Henryk Leszczyński

Iterative Methods for Generalized von Foerster Equations with Functional Dependence

We investigate when a natural iterative method converges to the exact solution of a differential-functional von Foerster-type equation which describes a single population depending on its past time and state densities, and on its total size. On the right-hand side, we assume either Perron comparison conditions or some monotonicity.

Blow-up and global existence for a kinetic equation of swarm formation

In the present paper we study possible blow–ups and global existence for a kinetic equation that describes swarm formations in the variable interacting rate case.

A simple kinetic equation of swarm formation: Blow-up and global existence

Abstract In the present paper we identify both blow-up and global existence behaviors for a simple but very rich kinetic equation describing of a swarm formation.

Computation of Trajectories and Displacement Fields in a Three-Dimensional Ternary Diffusion Couple: Parabolic Transform Method

The problem of Kirkendall’s trajectories in finite, three- and one-dimensional ternary diffusion couples is studied. By means of the parabolic transformation method, we calculate the solute field, the Kirkendall marker velocity, and displacement fields. The velocity field is generally continuous and can be integrated to obtain a displacement field that is continuous everywhere. Special features observed experimentally and reported in the literature are also studied: (i) multiple Kirkendall’s planes where markers placed on an initial compositional discontinuity of the diffusion couple evolve into two locations as a result of the initial distribution, (ii) multiple Kirkendall’s planes where markers placed on an initial compositional discontinuity of the diffusion couple move into two locations due to composition dependent mobilities, and (iii) a Kirkendall plane that coincides with the interphase interface. The details of the deformation (material trajectories) in these special situations are given using both methods and are discussed in terms of the stress-free strain rate associated with the Kirkendall effect. Our nonlinear transform generalizes the diagonalization method by Krishtal, Mokrov, Akimov, and Zakharov, whose transform of diffusivities was linear.

The Method of Lines for Ternary Diffusion Problems

The method of lines (MOL) for diffusion equations with Neumann boundary conditions is considered. These equations are transformed by a discretization in space variables into systems of ordinary differential equations. The proposed ODEs satisfy the mass conservation law. The stability of solutions of these ODEs with respect to discrete L2 norms and discrete norms is investigated. Numerical examples confirm the parabolic behaviour of this model and very regular dynamics.

Iterative methods for ternary diffusions

We apply iterative methods to three-component diffusion equations and study theirconvergence in L2 and in the Sobolev space W1,∞. The system is parabolic and mass-conservative.Newton’s method converges very fast and its iterations do not leave theset of admissible functions.MSC: 35K51, 35K57, 65M12, 65M80.

Self-organization with small range interactions: Equilibria and creation of bipolarity

Abstract We study a kinetic equation which describes self-organization of various complex systems, assuming the interacting rate with small support. This corresponds to interactions between an agent with a given internal state and agents having short distance states only. We identify all possible stationary (equilibrium) solutions and describe the possibility of creating of bipolar (bimodal) distribution that is able to capture interesting behavior in modeling systems, e.g. in political sciences.

Asymptotic Expansion Method with Respect to Small Parameter for Ternary Diffusion Models

Ternary diffusion models lead to strongly coupled systems of PDEs. We choose the smallest diffusion coefficient as a small parameter in a power series expansion whose components fulfill relatively simple equations. Although this series is divergent, one can use its finite sums to derive feasible numerical approximations, e.g. finite difference methods (FDMs).

Newton's method for nonlinear stochastic wave equations driven by one-dimensional Brownian motion.

We consider nonlinear stochastic wave equations driven by one-dimensional white noise with respect to time. The existence of solutions is proved by means of Picard iterations. Next we apply Newtons method. Moreover, a second-order convergence in a probabilistic sense is demonstrated.

Electrochemistry of Symmetrical Ion Channel: Three-Dimensional Nernst-Planck-Poisson Model

The paper provides a physical description of ionic transport through the rigid symmetrical channel. A three-dimensional mathematical model, in which the ionic transport is treated as the electrodiffusion of ions, is presented. The model bases on the solution of the 3D Nernst-Planck-Poisson system for cylindrical geometry. The total flux includes drift (convection) and diffusion terms. It allows simulating the transport characteristics at the steady-state and time evolution of the system. The numerical solutions of the coupled differential diffusion equation system are obtained by finite element method. Examples are presented in which the flow characteristics at the stationary state and during time evolution are compared. It is shown that the stationary state is achieved after about 2×10 -8 s since the process beginning. Various initial conditions (channel charging and dimensions) are considered as the key parameters controlling the selectivity of the channel. The model allows determining the flow characteristic, calculating the local concentration and potential across the channel. The model can be extended to simulate transport in polymer membranes and nanopores which might be useful in designing biosensors and nanodevices.