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Application of Rothe's method to a parabolic inverse problem with nonlocal boundary condition

Yong-Hyok Jo, Myong-Hwan Ri (2022)

Applications of Mathematics

We consider an inverse problem for the determination of a purely time-dependent source in a semilinear parabolic equation with a nonlocal boundary condition. An approximation scheme for the solution together with the well-posedness of the problem with the initial value u 0 ∈ H 1 ( Ω ) is presented by means of the Rothe time-discretization method. Further approximation scheme via Rothe’s method is constructed for the problem when u 0 ∈ L 2 ( Ω ) and the integral kernel in the nonlocal boundary condition is symmetric.

Approche visqueuse de solutions discontinues de systèmes hyperboliques semilinéaires

Franck Sueur (2006)

Annales de l’institut Fourier

On s’intéresse à des systèmes symétriques hyperboliques multidimensionnels en présence d’une semilinéarité. Il est bien connu que ces systèmes admettent des solutions discontinues, régulières de part et d’autre d’une hypersurface lisse caractéristique de multiplicité constante. Une telle solution u 0 étant donnée, on montre que u 0 est limite quand ε → 0 de solutions ( u ε ) ε ∈ ] 0 , 1 ] du système perturbé par une viscosité de taille ε . La preuve utilise un problème mixte parabolique et des développements de couches limites....

Approximation of a nonlinear thermoelastic problem with a moving boundary via a fixed-domain method

Jindřich Nečas, Tomáš Roubíček (1990)

Aplikace matematiky

The thermoelastic stresses created in a solid phase domain in the course of solidification of a molten ingot are investigated. A nonlinear behaviour of the solid phase is admitted, too. This problem, obtained from a real situation by many simplifications, contains a moving boundary between the solid and the liquid phase domains. To make the usage of standard numerical packages possible, we propose here a fixed-domain approximation by means of including the liquid phase domain into the problem (in...

Approximation of Parabolic Equations Using the Wasserstein Metric

David Kinderlehrer, Noel J. Walkington (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We illustrate how some interesting new variational principles can be used for the numerical approximation of solutions to certain (possibly degenerate) parabolic partial differential equations. One remarkable feature of the algorithms presented here is that derivatives do not enter into the variational principles, so, for example, discontinuous approximations may be used for approximating the heat equation. We present formulae for computing a Wasserstein metric which enters into the variational...

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