• Title/Summary/Keyword: Diffusion equation

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THE SPACE-TIME FRACTIONAL DIFFUSION EQUATION WITH CAPUTO DERIVATIVES

  • HUANG F.;LIU F.
    • Journal of applied mathematics & informatics
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    • v.19 no.1_2
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    • pp.179-190
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    • 2005
  • We deal with the Cauchy problem for the space-time fractional diffusion equation, which is obtained from standard diffusion equation by replacing the second-order space derivative with a Caputo (or Riemann-Liouville) derivative of order ${\beta}{\in}$ (0, 2] and the first-order time derivative with Caputo derivative of order ${\beta}{\in}$ (0, 1]. The fundamental solution (Green function) for the Cauchy problem is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. We derive explicit expression of the Green function. The Green function also can be interpreted as a spatial probability density function evolving in time. We further explain the similarity property by discussing the scale-invariance of the space-time fractional diffusion equation.

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.

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.

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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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.

Adaptive time-step control for modal methods to integrate the neutron diffusion equation

  • Carreno, A.;Vidal-Ferrandiz, A.;Ginestar, D.;Verdu, G.
    • Nuclear Engineering and Technology
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    • v.53 no.2
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    • pp.399-413
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    • 2021
  • The solution of the time-dependent neutron diffusion equation can be approximated using quasi-static methods that factorise the neutronic flux as the product of a time dependent function times a shape function that depends both on space and time. A generalization of this technique is the updated modal method. This strategy assumes that the neutron flux can be decomposed into a sum of amplitudes multiplied by some shape functions. These functions, known as modes, come from the solution of the eigenvalue problems associated with the static neutron diffusion equation that are being updated along the transient. In previous works, the time step used to update the modes is set to a fixed value and this implies the need of using small time-steps to obtain accurate results and, consequently, a high computational cost. In this work, we propose the use of an adaptive control time-step that reduces automatically the time-step when the algorithm detects large errors and increases this value when it is not necessary to use small steps. Several strategies to compute the modes updating time step are proposed and their performance is tested for different transients in benchmark reactors with rectangular and hexagonal geometry.