• Title/Summary/Keyword: Hyperbolic equation

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CONVERGENCE OF APPROXIMATE SOLUTIONS TO SCALAR CONSERVATION LAWS BY DEGENERATE DIFFUSION

  • Hwang, Seok
    • Communications of the Korean Mathematical Society
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    • v.22 no.1
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    • pp.145-155
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    • 2007
  • In this paper, we show the convergence of approximate solutions to the convective porous media equation using methodology developed in [8]. First, we obtain the approximate transport equation for the given convective porous media equation. Then using the averaging lemma, we obtain the convergence.

Transient heat transfer in thin films (초박막에서의 비정상 열전달)

  • Bai, C.H.;Chung, M.
    • Transactions of the Korean Society of Mechanical Engineers B
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    • v.22 no.1
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    • pp.1-11
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    • 1998
  • For the analysis of phonon heat transfer within short time and spatial scales, conventional macroscopic heat conduction equations with jump boundary conditions are tried and the results are compared to those of equation of phonon radiative transport(EPRT), which is one of microscopic transport equation. In transient state the macroscopic temperatures show far different behavior from EPRT. In steady state the hyperbolic temperatures with temperature jump at the wall from time relaxation model agrees well with EPRT temperatures. Since EPRT is also an approximate form of microscopic transport equation and there are no experimental results to verify the proposed model in this study, we can not conclude whether the approaching method from this study is valid or not. To the authors' knowledge, there are no experimental results available which can be used to test the validity of these models. Such an experiment, while difficult to conduct, would be invaluable.

Counter-Current Flow Limitation Model Based on the Hyperbolic Two-fluid Equations and Interface Shape Function (쌍곡선형 이상유동 방정식과 경계면 모양함수를 이용한 유체기계의 역류유동제한점 예측방법 개발)

  • 정지환
    • Journal of the Korea Academia-Industrial cooperation Society
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    • v.1 no.1
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    • pp.15-22
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    • 2000
  • There are lots of industrial machines of which functions are achieved by operation of multi-phase fluids. Some of them take advantage of the characteristics of counter-current two-phase flow The maximum flow rates of gas and liquid phases which flow in opposite-directions (counter-current flow) are limited by a phenomenon known as a Counter-Current Flow Limitation (CCFL or Flooding) The mass and momentum conservation equations for each Phase were established to build a first-order hyperbolic partial derivative equations system. A new CCFL model is developed based on the characteristic equation of the hyperbolic PDE system. The present model has its applicationto the case in which a non-uniform flow is developed around a square or sharp-edged entrance of liquid phase. The model is able to he used to Predict the operating-limit of components in which mass and heat transfer are taking place between liquid and gas phases.

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Solving Dynamic Equation Using Combination of Both Trigonometric and Hyperbolic Cosine Functions for Approximating Acceleration

  • Quoc Do Kien;Phuoc Nguyen Trong
    • Journal of Mechanical Science and Technology
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    • v.19 no.spc1
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    • pp.481-486
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    • 2005
  • This paper introduces a numerical method for integration of the linear and nonlinear differential dynamic equation of motion. The variation of acceleration in two time steps is approximated as a combination of both trigonometric cosine and hyperbolic cosine functions with weighted coefficient. From which all necessary formulae are elaborated for the direct integration of the governing equation. A number of linear and nonlinear dynamic problems with various degrees of freedom are analysed using both the suggested method and Newmark method for the comparison. The numerical results show high advantages and effectiveness of the new method.

Analytical solution of the Cattaneo - Vernotte equation (non-Fourier heat conduction)

  • Choi, Jae Hyuk;Yoon, Seok-Hun;Park, Seung Gyu;Choi, Soon-Ho
    • Journal of Advanced Marine Engineering and Technology
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    • v.40 no.5
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    • pp.389-396
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    • 2016
  • The theory of Fourier heat conduction predicts accurately the temperature profiles of a system in a non-equilibrium steady state. However, in the case of transient states at the nanoscale, its applicability is significantly limited. The limitation of the classical Fourier's theory was overcome by C. Cattaneo and P. Vernotte who developed the theory of non-Fourier heat conduction in 1958. Although this new theory has been used in various thermal science areas, it requires considerable mathematical skills for calculating analytical solutions. The aim of this study was the identification of a newer and a simpler type of solution for the hyperbolic partial differential equations of the non-Fourier heat conduction. This constitutes the first trial in a series of planned studies. By inspecting each term included in the proposed solution, the theoretical feasibility of the solution was achieved. The new analytical solution for the non-Fourier heat conduction is a simple exponential function that is compared to the existing data for justification. Although the proposed solution partially satisfies the Cattaneo-Vernotte equation, it cannot simulate a thermal wave behavior. However, the results of this study indicate that it is possible to obtain the theoretical solution of the Cattaneo-Vernotte equation by improving the form of the proposed solution.

ON AN EQUATION CONNECTED WITH THE THEORY FOR SPREADING OF ACOUSTIC WAVE

  • Zikirov, O.S.
    • East Asian mathematical journal
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    • v.27 no.1
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    • pp.51-65
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    • 2011
  • In the paper, we study questions on classical solvability of nonlocal problems for a third-order linear hyperbolic equation in a rectangular domain. The Riemann method is applied to the Goursat problem and solution is obtained in the integral form. Investigated problems are reduced to the uniquely solvable Volterra-type equation of second kind. Influence effects of coefficients at lowest derivatives on correctness of studied problems are detected.

RICCATI EQUATION IN QUADRATIC OPTIMAL CONTROL PROBLEM OF DAMPED SECOND ORDER SYSTEM

  • Ha, Junhong;Nakagiri, Shin-Ichi
    • Journal of the Korean Mathematical Society
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    • v.50 no.1
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    • pp.173-187
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    • 2013
  • This paper studies the properties of solutions of the Riccati equation arising from the quadratic optimal control problem of the general damped second order system. Using the semigroup theory, we establish the weak differential characterization of the Riccati equation for a general class of the second order distributed systems with arbitrary damping terms.

ON THE SUPERSTABILITY OF THE PEXIDER TYPE SINE FUNCTIONAL EQUATION

  • Kim, Gwang Hui
    • Journal of the Chungcheong Mathematical Society
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    • v.25 no.1
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    • pp.1-18
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    • 2012
  • The aim of this paper is to investigate the superstability of the pexider type sine(hyperbolic sine) functional equation $f(\frac{x+y}{2})^{2}-f(\frac{x+{\sigma}y}{2})^{2}={\lambda}g(x)h(y),\;{\lambda}:\;constant$ which is bounded by the unknown functions ${\varphi}(x)$ or ${\varphi}(y)$. As a consequence, we have generalized the stability results for the sine functional equation by P. M. Cholewa, R. Badora, R. Ger, and G. H. Kim.

Some Modifications of MacCormark's Methods (MacCormack 방법의 개량에 대한 연구)

  • Ha, Young-Soo;Yoo, Seung-Jae
    • Convergence Security Journal
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    • v.5 no.3
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    • pp.93-97
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    • 2005
  • MacCormack's method is an explicit, second order finite difference scheme that is widely used in the solution of hyperbolic partial differential equations. Apparently, however, it has shown entropy violations under small discontinuity. This non-physical shock grows fast and eventually all the meaningful information of the solution disappears. Some modifications of MacCormack's methods follow ideas of central schemes with an advantage of second order accuracy for space and conserve the high order accuracy for time step also. Numerical results are shown to perform well for the one-dimensional Burgers' equation and Euler equations gas dynamic.

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