# Alkalmazott Analízis Szeminárium

A BME Matematika Intézet Analízis és Differenciálegyenletek Tanszékének közös Alkalmazott Analízis Szemináriuma 2016. őszén indult Faragó István (Differenciálegyenletek Tanszék) kezdeményezésére az MTA-ELTE Numerikus Analízis és Nagy Hálózatok Kutatócsoporttal együttműködésben. A szeminárium célja, hogy elősegítse egy alkalmazott analízissel (funkcionálanalízis, differenciálegyenletek, numerikus módszerek) foglalkozó kutatói kör kialakítását az intézeten belül. A szemináriummal fórumot szeretnénk biztosítani az alkalmazott analízissel foglalkozó matematikusok és az analízist alkalmazó kutatók számára az együttgondolkodásra. További cél az érdeklődő hallgatók (MSc, PhD) bevonása a kutatói munkába.

Szemináriumunk 2017-től felvette a Farkas Miklós Alkalmazott Analízis Szeminárium nevet. Ezzel szeretnénk emléket állítani egyetemünk egykori tanszékvezető matematikaprofesszorának, aki elindította egyetemünkön a matematikus-mérnök képzést, és a stabilitáselmélet valamint a biomatematika terén elért jelentős tudományos eredményeivel ill. könyveivel nagyban hozzájárult az alkalmazott matematika erősödéséhez. (English version of the introduction.)

From the autumn semester of 2017 the talks will be in English on a regular basis. / 2017. őszi félévétől az előadásokat angol nyelven tartjuk.

Organizers / Szervezők: Faragó István1,2,3, Karátson János1,2,3 ,Horváth Róbert1,3 ,Mincsovics Miklós1,3 (1BME, 2ELTE, 3MTA-ELTE NUMNET)

Request for e-mail notifications and remarks to the organizers / Feliratkozás az e-mail listára ill. egyéb megjegyzések a szeminárium szervezőihez

Next seminar:

26 October 2017 (Thursday), 10:15, H306

Mónika Polner (Bolyai Institute, University of Szeged)

A space-time finite element method for neural field equations with transmission delays

Neural field equations are models that describe the spatio-temporal evolution of (spatially) coarse grained variables such as synaptic or firing rate activity in populations of neurons. We consider a single population of neurons, distributed over some bounded, connected, open region, whose state is described by their membrane potential. These potentials are assumed to evolve according to an integro-differential equation with space dependent delay.

Neural field models with transmission delay may be cast as abstract delay differential equations, which is the starting formulation for our numerical discretization. The numerical treatment of these systems is rare in the literature and has several restrictions on the space domain and the functions involved. The aim of this work is the development of an accurate numerical method without introducing limitations to its applicability. We present and analyze a novel time-discontinuous Galerkin finite element method. We give a theoretical analysis of the stability and order of accuracy of the numerical discretization and demonstrate the method on a number of neural field computations in one and two space dimensions.

Previous seminars:

19 October 2017

Miklós Horváth (BME, Institute of Mathematics)

Inverse scattering: Mathematical properties of the phase shifts

The inverse scattering is one of the most useful devices to obtain information about quantum mechanical forces. The solutions of the radial Schrödinger equation, describing mathematically the situation, behave asymptotically like sine functions with shifted phase. The sequence of these phase shifts is a central notion in scattering theory. In this talk some classical and recent results about phase shifts will be presented and some open questions will be discussed.

12 October 2017

Imre Fekete (Eötvös Loránd University & MTA-ELTE NUMNET)

On the zero-stability of multistep methods on smooth nonuniform grids

In order to be convergent, linear multistep methods must be zero stable. While constant step size theory was established in the 1950’s, zero stability on nonuniform grids is less well understood. Here we investigate zero stability on compact intervals and smooth nonuniform grids. The grid points are constructed as the image of an equidistant grid under a smooth deformation map. We show that for all strongly stable linear multistep methods, there is an $N^*$ such that a condition of zero stability is always fulfilled for $N > N^*$ under a smoothness condition. Examples are given for Adams and BDF type methods.

5 October 2017

András Zachár (University of Dunaújváros, Institute of Informatics)

An explicit analytic solution of a coupled first order partial and ordinary differential equation system for a discontinuous initial-boundary value problem

Non-polynomial series solution of a coupled first order partial and ordinary differential equation (PDE-ODE) system for a discontinuous initial and boundary condition has been developed. Linear equation systems are constructed to calculate the constant coefficients of the series solution. Explicit expressions have been found to the solution of these linear equation systems. Different forms of the solution have been compared to the numerical solution of the PDE-ODE system and the rate of the convergence is also investigated. The studied first order PDE-ODE system describes an unsteady convection dominated heat transfer process induced by a buoyant plume entrainment.

28 September 2017

Tamás Kalmár-Nagy

Devilish eigenvalues: hysteresis and mechanistic turbulence

We consider the adjacency matrix associated with a graph that describes transitions between 2^N states of the discrete Preisach memory model. This matrix can also be associated with the "last-in-first-out" inventory management rule. We present an explicit solution for the spectrum by showing that the characteristic polynomial is the product of Chebyshev polynomials. The eigenvalue distribution (density of states) is explicitly calculated and is shown to approach a scaled Devil's staircase. The eigenvectors of the adjacency matrix are also expressed analytically. This is joint work with Andreas Amann, Daniel Kim, and Dmitrii Rachinski. We also examine a mechanistic model of turbulence, a binary tree of masses connected by springs. We analyze the behavior of this linear model: a formula is presented for the analytical calculation of the eigenvalues and the optimal damping - at which the decay of the total mechanical energy is maximized. The discrete energy spectrum of the mechanistic model (defined as the total mechanical energy stored in each level) can be tuned to display the features of the Kolmogorov-spectrum. This is joint work with Bendegúz Dezső Bak.

21 September 2017

Qualitatively reliable numerical models of time-dependent problems

In the modeling process we construct mathematical and numerical models. Both models should preserve the basic (physically, biologically, etc. motivated) qualitative properties of the original phenomena. In this talk this problem will be discussed. We examine the different qualitative properties (maximum principles, non-negativity preservation, maximum norm contractivity) for both models and we show the relation between them for the linear problems. For the numerical models we give the condition for the construction of the mesh under which the above qualitative properties are valid. The results will be demonstrated in different real-life problems. The main attention will be focused to the heat conduction problem. Briefly we discuss the compartmental epidemic models which take into the account the space dependence, and also some simple discrete Lotka-Volterra models.

Presentations in 2016/17