• Journal of applied mathematics & informatics
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• v.12 no.1_2
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• pp.81-105
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• 2003

This paper deals with the very interesting problem about the influence of piecewise smooth boundary conditions on the distribution of the eigenvalues of the negative Laplacian in R$^3$. The asymptotic expansion of the trace of the wave operator (equation omitted) for small ｜t｜ and i=√-1, where (equation omitted) are the eigenvalues of the negative Laplacian (equation omitted) in the (x$^1$, x$^2$, x$^3$)-space, is studied for an annular vibrating membrane $\Omega$ in R$^3$together with its smooth inner boundary surface S$_1$and its smooth outer boundary surface S$_2$. In the present paper, a finite number of Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components (equation omitted)(i = 1,...,m) of S$_1$and on the piecewise smooth components (equation omitted)(i = m +1,...,n) of S$_2$such that S$_1$= (equation omitted) and S$_2$= (equation omitted) are considered. The basic problem is to extract information on the geometry of the annular vibrating membrane $\Omega$ from complete knowledge of its eigenvalues by analysing the asymptotic expansions of the spectral function (equation omitted) for small ｜t｜.

• Journal of the Korean Mathematical Society
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• v.44 no.6
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• pp.1417-1428
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• 2007

We obtain weighted $L^p$ estimates $(1{\leq}p<{\infty})\;for\;\bar{\partial}$ on convex domains with piecewise smooth boundaries in $\mathbb{C}^2$ by using explicit formulas of solutions introduced by Berndtsson and Andersson.

• Journal of the Korean Mathematical Society
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• v.40 no.6
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• pp.1075-1083
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• 2003

The purpose of this paper is to investigate the piecewise wise polynomial approximation for the curved boundary. We analyze the error of an approximated solution due to this approximation and then compare the approximation errors for the cases of polygonal and piecewise polynomial approximations for the curved boundary. Based on the results of analysis, p-version numerical methods for solving Dirichlet's problems are applied to any smooth curved domain.

• Bulletin of the Korean Mathematical Society
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• v.59 no.1
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• pp.45-72
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• 2022

In this work, we study Bernstein-Walsh-type estimations for the derivative of an arbitrary algebraic polynomial in regions with piecewise smooth boundary without cusps of the complex plane. Also, estimates are given on the whole complex plane.

• East Asian mathematical journal
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• v.30 no.5
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• pp.583-597
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• 2014

In this paper, we study an area-preserving piecewise linear map with the feature of dangerous border collision bifurcations. Using this map, we study dynamical properties occurred in the invariant set, specially related to the boundary of KAM-tori, and the existence and stabilities of periodic orbits. The result shows that elliptic regions having periodic orbits and chaotic region can be divided by smooth curve, which is an unexpected result occurred in area preserving smooth dynamical systems.

• Journal of applied mathematics & informatics
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• v.22 no.1_2
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• pp.49-65
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• 2006

A robust numerical method for a singularly perturbed second-order ordinary differential equation having two parameters with a discontinuous source term is presented in this article. Theoretical bounds are derived for the derivatives of the solution and its smooth and singular components. An appropriate piecewise uniform mesh is constructed, and classical upwind finite difference schemes are used on this mesh to obtain the discrete system of equations. Parameter-uniform error bounds for the numerical approximations are established. Numerical results are provided to illustrate the convergence of the numerical approximations.