### Element method for epitaxial growth with attachment-detachment kinetics

Eberhard Bänsch, Frank Haußer, Omar Lakkis, Bo Li & Axel Voigt
An adaptive finite element method is developed for a class of free or moving boundary problems modeling island dynamics in epitaxial growth. Such problems consist of an adatom (adsorbed atom) diffusion equation on terraces of different height, boundary conditions on terrace boundaries including the kinetic asymmetry in the adatom attachment and detachment, and the normal velocity law for the motion of such boundaries determined by a two-sided flux, together with the one-dimensional "surface" diffusion. The...

### Global bifurcation analysis of a class of planar systems

Alexander Grin & Klaus R. Schneider
We consider planar autonomous systems $dx/dt =P(x,y,la), ; dy/dt =Q(x,y,la)$ depending on a scalar parameter $la$. We present conditions on the functions $P$ and $Q$ which imply that there is a parameter value $la_0$ such that for $la > la_0$ this system has a unique limit cycle which is hyperbolic and stable. Dulac-Cherkas functions, rotated vector fields and singularly perturbed systems play an important role in the proof.

### Stochastic, analytic and numerical aspects of coagulation processes

Wolfgang Wagner
In this paper we review recent results concerning stochastic models for coagulation processes and their relationship to deterministic equations. Open problems related to the gelation effect are discussed. Finally we present some new conjectures based on numerical experiments performed with stochastic algorithms.

### Prescribed conditions at infinity for fractional parabolic and elliptic equations with unbounded coefficients

Fabio Punzo & Enrico Valdinoci
We investigate existence and uniqueness of solutions to a class of fractional parabolic equations satisfying prescribed pointwise conditions at infinity (in space), which can be time-dependent. Moreover, we study the asymptotic behaviour of such solutions. We also consider solutions of elliptic equations satisfying appropriate conditions at infinity.

### Reduced-order unscented Kalman filter in the frequency domain: Application to computational hemodynamics

Lucas O. Müller, Alfonso Caiazzo & Pablo J. Blanco
Objective: The aim of this work is to assess the potential of the reduced order unscented Kalman filter (ROUKF) in the context of computational hemodynamics, in order to estimate cardiovascular model parameters when employing real patient-specific data. Methods: The approach combines an efficient blood flow solver for one-dimensional networks (for the forward problem) with the parameter estimation problem cast in the frequency space. Namely, the ROUKF is used to correct model parameter after each cardiac...

### On the characterization of self-regularization properties of a fully discrete projection method for Symm's integral equation

Sergei V. Pereverzev & Siegfried Prößdorf
The influence of small perturbations in the kernel and the right-hand side of Symm's boundary integral equation, considered in an ill-posed setting, is analyzed. We propose a modification of a fully discrete projection method which is more economical in the sense of complexity and allows to obtain the optimal order of accuracy in the power scale with respect to the level of the noise in the kernel or in the parametric representation of the boundary.

### Dynamo action in cellular convection

Norbert Seehafer & Ayhan Demircan
The dynamo properties of square patterns in Boussinesq Rayleigh-Benard convection in a plane horizontal layer are studied numerically. Cases without rotation and with weak rotation about a vertical axis are considered, particular attention being paid to the relation between dynamo action and the kinetic helicity of the flow. While the fluid layer is symmetric with respect to up-down reflections, the square-pattern solutions may or may not possess this vertical symmetry. Vertically symmetric solutions, appearing in...

### Brownian motion in attenuated or renormalized inverse-square Poisson potential

Peter Nelson & Renato Soares Dos Santos
We consider the parabolic Anderson problem with random potentials having inverse-square singularities around the points of a standard Poisson point process in ℝ d, d ≥3. The potentials we consider are obtained via superposition of translations over the points of the Poisson point process of a kernel 픎 behaving as 픎 (x)≈ Θ|x|-2 near the origin, where &Theta ∈(0,(d-2)2/16]. In order to make sense of the corresponding path integrals, we require the potential to be...

### Maximum likelihood estimate for nonparametric signal in white noise by optimal control

Grigori N. Milstein & Michael Nussbaum
The paper is devoted to questions of constructing the maximum likelihood estimate for a nonparametric signal in white noise by considering corresponding problems of optimal control. For signals with bounded derivatives, sensitivity theorems are proved. The theorems state a stability of the maximum likelihood estimate with respect to changing output data. They make possible to reduce the original problem to a standard problem of optimal control which is solved by iterative procedure. For signals of...

### Regularizing aperiodic cycles of resonant radiation in filament light bullets

Carsten Brée, Ihar Babushkin, Uwe Morgner & Ayhan Demircan

### Homogenization of Scalar Wave Equations with Hysteresis

Jan Franců & Pavel Krejčí
The paper deals with a scalar wave equation of the form ρutt = (퓕[ux])x + ƒ, where 퓕 is a Prandtl-Ishlinskii operator and ρ, ƒ are given functions. This equation describes longitudinal vibrations of an elastoplastic rod. The mass density ρ and the Prandtl-Ishlinskii distribution function η are allowed to depend on the space variable x. We prove existence, uniqueness and regularity of solution to a corresponding initial-boundary value problem. The system is then homogenized...

### Coarse-graining via EDP-convergence for linear fast-slow reaction systems

Alexander Mielke & Artur Stephan
We consider linear reaction systems with slow and fast reactions, which can be interpreted as master equations or Kolmogorov forward equations for Markov processes on a finite state space. We investigate their limit behavior if the fast reaction rates tend to infinity, which leads to a coarse-grained model where the fast reactions create microscopically equilibrated clusters, while the exchange mass between the clusters occurs on the slow time scale. Assuming detailed balance the reaction system...

Ingo Bremer

### On the existence of classical solutions for a two phase flow through saturated porous media

Sabine Hengst
In this paper an elliptic-parabolic coupled system arising from a two-phase flow through a saturated porous medium is considered. The uniqueness and the existence of classical solutions are proved. The asymptotic behavior of solutions for large time is shown, too.

### Random approximation of finite sums.

Peter Mathé
This paper is devoted to a detailed study of the randomized approximation of finite sums, i.e., sums ∑mj=1 xj, x ∈ ℝm, where m is supposed to be large, shall be approximated with information on n coordinates, only. The error is measured on balls in lmp, 1 ≤ p ≤ ∞. Main emphasis is laid on the exact solution of the problems stated below. In most cases we obtain both, an optimal method for the...

### Error estimates for nonlinear reaction-diffusion systems involving different diffusion length scales

Sina Reichelt
We derive quantitative error estimates for coupled reaction-diffusion systems, whose coefficient functions are quasi-periodically oscillating modeling microstructure of the underlying macroscopic domain. The coupling arises via nonlinear reaction terms, and we allow for different diffusion length scales, i.e. whereas some species have characteristic diffusion length of order 1, other species may diffuse much slower, namely, with order of the characteristic microstructure-length scale. We consider an effective system, which is rigorously obtained via two-scale convergence, and...

### International Symposium "Operator Equations and Numerical Analysis" September 28 - October 2, 1992 Gosen (nearby Berlin).

Prößdorf, Siegfried (Ed.)

### Memory equations as reduced Markov processes

Holger Stephan & Artur Stephan
A large class of linear memory differential equations in one dimension, where the evolution depends on the whole history, can be equivalently described as a projection of a Markov process living in a higher dimensional space. Starting with such a memory equation, we give an explicit construction of the corresponding Markov process. From a physical point of view the Markov process can be understood as the change of the type of some quasiparticles along one-way...

### Global solutions to a Penrose-Fife phase-field model under flux boundary conditions for the inverse temperature

Werner Horn, Philippe Laurençot & Jürgen Sprekels
In this paper, we study an initial-boundary value problem for a system of phase-field equations arising from the Penrose-Fife approach to model the kinetics of phase transitions. In contrast to other recent works in this field, the correct form of the boundary condition for the temperature field is assumed which leads to additional difficulties in the mathematical treatment. It is demonstrated that global existence and, in the case of only one or two space dimensions,...

### Stochastic Lagrangian footprint calculations over a surface with an abrupt change of roughness height

Orazgeldi Kurbanmuradov, Ullar Rannik, Alexander I. Levykin, Karl Sabelfeld & Timo Vesala
Forward and backward stochastic Lagrangian trajectory simulation methods are developed to calculate the footprint and cumulative footprint functions of concentration and fluxes in the case when the ground surface has an abrupt change of the roughness height. The statistical characteristics to the stochastic model are extracted numerically from a closure model we developed for the atmospheric boundary layer. The flux footprint function is perturbed in comparison with the footprint function for surface without change in...

### On the smoothness of the solution to a boundary value problem for a differential-difference equation.

I. P. Ivanova & G. A. Kamenskij
This paper deals with the first boundary value problem (BVP) for equations which are a differential with respect to one variable (t) and difference with respect to the other variable (s) in a bounded domain. The initial value problem for differential-difference equations of this type was studied in [1], [2]. The theory of the BVP under investigation is connected with the theory of the BVP for strongly elliptic differential-difference equations which are difference and differential...

### On the velocity of the Biot slow wave in a porous medium: Uniform asymptotic expansion

Inna Edelman
Asymptotic behavior of the Biot slow wave is investigated. Formulae for short- and long-wave approximations of phase velocity of the P2 wave are presented. These asymptotic expansions are compared with exact solution, constructed numerically. It is shown that both expansions fit very well the real velocity of the P2 mode. Procedure for matching of short- and long-wave asymptotic expansions is suggested.

### Modeling diffusional coarsening in microelectronic solders

Wolfgang Dreyer & Wolfgang H. Müller
This paper presents a detailed numerical simulation of the coarsening phenomenon observed in microelectronic solder materials that are subjected to high homologeous temperatures in combination with thermo-mechanical stresses. The simulations are based on a phase field model which, for simplicity, is explicitly formulated for a binary alloy. To this end, the thermomechanical stresses originating within a Representative Volume Element (RVE) of the solder material are calculated first. This is achieved by means of a closed-form...

### The moment Lyapunov exponent for conservative systems with small periodic and random perturbations

Peter Imkeller & Grigori N. Milstein
Much effort has been devoted to the stability analysis of stationary points for linear autonomous systems of stochastic differential equations. Here we introduce the notions of Lyapunov exponent, moment Lyapunov exponent, and stability index for linear nonautonomous systems with periodic coefficients. Most extensively we study these problems for second order conservative systems with small random and periodic excitations. With respect to relations between the intrinsic period of the system and the period of perturbations we...

### On the Dynamics of Single Mode Lasers with Passive Dispersive Reflector

Fritz Henneberger, Klaus R. Schneider, Jan Sieber, Vasile Z. Tronciu & Hans-Jürgen Wünsche
For passive dispersive reflector (PDR) lasers we investigate a single mode model containing two functions characterizing the influence of the PDR. We study numerically the effect of the shape of these functions on the existence and robustness of self-pulsations. The possibility of tuning the frequency and modulation depth of the self-pulsations has been demonstrated.

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