper. To verify applications of copper slag to vertical drain material can substitute for the sands in ground improvement, laboratory soil tests and consolidation model tests were conducted. The results of consolidation model test was analyzed as the hyperbolic method. The hyperbolic method assumes that the settlement(s) versus time(t) behavior approaches a straight line describes a hyperbolic reaction. The inverse of the slope of the line would then yield the ultimate settlement. Through in this study, copper slag is compatible with vertical drain material as like sands. Copper slag compaction pile promote the consolidation settlement.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.17 no.4
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• pp.279-294
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• 2013

We are interested in an adaptive grid method for hyperbolic equations. A multiresolution analysis, based on a biorthogonal family of interpolating scaling functions and lifted interpolating wavelets, is used to dynamically adapt grid points according to the physical field profile in each time step. Traditional finite-difference schemes with fixed stencils produce high oscillations around sharp discontinuities. In this paper, we hybridize high-resolution schemes, which are suitable for capturing singularities, and apply a finite-difference approach to the scaling functions at non-singular points. We use a total variation diminishing Runge-Kutta method for the time integration. The computational cost is proportional to the number of points present after compression. We provide several numerical examples to verify our approach.