• Title, Summary, Keyword: diffusion equation

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A NOTE ON NUMERICAL APPROACHES FOR HEAT-DIFFUSION EQUATION WITH HETEROGENEOUS MEDIA AND ITS APPLICATIONS

  • Seo, Sat byul
    • East Asian mathematical journal
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    • v.35 no.1
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    • pp.99-108
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    • 2019
  • In this paper, we introduce a numerical approach to solve heat-diffusion equation with discontinuous diffusion coefficients in the three dimensional rectangular domain. First, we study the support operator method and suggest a new method, the continuous velocity method. Further, we apply both methods to a diffusion process for neurotransmitter release in an individual synapse and compare their results.

Diffusion synthetic acceleration with the fine mesh rebalance of the subcell balance method with tetrahedral meshes for SN transport calculations

  • Muhammad, Habib;Hong, Ser Gi
    • Nuclear Engineering and Technology
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    • v.52 no.3
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    • pp.485-498
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    • 2020
  • A diffusion synthetic acceleration (DSA) technique for the SN transport equation discretized with the linear discontinuous expansion method with subcell balance (LDEM-SCB) on unstructured tetrahedral meshes is presented. The LDEM-SCB scheme solves the transport equation with the discrete ordinates method by using the subcell balances and linear discontinuous expansion of the flux. Discretized DSA equations are derived by consistently discretizing the continuous diffusion equation with the LDEM-SCB method, however, the discretized diffusion equations are not fully consistent with the discretized transport equations. In addition, a fine mesh rebalance (FMR) method is devised to accelerate the discretized diffusion equation coupled with the preconditioned conjugate gradient (CG) method. The DSA method is applied to various test problems to show its effectiveness in speeding up the iterative convergence of the transport equation. The results show that the DSA method gives small spectral radii for the tetrahedral meshes having various minimum aspect ratios even in highly scattering dominant mediums for the homogeneous test problems. The numerical tests for the homogeneous and heterogeneous problems show that DSA with FMR (with preconditioned CG) gives significantly higher speedups and robustness than the one with the Gauss-Seidel-like iteration.

TRANSFORMATION OF DIMENSIONLESS HEAT DIFFUSION EQUATION FOR THE SOLUTION OF DYNAMIC DOMAIN IN PHASE CHANGE PROBLEMS

  • Ashraf, Muhammad;Avila, R.;Raza, S. S.
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.13 no.1
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    • pp.31-40
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    • 2009
  • In the present work transformation of dimensionless heat diffusion equation for the solution of moving boundary problems have been formulated. The formulation is based on 1-D, 2-D and 3-D, unsteady heat diffusion equations. These equations are rst turned int dimensionless form by using dimensionless quantities and their transformation was formulated in liquid and solid phases. The salient feature of this work is that during the transformation of dimensionless heat diffusion equation there arises a convective term $\tilde{v}$ which is responsible for the motion of interface in liquid as well as solid phase. In the transformed heat equation, a correction factor $\beta$ also arises naturally which gives the correct transformed flux at interface.

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Noise removal or video sequences with ,3-D anisotropic diffusion equation (3차원 이방성확산 방정식을 이용한 동영상의 영상잡음제거)

  • Lee, Seok-Ho;Choe, Eun-Cheol;Gang, Mun-Gi
    • Journal of the Institute of Electronics Engineers of Korea SP
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    • v.39 no.2
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    • pp.79-86
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    • 2002
  • Nowadays there is a trend to apply the diffusion equation to image Processing. The anisotropic diffusion equation is highly favoured as a noise removal algorithm because it can remove noise while enhancing edges. However if the two dimensional anisotropic diffusion equation is applied to the noise removal of video sequences, flickering artifact due to the luminance difference between frames and ghost artifact due to the interfiltering between frames occur. In this paper the two dimensional anisotropic diffusion equation is extended to the sequence axis. The Proposed three dimensional anisotropic diffusion equation removes noise more efficiently than the two dimensional equation, and furthermore removes the flickering and ghost artifact as well.

BIFURCATIONS IN A DISCRETE NONLINEAR DIFFUSION EQUATION

  • Kim, Yong-In
    • Bulletin of the Korean Mathematical Society
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    • v.35 no.4
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    • pp.689-700
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    • 1998
  • We consider an infinite dimensional dynamical system what is called Lattice Dynamical System given by a discrete nonlinear diffusion equation. By assuming the nonlinearity to be a general nonlinear function with mild restrictions, we show that as the diffusion parameter changes the stationery state of the given system undergoes bifurcations from the zero state to a bounded invariant set or a 3- or 4-periodic state in the global phase space of the given system according to the values of the coefficients of the linear part of the given nonlinearity.

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Treatment of Melamine by GAC Adsorption According to Adsorbent Size: Kinetics and Dispersion-Diffusion (흡착제 크기에 따른 GAC의 멜라민 흡착 처리 : 반응속도와 분산-확산)

  • Lee, Jai-Yeop;Lee, Sangjung;Han, Ihnsup
    • Journal of Soil and Groundwater Environment
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    • v.18 no.3
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    • pp.65-72
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    • 2013
  • Adsorption of melamine was examined using columns packed with granular activated carbon (GAC). Raw GAC was sieved with 20, 40, 60 and 80 mesh to determine the influence of adsorbent particle size on reaction and diffusion. The mass ratio of the adsorption capacity of GAC for melamine ranged from 9.19 to 11.06%, and adsorption rates increased with decreasing particle size within this range. Rate constants between 3.295 ~ 4.799 $min^{-1}$ were obtained using a pseudofirst-order equation that was used to determine adsorption kinetics. A surface diffusion model was adapted to take into account the unsteady-state equation of a spherical adsorbent by converting the surface concentration from a constant to a variable governed by a dispersion equation. The calculated values were fit with the experimental results by using the diffusion coefficients as regression parameters. The modified equation exhibited a more precise agreement with respect to the sum of the absolute error (SAE).

A Model for Activation Energy of Moisture Diffusion in Wood (수분확산(水分擴散)의 활성화(活性化)에너지 모델)

  • Kang, Ho-Yang
    • Journal of the Korean Wood Science and Technology
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    • v.20 no.4
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    • pp.21-30
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    • 1992
  • An activation energy equation for moisture diffusion in wood was developed with an assumption that activation energy is directly proportional to wood specific gravity. Theoretical activation energies obtained from the activation energy equation were revealed to be always lower than actual activation energies, which implies that activation energy isn't affected only by wood specific gravity. The other affecting factors are possibly anatomical structures of wood which determine a ratio of vapor diffusion to bound water diffusion in wood. For the convenience of estimating actual activation energy by using the activation energy equation, thirteen kinds of species were categorized into three groups according to their anatomical structures.

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Thin-layer Rewetting Equation for Short Grain Rough Rice (단립종(短粒種)벼의 박층흡습방정식(薄層吸濕方程式))

  • Jung, C.S.;Keum, D.H.;Park, S.J.
    • Journal of Biosystems Engineering
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    • v.12 no.2
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    • pp.38-43
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    • 1987
  • An experimental study was conducted to develop a thin-layer rewetting equation of short grain rough rice of Akihikari variety. Four thin-layer rewetting equations were experimentally determined from $25^{\circ}C$ to $45^{\circ}C$ and 70%RH to 85%RH conditions. Diffusion, Henderson, Page, and Thompson equations widely used as thin-layer drying equations were selected. Experimental data were fitted to these equations using linear regression analysis except diffusion equation. The diffusivity in the diffusion equation was determined by optimization method. Four equations were highly significant. In order to compare the goodness of fit of each equation, the error mean square of each equawas calculated. The diffusion model was not a very good model because the error mean square was very large. The other three models showed the same level or error mean square and could predict satisfactorily the rewetting rate or short grain rough rice.

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Analytical Solutions of Unsteady Reaction-Diffusion Equation with Time-Dependent Boundary Conditions for Porous Particles

  • Cho, Young-Sang
    • Korean Chemical Engineering Research
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    • v.57 no.5
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    • pp.652-665
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    • 2019
  • Analytical solutions of the reactant concentration inside porous spherical catalytic particles were obtained from unsteady reaction-diffusion equation by applying eigenfunction expansion method. Various surface concentrations as exponentially decaying or oscillating function were considered as boundary conditions to solve the unsteady partial differential equation as a function of radial distance and time. Dirac delta function was also used for the instantaneous injection of the reactant as the surface boundary condition to calculate average reactant concentration inside the particles as a function of time by Laplace transform. Besides spherical morphology, other geometries of particles, such as cylinder or slab, were considered to obtain the solution of the reaction-diffusion equation, and the results were compared with the solution in spherical coordinate. The concentration inside the particles based on calculation was compared with the bulk concentration of the reactant molecules measured by photocatalytic decomposition as a function of time.