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- ItemDFG Final Report: Validated computation of patterns in recurrent neural networks(Hannover : Technische Informationsbibliothek, 2025) Queirolo, Elena[no abstract available]
- ItemFinal Report on the DFG project "On the analysis of a class of cross-diffusion Cahn-Hilliard systems"(Hannover : Technische Informationsbibliothek, 2025) Marino, GretaIn this project we considered a class of cross-diffusion systems involving Cahn-Hilliard terms and gave exhaustive answers to some main questions related to it. This class arises when modeling mixtures composed of several species that interact with one another with cross-diffusion effects and also have the tendency to separate from each other. In the case under consideration, there is only one species (that accounts for the void) which does separate from all the others. The interest for such a model stems from the fact that in many real world applications there exist multiphase systems where miscible entities do coexist in one single phase of the system. Our project started from an existing model and, by making a combination of both numerical and theoretical approaches, provided a systematical analysis to some core topics related to it, in order to gain a better understanding of the dynamics of the model and of its stationary solutions.
- ItemDerivation of Moment Equations for the Theoretical Description of Electrons in Nonthermal Plasmas(Irvine, Calif. : Scientific Research Publ., 2013) Becker, Markus M.; Loffhagen, DetlefThe derivation of moment equations for the theoretical description of electrons is of interest for modelling of gas discharge plasmas and semiconductor devices. Usually, certain artificial closure assumptions are applied in order to derive a closed system of moment equations from the electron Boltzmann equation. Here, a novel four-moment model for the description of electrons in nonthermal plasmas is derived by an expansion of the electron velocity distribution function in Legendre polynomials. The proposed system of partial differential equations is consistently closed by definition of transport coefficients that are determined by solving the electron Boltzmann equation and are then used in the fluid calculations as function of the mean electron energy. It is shown that the four-moment model can be simplified to a new drift-diffusion approximation for electrons without loss of accuracy, if the characteristic frequency of the electric field alteration in the discharge is small in comparison with the momentum dissipation frequency of the electrons. Results obtained by the proposed fluid models are compared to those of a conventional drift-diffusion approximation as well as to kinetic results using the example of low pressure argon plasmas. It is shown that the results provided by the new approaches are in good agreement with kinetic results and strongly improve the accuracy of fluid descriptions of gas discharges.
- ItemHigh-order time-accurate schemes for singularly perturbed parabolic convection-diffusion problems with robin boundary conditions(Berlin : De Gruyter, 2002) Hemker, Pieter W.; Shishkin, Grigorii I.; Shishkina, Lidia P.The boundary-value problem for a singularly perturbed parabolic PDE with convection is considered on an interval in the case of the singularly perturbed Robin boundary condition; the highest space derivatives in the equation and in the boundary condition are multiplied by the perturbation parameterε. In contrast to the Dirichlet boundary-value problem, for the problem under consideration the errors of the well-known classical methods, generally speaking, grow without bound as ε≪N-1 where N defines the number of mesh points with respect to x. The order of convergence for the known ε-uniformly convergent schemes does not exceed 1. In this paper, using a defect correction technique, we construct ε-uniformly convergent schemes of highorder time-accuracy. The efficiency of the new defect-correction schemes is confirmed by numerical experiments. A new original technigue for experimental studying of convergence orders is developed for the cases where the orders of convergence in the x-direction and in the t-direction can be substantially different. © 2002, Institute of Mathematics, NAS of Belarus. All rights reserved.
- ItemDFG project report "A Unified Approach to Limit Theorems for Dual Objects in Probability and Number Theory"(Hannover : Technische Informationsbibliothek, 2024-09-24) Indlekofer, Karl-Heinz; Klesov, Oleg I.; Steinebach, Josef G.Several situations are known in mathematics, where similar statements hold for different mathematical objects. However, since the objects are different, the proofs of these statements as well as their assumptions are different. One of the basic ideas of the project is to suggest a general approach for fi nding certain similarities between two mathematical objects that allow one to provide the proof only for one of them and to transfer this to a similar result for the another one. Such objects are called dual in our language. An integral part of this approach is to develop tools for transforming necessary conditions for one of the objects into corresponding conditions for the another one.