• Title/Summary/Keyword: parabolic equation

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Stability Improved Split-step Parabolic Equation Model

  • Kim, Tae-Hyun;Seong, Woojae
    • The Journal of the Acoustical Society of Korea
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    • v.21 no.3E
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    • pp.105-111
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    • 2002
  • The parabolic equation technique provides an excellent model to describe the wave phenomena when there exists a predominant direction of propagation. The model handles the square root wave number operator in paraxial direction. Realization of the pseudo-differential square root operator is the essential part of the parabolic equation method for its numerical accuracy. The wide-angled approximation of the operator is made based on the Pade series expansion, where the branch line rotation scheme can be combined with the original Pade approximation to stabilize its computational performance for complex modes. The Galerkin integration has been employed to discretize the depth-dependent operator. The benchmark tests involving the half-infinite space, the range independent and dependent environment will validate the implemented numerical model.

PARABOLIC QUATERNIONIC MONGE-AMPÈRE EQUATION ON COMPACT MANIFOLDS WITH A FLAT HYPERKÄHLER METRIC

  • Zhang, Jiaogen
    • Journal of the Korean Mathematical Society
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    • v.59 no.1
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    • pp.13-33
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    • 2022
  • The quaternionic Calabi conjecture was introduced by Alesker-Verbitsky, analogous to the Kähler case which was raised by Calabi. On a compact connected hypercomplex manifold, when there exists a flat hyperKähler metric which is compatible with the underlying hypercomplex structure, we will consider the parabolic quaternionic Monge-Ampère equation. Our goal is to prove the long time existence and C convergence for normalized solutions as t → ∞. As a consequence, we show that the limit function is exactly the solution of quaternionic Monge-Ampère equation, this gives a parabolic proof for the quaternionic Calabi conjecture in this special setting.

A GLOBALITY OF A HOPF BIFURCATION IN A FREE BOUNDARY PROBLEM

  • Ham, Yoon-Mee
    • 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.

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POLLUTION DETECTION FOR THE SINGULAR LINEAR PARABOLIC EQUATION

  • IQBAL M. BATIHA;IMAD REZZOUG;TAKI-EDDINE OUSSAEIF;ADEL OUANNAS;IQBAL H. JEBRIL
    • Journal of applied mathematics & informatics
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    • v.41 no.3
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    • pp.647-656
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    • 2023
  • In this work, we are concerned by the problem of identification of noisy terms which arise in singular problem as for remote sensing problems, and which are modeled by a linear singular parabolic equation. For the reason of missing some data that could be arisen when using the traditional sentinel method, the later will be changed by a new sentinel method for attaining the same purpose. Such new method is a particular least square-like method which permits one to distinguish between the missing terms and the pollution terms. In particular, a sentinel method will be given here in its more realistic setting for singular parabolic problems, where in this case, the observation and the control have their support in different open sets. The problem of finding a new sentinel is equivalent to finding singular optimality system of the least square control for the parabolic equation that we solve.

Model Reference Adaptive Control of a Time-Varying Parabolic System

  • Hong, Keum-Shik;Yang, Kyung-Jinn;Kang, Dong-Hunn
    • Journal of Mechanical Science and Technology
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    • v.14 no.2
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    • pp.168-176
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    • 2000
  • Related to the error dynamics of an adaptive system, averaging theorems are developed for coupled differential equations which consist of ordinary differential equations and a parabolic partial differential equation. The results are then applied to the convergence analysis of the parameter estimate errors in the model reference adaptive control of a nonautonomous parabolic partial differential equation with lowly time-varying parameters.

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Comparison of Parabolic Mild-Slope Equations in View of Wave Diffraction (회절현상의 관점에서 본 포물선형 완경사방정식의 비교)

  • 이해균;이길성;이창훈
    • Journal of Korean Society of Coastal and Ocean Engineers
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    • v.10 no.1
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    • pp.1-9
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    • 1998
  • Among the phenomena of water-wave transformation, the wave diffraction is prominent for waves insidc the harbor. It is important to study how accurately the diffraction can be resolved by the numerical model. Three parabolic mild-slope equations, i.e., simple, wide-ang1e, three-parameter parabolic equations, are compared in view of the diffraction of water-waves around a semi-infinite breakwater. To avoid reflections at lateral boundaries, we apply the perfect boundary condition of Dalrymple and Martin (1992) in case of simple parabolic equation. The numerical results for the case of a semi-infinite breakwater are compared with the analytical solution of Penney and Price (1952). All the results are very accurate when waves attack the breakwater normally. When waves attack the breakwater obliquely, however, the simple parabolic equation yields the worst solution and the three-parameter parabolic equation yields the most accurate solution.

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Analysis of Acoustic Propagation using Spectral Parabolic Equation Method (스펙트럴 포물선 방정식 법을 이용한 수중음파 전달해석)

  • Kim, Kook-Hyun;Seong, Woo-Jae
    • The Journal of the Acoustical Society of Korea
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    • v.15 no.2
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    • pp.72-78
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    • 1996
  • This thesis deals with a method to solve a two-and-one-half-dimensional ($2\frac12$ D) problem, which means that the ocean environment is two-dimensional whereas the source is fully three-dimensionally propagating, including three-dimensional refraction phenomena and three-dimensional back-scattering, using two-dimensional two-way parabolic equation method combined with Fourier synthesis. Two dimensional two-way parabolic equation method uses Galerkin's method for depth and Crank-Nicolson method and alternating direction for range and provides a solution available to range-dependent problem with wave-field back-scattered from discontinuous interface. Since wavenumber, k, is the function of depth and vertical or horizontal range, we can reduce a dimension of three-dimensional Helmholtz equation by Fourier transforming in the range direction. Thus transformed two-dimensional Helmholtz equation is solved through two-way parabolic equation method. Finally, we can have the $2\frac12$ D solution by inverse Fourier transformation of the spectral solution gained from in the last step. Numerical simulation has been carried out for a canonical ocean environment with stair-step bottom in order to test its accuracy using the present analysis. With this spectral parabolic equation method, we have examined three-dimensional acoustic propagation properties in a specified site in the Korean Straits.

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Numerical Analysis of Waves from Point Source in Variable Depth Using Parabolic Wave Equation in Polar Coordinates (極座標 抛物形 波動方程式을 이용한 變數深 点源波의 數値解析)

  • 곽문수;편종근
    • Journal of Korean Society of Coastal and Ocean Engineers
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    • v.11 no.1
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    • pp.68-74
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    • 1999
  • The Green function method is widely used for the analysis of waves in a harbor with a constant depth. In extending this method to a wave field over arbitrary depth, a generalized and convenient method is needed to obtain unit solutions for waves emerging from a point source. For this purpose, a parabolic wave equation is derived to approximate the mild-slope equation written in terms of polar coordinates. Usefulness of the equation obtained is examined through trial computations.

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UNIQUENESS OF SOLUTIONS FOR A DEGENERATE PARABOLIC EQUATION WITH ABSORPTION

  • Lee, Jin Ho;Jang, Seong Hee
    • Korean Journal of Mathematics
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    • v.5 no.2
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    • pp.151-167
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    • 1997
  • We estimate the interior Lipschitz norm and maximum of the solution for degenerate parabolic equations with absorption. Also obtain the growth rate of the solution $u$ in terms of time $t$. From this we show the uniqueness of solution with respect to the initial trace.

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