• Title/Summary/Keyword: parabolic equation

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Effects of Stem Wave on the Vertical Breakwater (해안구조물 전면의 Stem Wave 특성에 관한 연구)

  • 박효봉;윤한삼;류청로
    • Proceedings of the Korea Committee for Ocean Resources and Engineering Conference
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    • 2001.10a
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    • pp.138-143
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    • 2001
  • Based on mild slope equation and parabolic approximation the forward diffraction of monochromatic waves by a straight breakwater are studied numerically. The characteristics and effects of stem wave along breakwater and the relations between the stem wave and incident wave angle are discussed.

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TIME DISCRETIZATION WITH SPATIAL COLLOCATION METHOD FOR A PARABOLIC INTEGRO-DIFFERENTIAL EQUATION WITH A WEAKLY SINGULAR KERNEL

  • Kim Chang-Ho
    • The Pure and Applied Mathematics
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    • v.13 no.1 s.31
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    • pp.19-38
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    • 2006
  • We analyze the spectral collocation approximation for a parabolic partial integrodifferential equations(PIDE) with a weakly singular kernel. The space discretization is based on the spectral collocation method and the time discretization is based on Crank-Nicolson scheme with a graded mesh. We obtain the stability and second order convergence result for fully discrete scheme.

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ERROR ANALYSIS OF FINITE ELEMENT APPROXIMATION OF A STEFAN PROBLEM WITH NONLINEAR FREE BOUNDARY CONDITION

  • Lee H.Y.
    • Journal of applied mathematics & informatics
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    • v.22 no.1_2
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    • pp.223-235
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    • 2006
  • By applying the Landau-type transformation, we transform a Stefan problem with nonlinear free boundary condition into a system consisting of a parabolic equation and the ordinary differential equations. Fully discrete finite element method is developed to approximate the solution of a system of a parabolic equation and the ordinary differential equations. We derive optimal orders of convergence of fully discrete approximations in $L_2,\;H^1$ and $H^2$ normed spaces.

Numerical Models of Water Wave with Parabolic and Hyperbolic Forms

  • Lee, Jong-Kyu;Lee, Chang-Hae
    • Korean Journal of Hydrosciences
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    • v.2
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    • pp.25-37
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    • 1991
  • The numerical models of the parabolic equation, applicable only to the progressive wave, and hyperblic equation, which may consider even the reflected wave, were developed and applied to the area of the submerged circular shoal and then results obtained from both models were compared with experimental measurements and each other. The hyperbolic model was further applied to both the detaced breakwater and the breakwater with a gap. The numerical results were plotted and compated with the existing data. Numerical solutions were obtained with the finite difference method.

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HEAT EQUATION WITH A GEOMETRIC ROUGH PATH POTENTIAL IN ONE SPACE DIMENSION: EXISTENCE AND REGULARITY OF SOLUTION

  • Kim, Hyun-Jung;Lototsky, Sergey V.
    • Communications of the Korean Mathematical Society
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    • v.34 no.3
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    • pp.757-769
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    • 2019
  • A solution of the heat equation with a distribution-valued potential is constructed by regularization. When the potential is the generalized derivative of a $H{\ddot{o}}lder$ continuous function, regularity of the resulting solution is in line with the standard parabolic theory.

NEW ALGORITHM FOR THE DETERMINATION OF AN UNKNOWN PARAMETER IN PARABOLIC EQUATIONS

  • Yue, Sufang;Cui, Minggen
    • The Pure and Applied Mathematics
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    • v.15 no.1
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    • pp.19-34
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    • 2008
  • A new algorithm for the solution of an inverse problem of determining unknown source parameter in a parabolic equation in reproducing kernel space is considered. Numerical experiments are presented to demonstrate the accuracy and the efficiency of the proposed algorithm.

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NONTRIVIAL PERIODIC SOLUTION FOR THE SUPERQUADRATIC PARABOLIC PROBLEM

  • Jung, Tacksun;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • v.17 no.1
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    • pp.53-66
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    • 2009
  • We show the existence of a nontrivial periodic solution for the superquadratic parabolic equation with Dirichlet boundary condition and periodic condition with a superquadratic nonlinear term at infinity which have continuous derivatives. We use the critical point theory on the real Hilbert space $L_2({\Omega}{\times}(0 2{\pi}))$. We also use the variational linking theorem which is a generalization of the mountain pass theorem.

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LERAY-SCHAUDER DEGREE THEORY APPLIED TO THE PERTURBED PARABOLIC PROBLEM

  • Jung, Tacksun;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • v.17 no.2
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    • pp.219-231
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    • 2009
  • We show the existence of at least four solutions for the perturbed parabolic equation with Dirichlet boundary condition and periodic condition when the nonlinear part cross two eigenvalues of the eigenvalue problem of the Laplace operator with boundary condition. We obtain this result by using the Leray-Schauder degree theory, the finite dimensional reduction method and the geometry of the mapping. The main point is that we restrict ourselves to the real Hilbert space instead of the complex space.

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NOTE ON LOCAL ESTIMATES FOR WEAK SOLUTION OF BOUNDARY VALUE PROBLEM FOR SECOND ORDER PARABOLIC EQUATION

  • Choi, Jongkeun
    • Bulletin of the Korean Mathematical Society
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    • v.53 no.4
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    • pp.1123-1148
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    • 2016
  • The aim of this note is to provide detailed proofs for local estimates near the boundary for weak solutions of second order parabolic equations in divergence form with time-dependent measurable coefficients subject to Neumann boundary condition. The corresponding parabolic equations with Dirichlet boundary condition are also considered.

ERROR ESTIMATES FOR FULLY DISCRETE DISCONTINUOUS GALERKIN METHOD FOR NONLINEAR PARABOLIC EQUATIONS

  • Ohm, Mi-Ray;Lee, Hyun-Yong;Shin, Jun-Yong
    • Journal of applied mathematics & informatics
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    • v.28 no.3_4
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    • pp.953-966
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    • 2010
  • In this paper, we develop discontinuous Galerkin methods with penalty terms, namaly symmetric interior penalty Galerkin methods to solve nonlinear parabolic equations. By introducing an appropriate projection of u onto finite element spaces, we prove the optimal convergence of the fully discrete discontinuous Galerkin approximations in ${\ell}^2(L^2)$ normed space.