Janis S. Rimshans

Polarization kinetics in ferroelectrics with regard to fluctuations

Polarization in ferroelectrics, described by the Landau–Ginzburg Hamiltonian, is considered, based on a multi-dimensional Fokker–Planck equation. This formulation describes the time evolution of the probability distribution function over the polarization field configurations in the presence of a time-dependent external field. The Fokker–Planck equation in a Fourier representation is obtained, which can then be solved numerically for a finite number of modes. Calculation results are presented for one and three modes. These results show the hysteresis of the mean polarization as well as that of the mean squared gradient of the polarization.

Analytical and Numerical Solutions of Temperature Transients in Electrochemical Machining

A new semi-implicit difference scheme based on our propagator method is elaborated for solution of initial-boundary value problem of electrochemical machining. The scheme is unconditionally monotonic and has truncation errors of the first order in time and of the second order in space. Analytical and numerical solutions for temperature distribution in 2D electrolytic cell are presented.

ON THE GINZBURG-FEINBERG PROBLEM OF FREQUENCY ELECTROMAGNETIC SOUNDING FOR UNAMBIGUOUS DETERMINATION OF THE ELECTRON DENSITY IN THE IONOSPHERE

In the present work, we investigate an inverse problem of frequency electromagnetic sounding for unambiguous determination of the electron density in the ionosphere. Direct statement of this problem is known as the Ginzburg-Feinberg problem that has, in general case, an essential nonlinearity. Inverse statement of the Ginzburg-Feinberg problem has the boundary-value formulation relative to two functions: the sought-for electric-field strength and the distribution of the electron density (or rather two-argument function appearing in the additive decomposition formula for distribution of the electron density) in the ionosphere. In the present work, we prove the existence and uniqueness of the solution of the Ginzburg-Feinberg problem as well as we propose the analytical method, permitting: first, to reduce it to the problem of integral geometry, and, thereupon, having applied the adjusted variant of the Lavrentievs theorem, to reduce the obtained problem of integral geometry to the first kind matrix integral equation of Volterra type with a weak singularity.

Modeling of Surface Structure Formation after Laser Irradiation

The Stefan problem in a semi-infinite media under laser irradiation is considered. It is related to the melting and solidification processes, resulting in certain surface structure after the solidification. A simple model, as well as a more sophisticated one is proposed to describe this process. The latter model allows us to calculate the surface profile by solving a system of two nonlinear differential equations, if the shape of the solid-liquid interface is known. It has to be found as a solution of two-phases Stefan problem. The results of example calculations by the fourth-order Runge-Kutta method are presented, assuming that the solid-liquid interface has a parabolic shape. The calculated crossection of the surface structure shows a characteristic cone in the center, in agreement with experimental observations.

Propagator Method for Numerical Solution of Convective Fisher Equation

Propagator numerical method was developed as an effective tool for modeling of linear advective dispersive reactive (ADR) processes [1]. In this work implicit propagator difference scheme for Fisher equation with nonlinear convection (convective Fisher equation) is elaborated. Our difference scheme has truncation errors of the second order in space and of the first order in time. Iteration process for implicit difference scheme is proposed by introducing forcing terms in the left and right sides of the difference equation. Convergence and stability criterions for the elaborated implicit propagator difference scheme are obtained.

A Method of Conversion of some Coefficient Inverse Parabolic Problems to a Unified Type of Integral-Differential Equation

Coefficient inverse problems are reformulated to a unified integral differential equation. The presented method of conversion of the considered inverse problems to a unified Volterra integral-differential equation gives an opportunity to distribute the acquired results also to analogous inverse problems for non-linear parabolic equations of different types.

Nodal Numerical 2D Helmholtz Equation: Truncation Analysis

A functional nodal method for numerical solution of a two-dimensional Helmholtz equation is considered. The method uses a compact 7-point difference scheme and provides continuity of averaged fluxes and the numerical solution. Truncation analysis of the difference scheme is done and it is shown that the considered difference scheme is of the 2nd order of precision inside the domain.