• Bulletin of the Korean Mathematical Society
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• v.36 no.4
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• pp.637-647
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• 1999

In this paper, we consider a free boundary problem with Pushchino dynamics satisfying the Dirichlet boundary condition. We shall the stationary solutions ($v^{*}(x),s^{*}$) exist and the Hopf bifurcation occurs at a critical point when s $\in$ (1/3,1).

• Proceedings of the KSME Conference
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• 2003.04a
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• pp.740-745
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• 2003

The so-called boundary node method (or NDIF method) that was developed by the authors has been extended for free vibration analysis of arbitrarily shaped plates with free edges. Since the proposed method is based on the collocation method, no integration procedure is needed on boundary edges of the plates and only a small amount of numerical calculation is required. A special coordinate transformation has been devised to consider the complicated free boundary conditions at boundary nodes. By the use of the special coordinate transformation, the radius of curvature involved in the free boundary conditions can be successfully dealt with. Finally, verification examples show that natural frequencies obtained by the present method agree well with those given by exact method and other analytical methods.

• Journal of the Korean Mathematical Society
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• v.32 no.2
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• pp.237-250
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• 1995

A hopf bifurcation of a free boundary (or an internal layer) occurs in solidification, chemical reactions and combustion. It is a well-known fact that a free boundary usually appear as sharp transitions with narrow width between two materials ([2]).

• Korean Journal of Mathematics
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• v.6 no.2
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• pp.173-181
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• 1998

The local behaviors of the Hopf bifurcation in the free boundary problem satisfying the zero flux boundary condition was examined in [3]. In this paper, we shall examine the global behaviors for this problem and shall apply the center-index theory to show the globality.

• Steel and Composite Structures
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• v.28 no.6
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• pp.655-670
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• 2018

A coupled method, that combines the Ritz method and the finite element (FE) method, is proposed to solve the vibration problem of rectangular thin and thick plates with general boundary conditions. The eigenvalue partial differential equation(s) of the plate is (are) first reduced to a set of eigenvalue ordinary differential equations by the application of the Ritz method. The resulting eigenvalue differential equations are then reduced to an eigenvalue algebraic equation system using the finite element method. The natural boundary conditions of the plate problem including the free edge and free corner boundary conditions are also implemented in a simple and accurate manner. Various boundary conditions including simply supported, clamped and free boundary conditions are considered. Comparisons with existing numerical and analytical solutions show that the proposed mixed method can produce highly accurate results for the problems considered using a small number of Ritz terms and finite elements. The proposed mixed Ritz-FE formulation is also compared with the mixed FE-Ritz formulation which has been recently proposed by the present author and his co-author. It is found that the proposed mixed Ritz-FE formulation is more efficient than the mixed FE-Ritz formulation for free vibration analysis of rectangular plates with Levy-type boundary conditions.

• Communications of the Korean Mathematical Society
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• v.14 no.2
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• pp.441-447
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• 1999

A free boundary problem is derived from a singular limit system of a reaction diffusion equation whose reaction terms are bistable type. In this paper, we shall consider a free boundary problem with two layers satisfying the zero flux boundary condition and shall show that the Hopf bifurcation can not occur as a parameter varies.

• Selected Papers of The Society of Naval Architects of Korea
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• v.2 no.1
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• pp.63-78
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• 1994

This paper deals with the treatment of the open boundary in two-dimensional free-surface wave problems. Two numerical schemes are investigated for the implementation of the open boundary condition. One is to add the artificial damping term to the dynamic free-surface boundary condition, in which the determination of suitable damping coefficient and the damping zone is the most important. The other is a modified Orlanski's method, which is known to be very useful for the uni-directional waves. Using these two schemes, numerical tests have been conducted for a few typical free-surface wave problems. To obtain the numerical solution of the free-surface boundary value problem, the fundamental source-distribution method is used and the fully nonlinear free-surface boundary conditions are applied. The computed results are presented in comparison with those of others for the proof of practicality of these two schemes.

• Structural Engineering and Mechanics
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• v.78 no.4
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• pp.497-518
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• 2021

In this study, a new method for treating the wall boundary in smoothed particle hydrodynamics (SPH) is proposed to simulate free surface flows effectively. Unlike conventional methods of wall boundary treatment through boundary particles, in the proposed method, the wall boundary condition is directly imposed by adding boundary truncation terms to the mass and momentum conservation equations. Thus, boundary particles are not used in boundary modeling. Doing so, the wall boundary condition is accurately imposed, boundary modeling is simplified, and computation is made efficient without losing stability in SPH. Performance of the proposed method is demonstrated through several numerical examples: dam break, dam break with a wedge, sloshing, inclined bed, cross-lever rotation, pulsating tank and sloshing with a flexible baffle. These results are compared with available experimental results, analytical solutions, and results obtained using the boundary particle method.

• Bulletin of the Korean Mathematical Society
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• v.33 no.2
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• pp.319-328
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• 1996

In [3], they deal with the free boundary problem with Pushchino dynamics. They showed the existence of solutions and the occurence of a Hopf bifurcation.

• Journal of the Korean Mathematical Society
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• v.34 no.2
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• pp.395-405
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• 1997

A globality of the Hopf bifurcation in a free boundary problem for a parabolic partial differential equation is investigated in this paper. We shall examine the global behavior of the Hopf critical eigenvalues and and apply the center-index theory to show the globality.