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Exponential decay of a solution for some parabolic equation involving a time nonlocal term

Kota Kumazaki (2015)

Mathematica Bohemica

We consider the large time behavior of a solution of a parabolic type equation involving a nonlocal term depending on the unknown function. This equation is proposed as a mathematical model of carbon dioxide transport in concrete carbonation process, and we proved the existence, uniqueness and large time behavior of a solution of this model. In this paper, we derive the exponential decay estimate of the solution of this model under restricted boundary data and initial data.

Finite difference scheme for the Willmore flow of graphs

Tomáš Oberhuber (2007)

Kybernetika

In this article we discuss numerical scheme for the approximation of the Willmore flow of graphs. The scheme is based on the finite difference method. We improve the scheme we presented in Oberhuber [Obe-2005-2,Obe-2005-1] which is based on combination of the forward and the backward finite differences. The new scheme approximates the Willmore flow by the central differences and as a result it better preserves symmetry of the solution. Since it requires higher regularity of the solution, additional...

Finite element approximation of a Stefan problem with degenerate Joule heating

John W. Barrett, Robert Nürnberg (2004)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider a fully practical finite element approximation of the following degenerate system t ρ ( u ) - . ( α ( u ) u ) σ ( u ) | φ | 2 , . ( σ ( u ) φ ) = 0 subject to an initial condition on the temperature, u , and boundary conditions on both u and the electric potential, φ . In the above ρ ( u ) is the enthalpy incorporating the latent heat of melting, α ( u ) > 0 is the temperature dependent heat conductivity, and σ ( u ) 0 is the electrical conductivity. The latter is zero in the frozen zone, u 0 , which gives rise to the degeneracy in this Stefan system. In addition to showing stability...

Finite element approximation of a Stefan problem with degenerate Joule heating

John W. Barrett, Robert Nürnberg (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a fully practical finite element approximation of the following degenerate system t ρ ( u ) - . ( α ( u ) u ) σ ( u ) | φ | 2 , . ( σ ( u ) φ ) = 0 subject to an initial condition on the temperature, u, and boundary conditions on both u and the electric potential, ϕ. In the above p(u) is the enthalpy incorporating the latent heat of melting, α(u) > 0 is the temperature dependent heat conductivity, and σ(u) > 0 is the electrical conductivity. The latter is zero in the frozen zone, u ≤ 0, which gives rise to the degeneracy in this Stefan...

Finite element approximation of a two-layered liquid film in the presence of insoluble surfactants

John W. Barrett, Linda El Alaoui (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a system of degenerate parabolic equations modelling a thin film, consisting of two layers of immiscible Newtonian liquids, on a solid horizontal substrate. In addition, the model includes the presence of insoluble surfactants on both the free liquid-liquid and liquid-air interfaces, and the presence of both attractive and repulsive van der Waals forces in terms of the heights of the two layers. We show that this system formally satisfies a Lyapunov structure, and a second energy...

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