• Title/Summary/Keyword: Galerkin Approximate Method

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NUMERICAL METHODS FOR A STIFF PROBLEM ARISING FROM POPULATION DYNAMICS

  • Kim, Mi-Young
    • Korean Journal of Mathematics
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    • v.13 no.2
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    • pp.161-176
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    • 2005
  • We consider a model of population dynamics whose mortality function is unbounded. We note that the regularity of the solution depends on the growth rate of the mortality near the maximum age. We propose Gauss-Legendre methods along the characteristics to approximate the solution when the solution is smooth enough. It is proven that the scheme is convergent at fourth-order rate in the maximum norm. We also propose discontinuous Galerkin finite element methods to approximate the solution which is not smooth enough. The stability of the method is discussed. Several numerical examples are presented.

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hp-DISCONTINUOUS GALERKIN METHODS FOR THE LOTKA-MCKENDRICK EQUATION: A NUMERICAL STUDY

  • Jeong, Shin-Ja;Kim, Mi-Young;Selenge, Tsendanysh
    • Communications of the Korean Mathematical Society
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    • v.22 no.4
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    • pp.623-640
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    • 2007
  • The Lotka-McKendrick model which describes the evolution of a single population is developed from the well known Malthus model. In this paper, we introduce the Lotka-McKendrick model. We approximate the solution to the model using hp-discontinuous Galerkin finite element method. The numerical results show that the presented hp-discontinuous Galerkin method is very efficient in case that the solution has a sharp decay.

Dynamic Analysis of a Deploying Beam with Geometric Non-Linearity and Translating Acceleration (기하학적 비선형과 이송 가속도를 갖는 전개하는 보의 동적해석)

  • Song, Deok-Ki;Chung, Jin-Tai
    • Proceedings of the KSME Conference
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    • 2001.06b
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    • pp.658-663
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    • 2001
  • The dynamic response of an axially deploying beam is studied when the beam has geometric non-linearity and translating acceleration. Based upon the von Karman strain theory, the governing equations and the boundary conditions of a deploying beam are derived by using extended Hamilton's principle considering the longitudinal and transverse deflections. The equations of motion are discretized by using the Galerkin approximate method. From the discretized equations, the dynamic responses are computed by the Newmark time integration method.

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Higher Order Parabolic Equation Modeling Using Galerkin's Method (Galerkin방법을 이용한 고차 포물선 방정식 수중음 전달 해석)

  • 이철원;성우제;정문섭
    • The Journal of the Acoustical Society of Korea
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    • v.18 no.4
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    • pp.71-77
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    • 1999
  • Exact forward modeling of acoustic propagation is crucial in MFP such as inverse problems and various other acoustic applications. As acoustic propagation in shallow water environments become important, range dependent modeling has to be considered of which PE method is considered as one of the most accurate and relatively fast. In this paper higher order numerical rode employing the PE method is developed. To approximate the depth directional operator, Galerkin's method is used with partial collocation to lessen necessary calculations. Linearization of tile depth directional operator is achieved via expansion into a multiplication form of (equation omitted) approximation. To approximate the range directional equation, Crank-Nicolson's method is used. Final1y, numerical self stater is employed. Numerical tests are performed for various occan environment scenarios. The results of these tests are compared to exact solutions, OASES and RAM results.

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Brief and accurate analytical approximations to nonlinear static response of curled cantilever micro beams

  • Sun, Youhong;Yu, Yongping;Liu, Baochang
    • Structural Engineering and Mechanics
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    • v.56 no.3
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    • pp.461-472
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    • 2015
  • In this paper, the nonlinear static response of curled cantilever beam actuators subjected to the one-sided electrostatic field is focused on. By assuming the deflection function of electrostatically actuated beam, analytical approximate solutions are established via using Galerkin method to solve the equilibrium equation. The Pull-In voltages which determine the stability of the curled beam actuators are also obtained. These approximate solutions show excellent agreements with numerical solutions obtained by the shooting method and the experimental data for a wide range of beam length. Expressions of these analytical approximate solutions are brief and could easily be used to derive the effects of various physical parameters on MEMS structures.

Longitudinal Vibration Analysis of Deploying Rods (전개하는 막대의 종진동 해석)

  • Cho, Eun-Hyoung;Chung, Jin-Tai
    • Proceedings of the KSME Conference
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    • 2000.04a
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    • pp.625-630
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    • 2000
  • In this paper, the governing equation and the boundary conditions of deploying rods are derived by using Hamilton's principle. The Galerkin method using the comparison function of the instantaneous natural modes is adopted by which the governing equation is discretized. Based on the discretized equations, the time integration analysis is performed and the longitudinal vibrations for the deploying and the retrieving velocity are analyzed.

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A DISCONTINUOUS GALERKIN METHOD FOR A MODEL OF POPULATION DYNAMICS

  • Kim, Mi-Young;Yin, Y.X.
    • Communications of the Korean Mathematical Society
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    • v.18 no.4
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    • pp.767-779
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    • 2003
  • We consider a model of population dynamics whose mortality function is unbounded. We approximate the solution of the model using a discontinuous Galerkin finite element for the age variable and a backward Euler for the time variable. We present several numerical examples. It is experimentally shown that the scheme converges at the rate of $h^{3/2}$ in the case of piecewise linear polynomial space.

Simplified dynamic analysis of slender tapered thin-walled towers with additional mass and rigidity

  • Takabatake, Hideo;Mizuki, Akira
    • Structural Engineering and Mechanics
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    • v.3 no.1
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    • pp.61-74
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    • 1995
  • A linearly tapered, doubly symmetric thin-walled closed member, such as power-transmission towers and tourist towers, are often characterized by local variation in mass and/or rigidity, due to additional mass and rigidity. On the preliminary stage of design the closed-form solution is more effective than the finite element method. In order to propose approximate solutions, the discontinuous and local variation in mass and/or rigidity is treated continuously by means of a usable function proposed by Takabatake(1988, 1991, 1993). Thus, a simplified analytical method and approximate solutions for the free and forced transverse vibrations in linear elasticity are demonstrated in general by means of the Galerkin method. The solutions proposed here are examined from the results obtained using the Galerkin method and Wilson-${\theta}$ method and from the results obtained using NASTRAN.

HIGHER ORDER DISCONTINUOUS GALERKIN FINITE ELEMENT METHODS FOR NONLINEAR PARABOLIC PROBLEMS

  • Ohm, Mi Ray;Lee, Hyun Young;Shin, Jun Yong
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.18 no.4
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    • pp.337-350
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    • 2014
  • In this paper, we consider discontinuous Galerkin finite element methods with interior penalty term to approximate the solution of nonlinear parabolic problems with mixed boundary conditions. We construct the finite element spaces of the piecewise polynomials on which we define fully discrete discontinuous Galerkin approximations using the Crank-Nicolson method. To analyze the error estimates, we construct an appropriate projection which allows us to obtain the optimal order of a priori ${\ell}^{\infty}(L^2)$ error estimates of discontinuous Galerkin approximations in both spatial and temporal directions.

A NEW MIXED FINITE ELEMENT METHOD FOR BURGERS' EQUATION

  • Pany Ambit Kumar;Nataraj Neela;Singh Sangita
    • Journal of applied mathematics & informatics
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    • v.23 no.1_2
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    • pp.43-55
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    • 2007
  • In this paper, an $H^1-Galerkin$ mixed finite element method is used to approximate the solution as well as the flux of Burgers' equation. Error estimates have been derived. The results of the numerical experiment show the efficacy of the mixed method and justifies the theoretical results obtained in the paper.