• Title/Summary/Keyword: nonlinear diffusion

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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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FINITE DIFFERENCE SCHEMES FOR A GENERALIZED NONLINEAR CALCIUM DIFFUSION EQUATION

  • Choo, S.M.
    • Journal of applied mathematics & informatics
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    • v.27 no.5_6
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    • pp.1247-1256
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    • 2009
  • Finite difference schemes are considered for a nonlinear $Ca^{2+}$ diffusion equations with stationary and mobile buffers. The scheme inherits mass conservation as for the classical solution. Stability and $L^{\infty}$ error estimates of approximate solutions for the corresponding schemes are obtained. using the extended Lax-Richtmyer equivalence theorem.

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PERELMAN TYPE ENTROPY FORMULAE AND DIFFERENTIAL HARNACK ESTIMATES FOR WEIGHTED DOUBLY NONLINEAR DIFFUSION EQUATIONS UNDER CURVATURE DIMENSION CONDITION

  • Wang, Yu-Zhao
    • Bulletin of the Korean Mathematical Society
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    • v.58 no.6
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    • pp.1539-1561
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    • 2021
  • We prove Perelman type 𝒲-entropy formulae and differential Harnack estimates for positive solutions to weighed doubly nonlinear diffusion equation on weighted Riemannian manifolds with CD(-K, m) condition for some K ≥ 0 and m ≥ n, which are also new for the non-weighted case. As applications, we derive some Harnack inequalities.

INVERSE PROBLEM FOR STOCHASTIC DIFFERENTIAL EQUATIONS ON HILBERT SPACES DRIVEN BY LEVY PROCESSES

  • N. U., Ahmed
    • Nonlinear Functional Analysis and Applications
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    • v.27 no.4
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    • pp.813-837
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    • 2022
  • In this paper we consider inverse problem for a general class of nonlinear stochastic differential equations on Hilbert spaces whose generating operators (drift, diffusion and jump kernels) are unknown. We introduce a class of function spaces and put a suitable topology on such spaces and prove existence of optimal generating operators from these spaces. We present also necessary conditions of optimality including an algorithm and its convergence whereby one can construct the optimal generators (drift, diffusion and jump kernel).

A HYBRID METHOD FOR HIGHER-ORDER NONLINEAR DIFFUSION EQUATIONS

  • KIM JUNSEOK;SUR JEANMAN
    • Communications of the Korean Mathematical Society
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    • v.20 no.1
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    • pp.179-193
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    • 2005
  • We present results of fully nonlinear time-dependent simulations of a thin liquid film flowing up an inclined plane. Equations of the type $h_t+f_y(h) = -{\in}^3{\nabla}{\cdot}(M(h){\nabla}{\triangle}h)$ arise in the context of thin liquid films driven by a thermal gradient with a counteracting gravitational force, where h = h(x, t) is the fluid film height. A hybrid scheme is constructed for the solution of two-dimensional higher-order nonlinear diffusion equations. Problems in the fluid dynamics of thin films are solved to demonstrate the accuracy and effectiveness of the hybrid scheme.

An Image Segmentation method using Morphology Reconstruction and Non-Linear Diffusion (모폴로지 재구성과 비선형 확산을 적용한 영상 분할 방법)

  • Kim, Chang-Geun;Lee, Guee-Sang
    • Journal of KIISE:Software and Applications
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    • v.32 no.6
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    • pp.523-531
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    • 2005
  • Existing methods for color image segmentation using diffusion can't preserve contour information, or noises with high gradients become more salient as the number of times of the diffusion increases, resulting in over-segmentation when applied to watershed. This paper proposes a method for color image segmentation by applying morphological operations together with nonlinear diffusion For an input image, transformed into LUV color space, closing by reconstruction and nonlinear diffusion are applied to obtain a simplified image which preserves contour information with noises removed. With gradients computed from this simplified image, watershed algorithm is applied. Experiments show that color images are segmented very effectively without over-segmentation.

Ussing's flux ratio theorem for nonlinear diffusive transport with chemical interactions

  • Bracken, A.J.;McNabb, A.;Suzuki, M.
    • 제어로봇시스템학회:학술대회논문집
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    • 1994.10a
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    • pp.747-752
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    • 1994
  • Ussing's flux ratio theorem (1978) reflects a reciprocal relationship behavior between the unidirectional fluxes in asymmetric steady diffusion-convection in a membrane slab. This surprising result has led to many subsequent studies in a wide range of applications, in particular involving linear models of time dependent problems in biology and physiology. Ussing's theorem and its extensions are inherently linear in character. It is of considerable interest to ask to what extent these results apply, if at all, in situations involving, for example, nonlinear reaction. A physiologically interesting situation has been considered by Weisiger et at. (1989, 1991, 1992) and by McNabb et al. (1990, 1991) who studied the role of albumin in the transport of ligands across aqueous diffusion barriers in a liver membrane slab. The results are that there exist reciprocal relationships between unidirectional fluxes in the steady state, although albumin is chemically interacting in a nonlinear way of the diffusion processes. However, the results do not hold in general at early times. Since this type of study first started, it has been speculated about when and how the Ussing's flux ratio theorem fails in a general diffusion-convection-reaction system. In this paper we discuss the validity of Ussing-type theorems in time-dependent situations, and consider the limiting time behavior of a general nonlinear diffusion system with interaction.

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Drift Diffusion of Radiation-produced Point Defects to Edge Dislocation

  • Park, S.S.;Chang, K.O.;Choi, S.P.;Kim, C.O.
    • Nuclear Engineering and Technology
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    • v.31 no.2
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    • pp.151-156
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    • 1999
  • Under the heavy irradiation of crystalline materials when the production and the recombination of interstitials and vacancies are included, the diffusion equations become nonlinear. An effort has been made to arrange an appropriate transformation of these nonlinear differential equations to more solvable Poisson's equations, finally analytical solutions for simultaneously calculating the concentrations of interstitials and vacancies in the angular dependent Cottrell's potential of the edge dislocation have been derived from the well-known Green's theorem and perturbation theory.

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Modeling of Moisture Diffusion Coefficient with Porosity in Concrete (공극률 변화를 고려한 콘크리트의 수분확산계수 모델)

  • 강수태;전상은;김진근;김성욱
    • Proceedings of the Korea Concrete Institute Conference
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    • 2003.05a
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    • pp.321-326
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    • 2003
  • The nonlinear humidity distribution occurs due to the moisture diffusion when a concrete is exposed to an ambient air. These nonlinear humidity distribution induces shrinkage cracks on surfaces of the concrete. Because shrinkage cracks largely affect the durability and serviceability of concrete structures, the moisture diffusion in concrete must be investigated. The purpose of this paper is to propose a model of the moisture diffusion coefficient that governs moisture diffusion within concrete structures. To propose the model, numerical analysis were performed based on several experiments. Because the moisture diffusion coefficient is changed with aging, especially at early ages, the proposed model includes aging effect by terms of the porosity as well as the humidity of concrete.

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Parameter Estimation in a Complex Non-Stationary and Nonlinear Diffusion Process

  • So, Beong-Soo
    • Journal of the Korean Statistical Society
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    • v.29 no.4
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    • pp.489-499
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    • 2000
  • We propose a new instrumental variable estimator of the complex parameter of a class of univariate complex-valued diffusion processes defined by the possibly non-stationary and/or nonlinear stochastic differential equations. On the basis of the exact finite sample distribution of the pivotal quantity, we construct the exact confidence intervals and the exact tests for the parameter. Monte-Carlo simulation suggests that the new estimator seems to provide a viable alternative to the maximum likelihood estimator (MLE) for nonlinear and/or non-stationary processes.

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