Hyperbolicity of linear partial differential equations with delay

Robust hyperbolicity and stability results for linear partial differential equations with delay will be given and, as an application, the effect of small delays to the asymptotic properties of feedback systems will be analyzed

Parciális Funkcionáldifferenciálegyenletek = Partial differential equations with delay

A kutatási időszak első részében a monográfia befejezésével foglalkoztam. A matematikai eredmények legnagyobb része a késleltetett egyenletekhez tartozó megoldó-félcsoport aszimptotikus tulajdonságaival kapcsolatos. A Klaus-Jochen Engellel közösen írt cikkben szorzattereken ható operátorfélcsoportok exponenciális stabilitására sikerült absztrakt eredményeket elérni valamint a növekedési rátára becslést adni. Sikerült továbbá Charles Batty egy egy kérdését pozitívan megválaszolni a késleltetett félcsoportok differenciálhatóságával kapcsolatban. Csomós Petra doktorandusszal és Gregor Nickel (Siegen) professzorral a késleltetett és inhomogén egyenletek numerikus módszereivel foglalkoztunk. Megmutattuk a splitting konvergenciáját térbeli diszkretizáció jelenlétében. M. S. Elbialy (Toledo, Ohio) professzorral késleltetett egyenletek invariáns sokaságainek exisztenciáját vizsgáltuk általában ''gap condition'' jelenlétekor. Fontosabb szervezett konferenciák: - 2nd Dynamical Networks days, 2005 május, Róma - 3rd Dynamical Network Days, 2005 október, Horb, Németország - Workshop int the honor of Prof. Ulf Schlotterbeck, 2006 július, Tübingen, Németország - German-Hungarian Workshop, Dobogókő, 2007 - Encounters between discrete and continuous mathematics, Blaubeuren, 2008 - 35 Jahre AGFA : Conference int he honor of Rainer Nagel, 2008 november | I succeeded in finishing a monograph on operator semigroup methods applied to delay equations. In a paper with K.-J. Engel we described the asymptics of some important special wave equations. A question of Charles Batty on the differentiability of delay semigroups was solved. We also investigated the operator splitting method in the presence of a spatial approximation and applied the method to delay equations. Spectral mapping properties of the delay semigroup and invariant manifolds for some nonlinear delay equations were also investigated

Operator splitting for nonautonomous delay equations

We provide a general product formula for the solution of nonautonomous abstract delay equations. After having shown the convergence we obtain estimates on the order of convergence for differentiable history functions. Finally, the theoretical results are demonstrated on some typical numerical examples.Comment: to appear in "Computers & Mathematics with Applications (CAMWA)

Operator splitting for dissipative delay equations

We investigate Lie-Trotter product formulae for abstract nonlinear evolution equations with delay. Using results from the theory of nonlinear contraction semigroups in Hilbert spaces, we explain the convergence of the splitting procedure. The order of convergence is also investigated in detail, and some numerical illustrations are presented.Comment: to appear in Semigroup Foru

Operator splitting with spatial-temporal discretization

Continuing earlier investigations, we analyze the convergence of operator splitting procedures combined with spatial discretization and rational approximations

Mathematik und Ethik: eine \"Uberlegung f\"ur zuk\"unftige Lehrpersonen

Mathematical modelling and ethics have more touching points than most of us would like to admit. Everyday decisions are often reasoned by mathematical arguments. Mathematics teachers belong to those mathematically literate, who must point out mistakes in public discussions and reflect on them in their teaching to educate students to become critical thinkers. Ethical questions arise in mathematical modelling problems from two possible contexts. The first question is what values are reflected in the modelling problem. The other question is about the ethical implications of applying mathematical models to real problems.Comment: in German languag

Stability and Convergence of Product Formulas for Operator Matrices

We present easy to verify conditions implying stability estimates for operator matrix splittings which ensure convergence of the associated Trotter, Strang and weighted product formulas. The results are applied to inhomogeneous abstract Cauchy problems and to boundary feedback systems.Comment: to appear in Integral Equations and Operator Theory (ISSN: 1420-8989

Differential equation approximations of stochastic network processes: an operator semigroup approach

The rigorous linking of exact stochastic models to mean-field approximations is studied. Starting from the differential equation point of view the stochastic model is identified by its Kolmogorov equations, which is a system of linear ODEs that depends on the state space size ($N$) and can be written as $\dot u_N=A_N u_N$. Our results rely on the convergence of the transition matrices $A_N$ to an operator $A$. This convergence also implies that the solutions $u_N$ converge to the solution $u$ of $\dot u=Au$. The limiting ODE can be easily used to derive simpler mean-field-type models such that the moments of the stochastic process will converge uniformly to the solution of appropriately chosen mean-field equations. A bi-product of this method is the proof that the rate of convergence is $\mathcal{O}(1/N)$. In addition, it turns out that the proof holds for cases that are slightly more general than the usual density dependent one. Moreover, for Markov chains where the transition rates satisfy some sign conditions, a new approach for proving convergence to the mean-field limit is proposed. The starting point in this case is the derivation of a countable system of ordinary differential equations for all the moments. This is followed by the proof of a perturbation theorem for this infinite system, which in turn leads to an estimate for the difference between the moments and the corresponding quantities derived from the solution of the mean-field ODE