Jacek Leszczynski

A Numerical Method for Solution of Ordinary Differential Equations of Fractional Order

In this paper we propose an algorithm for the numerical solution of arbitrary differential equations of fractional order. The algorithm is obtained by using the following decomposition of the differential equation into a system of differential equation of integer order connected with inverse forms of Abel-integral equations. The algorithm is used for solution of the linear and non-linear equations.

Numerical solution of fractional oscillator equation

Abstract We focus on a numerical scheme applied for a fractional oscillator equation in a finite time interval. This type of equation includes a complex form of left- and right-sided fractional derivatives. Its analytical solution is represented by a series of left and right fractional integrals and therefore is difficult in practical calculations. Here we elaborated two numerical schemes being dependent on a fractional order of the equation. The results of numerical calculations are compared with analytical solutions. Then we illustrate convergence and stability of our schemes.

Numerical treatment of an initial-boundary value problem for fractional partial differential equations

This paper deals with numerical solutions to a partial differential equation of fractional order. Generally this type of equation describes a transition from anomalous diffusion to transport processes. From a phenomenological point of view, the equation includes at least two fractional derivatives: spatial and temporal. In this paper we proposed a new numerical scheme for the spatial derivative, the so-called Riesz-Feller operator. Moreover, using the finite difference method, we show how to employ this scheme in the numerical solution of fractional partial differential equations. In other words, we considered an initial-boundary value problem in one-dimensional space. In the final part of this paper some numerical results and plots of simulations are shown as examples.

Using the fractional interaction law to model the impact dynamics of multiparticle collisions in arbitrary form.

Using the molecular dynamics method, we examine a discrete deterministic model for the motion of spherical particles in three-dimensional space. The model takes into account multiparticle collisions in arbitrary forms. Using fractional calculus we proposed an expression for the repulsive force, which is the so called fractional interaction law. We then illustrate and discuss how to control (correlate) the energy dissipation and the collisional time for an individual article within multiparticle collisions. In the multiparticle collisions we included the friction mechanism needed for the transition from coupled torsion-sliding friction through rolling friction to static friction. Analysing simple simulations we found that in the strong repulsive state binary collisions dominate. However, within multiparticle collisions weak repulsion is observed to be much stronger. The presented numerical results can be used to realistically model the impact dynamics of an individual particle in a group of colliding particles.

Nanocrystalline Block Cores for High-Frequency Chokes

Nanocrystalline Fe-based soft magnetic materials are more and more common not only in power electronic, but also in power energy conversion systems. The use of different treatments (the methods of stacking and annealing at/without magnetic field presence) on magnetic cores allow to achieve their predefined magnetic properties. This fact increases the level of the applicability such magnetic cores. Different concepts of nanocrystalline magnetic block cores were discussed in this paper. Obtained results were compared with magnetic cores used up to nowadays (made of amorphous and Fe-Si steel).

Computer simulations of multiparticle-contacts dynamics

We considered the complex problem of how to simulate dynamics of multiparticle contacts under the molecular dynamics method. The understanding of interaction process is therefore crucial in order to develop theoretical studies and also to perform simulations of motion of a granular material. In opposite to binary collisions, where several contacts between particles are independent, multiparticle contacts depend on some history including several two-particle contacts. To solve this problem we applied fractional interaction law, where fractional derivatives accumulate the whole history of the function in weighted form. We proposed a novel algorithm which allows to perform calculations for an arbitrary form of multiparticle contacts.

Evaluation of structure and particle velocity distribution in circulating fluidised beds

Abstract This paper deals with the complex problem of how to evaluate the parameters of the structure in circulating fluidised beds. Some aspects of this problem have arisen from the introduction of circulating fluidised bed boilers. The distributions of particle concentration and particle velocity reflect the random behaviour of particles in a combustion chamber. The level of particle velocity—while the boiler is in operation—has an impact on the combustion process, kinetics and heat transfer, and also on the erosion processes on the heat surfaces. A linear combination of double Maxwellian distributions has been assumed to describe a component of the distribution of particle velocity. This assumption allows us to calculate statistical moments in a different way, e.g. the mean particle velocity, granular temperature, etc. Additionally, the local structure of a circulating fluidised bed has been presented.

Effective Algorithm for Detection of a Collision between Spherical Particles

In this work we present a novel algorithm which detects contacts between spherical particles in 2D and 3D. We estimate efficiency of this algorithm throughout analysis of the execution time. We also compare our results with the Linked Cell Method.

Object Oriented Implementation of Modelling Bi-phase Gas-Particle Flows

This paper deals with the problem of modeling bi–phase gas-particle flows taking into account the basic mathematical model, numerical methods, problem geometry and object oriented model of an application used for modeling. The object-oriented approach is used to build an efficient software package based on DEM and MP-PIC methods.

Properties of Fe-based nanocrystalline magnetic powder cores (MPC) and structure of particle size distribution (PSD)

Abstract This paper discusses the influence of the Particle Size Distribution (PSD) of the nanocrystalline Fe-based granular-soft-magnetic material on the final magnetic properties of a Magnetic Powder Core (MPC). Here we show how PSD impacts the final magnetic properties. Mixing fine and coarse particles, with a dominance of coarse particles, significantly influences the magnetic permeability increase of the core. Better magnetic features are noted for MPCs constructed with certain mass ratio of fine and coarse particles due to improvement in the magnetic path in the cores. This allows to offer new induction components to industry.