• Title, Summary, Keyword: Sobolev equation

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A CHARACTERIZATION OF SOBOLEV SPACES BY SOLUTIONS OF HEAT EQUATION AND A STABILITY PROBLEM FOR A FUNCTIONAL EQUATION

  • Chung, Yun-Sung;Lee, Young-Su;Kwon, Deok-Yong;Chung, Soon-Yeong
    • Communications of the Korean Mathematical Society
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    • v.23 no.3
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    • pp.401-411
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    • 2008
  • In this paper, we characterize Sobolev spaces $H^s(\mathbb{R}^n),\;s{\in}\mathbb{R}$ by the initial value of solutions of heat equation with a growth condition. By using an idea in its proof, we also discuss a stability problem for Cauchy functional equation in the Sobolev spaces.

SOBOLEV ORTHOGONAL POLYNOMIALS RELATIVE TO ${\lambda}$p(c)q(c) + <${\tau}$,p'(x)q'(x)>

  • Jung, I.H.;Kwon, K.H.;Lee, J.K.
    • Communications of the Korean Mathematical Society
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    • v.12 no.3
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    • pp.603-617
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    • 1997
  • Consider a Sobolev inner product on the space of polynomials such as $$ \phi(p,q) = \lambda p(c)q(c) + <\tau,p'(x)q'(x)> $$ where $\tau$ is a moment functional and c and $\lambda$ are real constants. We investigate properties of orthogonal polynomials relative to $\phi(\cdot,\cdot)$ and give necessary and sufficient conditions under which such Sobolev orthogonal polynomials satisfy a spectral type differential equation with polynomial coefficients.

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AN EXTRAPOLATED CRANK-NICOLSON CHARACTERISTIC FINITE ELEMENT METHOD FOR SOBOLEV EQUATIONS

  • Ohm, Mi Ray;Shin, Jun Yong
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.4
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    • pp.1409-1419
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    • 2017
  • We introduce an extrapolated Crank-Nicolson characteristic finite element method to approximate solutions of a convection dominated Sobolev equation. We obtain the higher order of convergence in both the spatial direction and the temporal direction in $L^2$ normed space for the extrapolated Crank-Nicolson characteristic finite element method.

AN EXTRAPOLATED HIGHER ORDER CHARACTERISTIC FINITE ELEMENT METHOD FOR SOBOLEV EQUATIONS

  • Ohm, Mi Ray;Shin, Jun Yong
    • East Asian mathematical journal
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    • v.33 no.5
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    • pp.511-525
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    • 2017
  • We introduce an extrapolated higher order characteristic finite element method to construct approximate solutions of a Sobolev equation with a convection term. The higher order of convergence in both the temporal direction and the spatial direction in $L^2$ normed space is established and some computational results to support our theoretical results are presented.

A CRANK-NICOLSON CHARACTERISTIC FINITE ELEMENT METHOD FOR NONLINEAR SOBOLEV EQUATIONS

  • Ohm, Mi Ray;Shin, Jun Yong
    • East Asian mathematical journal
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    • v.33 no.3
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    • pp.295-308
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    • 2017
  • We introduce a Crank-Nicolson characteristic finite element method to construct approximate solutions of a nonlinear Sobolev equation with a convection term. And for the Crank-Nicolson characteristic finite element method, we obtain the higher order of convergence in the temporal direction and in the spatial direction in $L^2$ normed space.

SOBOLEV ESTIMATES FOR THE LOCAL EXTENSION OF BOUNDARY HOLOMORPHIC FORMS ON REAL HYPERSURFACES IN ℂn

  • Cho, Sanghyun
    • Journal of the Korean Mathematical Society
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    • v.50 no.3
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    • pp.479-491
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    • 2013
  • Let M be a smooth real hypersurface in complex space of dimension $n$, $n{\geq}3$, and assume that the Levi-form at $z_0$ on M has at least $(q+1)$-positive eigenvalues, $1{\leq}q{\leq}n-2$. We estimate solutions of the local $\bar{\partial}$-closed extension problem near $z_0$ for $(p,q)$-forms in Sobolev spaces. Using this result, we estimate the local solution of tangential Cauchy-Riemann equation near $z_0$ in Sobolev spaces.

A CRANK-NICOLSON CHARACTERISTIC FINITE ELEMENT METHOD FOR SOBOLEV EQUATIONS

  • Ohm, Mi Ray;Shin, Jun Yong
    • East Asian mathematical journal
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    • v.32 no.5
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    • pp.729-744
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    • 2016
  • A Crank-Nicolson characteristic finite element method is introduced to construct approximate solutions of a Sobolev equation with a convection term. The higher order of convergences in the temporal direction and in the spatial direction in $L^2$ normed space are verified for the Crank-Nicolson characteristic finite element method.