Harijs Kalis

Method of Lines and Finite Difference Schemes with the Exact Spectrum for Solution the Hyperbolic Heat Conduction Equation

Abstract This paper is concerning with the 1-D initial–boundary value problem for the hyperbolic heat conduction equation. Numerical solutions are obtained using two discretizations methods – the finite difference scheme (FDS) and the difference scheme with the exact spectrum (FDSES). Hyperbolic heat conduction problem with boundary conditions of the third kind is solved by the spectral method. Method of lines and the Fourier method are considered for the time discretization. Finite difference schemes with central difference and exact spectrum are analyzed. A novel method for solving the discrete spectral problem is used. Special matrix with orthonormal eigenvectors is formed. Numerical results are obtained for steel quenching problem in the plate and in the sphere with holes. The hyperbolic heat conduction problem in the sphere with holes is reduced to the problem in the plate. Some examples and numerical results for two typical problems related to hyperbolic heat conduction equation are presented.

Dynamics of superparamagnetic filaments with finite magnetic relaxation time

Abstract.In this paper we formulate a model of superparamagnetic filaments with internal dissipative torques due to the action of a rotating magnetic field. It is shown that spirals are formed at both ends of the filament due to the action of the internal torques. These spirals propagate to the center of the filament and collide, forming a compact cluster that rotates in accordance with the rotating magnetic field. These results are in agreement with recent experiments with chains of superparamagnetic beads in a rotating magnetic field.

Numerical Investigations of Single Mode Gyrotron Equation

Abstract A stationary problem with the integral boundary condition arising in the mathematical modelling of a gyrotron is numerically investigated. The Chebyshevs polynomials of the second kind are used as the tool of calculations. The main result with physical meaning is the possibility to determine the maximal value of electrons efficiency.

A Certain Mathematical Model of the Glass Fibre Material Production

There is considered the full mathematical model of chemical reactions on the surface of glass fibre material that was imbedded in the flow of acid solution and was pulled longitudionally. Self-similar forms of this model are obtained and their approximations by monotone schemes of differences are proposed. Some special cases which make possible to get the analytic solutions are underlined. The self-similar forms of the differential equations of the substances transport allow to calculate the emission of the alkaline oxide from the glass fibre material under the influence of the acid solution flow. Some conclusions with practical significance for the technological process is made up according to the provided computational experiments.

Special Splines of Exponential Type for the Solutions of Mass Transfer Problems in Multilayer Domains

AbstractWe consider averaging methods for solving the 3-D boundary-value problem of second order in multilayer domain.The special hyperbolic and exponential type splines, with middle integral values of piece-wise smooth function interpolation are considered. With the help of these splines the problems of mathematical physics in 3-D with piece-wise coefficients are reduced with respect to one coordinate to 2-D problems. This procedure also allows to reduce the 2-D problems to 1-D problems and the solution of the approximated problemsa can be obtained analytically. In the case of constant piece-wise coefficients we obtain the exact discrete approximation of a steady-state 1-D boundary-value problem.The solution of corresponding averaged 3-D initial-boundary value problem is also obtained numerically, using the discretization in space with the central diferences. The approximation of the 3-D nonstationary problem is based on the implicit finite-difference and alternating direction (ADI) methods. The numerica...

Mathematical Modelling of an Elongated Magnetic Droplet in a Rotating Magnetic Field

Dynamics of an elongated droplet under the action of a rotating magnetic field is considered by mathematical modelling. The actual shape of a droplet is obtained by solving the initial-boundary value problem of a nonlinear singularly perturbed partial differential equation (PDE). For the discretization in space the finite difference scheme (FDS) is applied. Time evolution of numerical solutions is obtained with MATLAB by solving a large system of ordinary differential equations (ODE).

Creation of Temperature Field in a Finite Cylinder by Alternated Electromagnetic Force

One of the modern areas of applications developed during last years is effective use of electrical energy produced by alternating current in production of heat energy. This process is ecologically clean.The water is weakly electrically conducting medium (electrolyte). Devices based on this principle are developed during last ten years. Compared to classical devices with heating elements, new devices are more compact.

Non-Isothermal Mathematical Model of Wood and Paper Drying

A mathematical model of wood or paper drying based on a detailed consideration of both heat and moisture transport phenomena is proposed. By averaging we express the model as a sequence of initial value problems for systems of two first order nonlinear ordinary differential equations. This mathematical model makes it possible to efficiently investigate the drying process of a thin wood plate or paper sheet for varying temperature and humidity conditions in the surroundings. In particular, we have considered the optimization of the heat regime over a series of steam-heated cylinders in a papermaking machine.

Numerical Modelling of Heat and Magnetohydrodynamic Flows in a Finite Cylinder

Abstract The distribution of electromagnetic fields, forces and temperatures induced by a three- phase axially-symmetric system of electric current in a conducting cylinder of finite length has been calculated. An original method was used to calculate the mean values of electromagnetic forces. The magnetohydrodynamic (MHD) flow of viscous incompressible liquid and distribution of temperature are obtained by the finite-difference method, using monotonic finite difference schemes.

Numerical Experiments with Single Mode Gyrotron Equations

Gyrotrons are microwave sources whose operation is based on the stimulated cyclotron radiation of electrons oscillating in a static magnetic field. This process is described by the system of two complex differential equations: nonlinear first order ordinary differential equation with parameter (averaged equation of electron motion) and second order partial differential equation for high frequency field (RF field) in resonator (Schrodinger type equation for the wave amplitude). The stationary problem of the single mode gyrotron equation in short time interval with real initial conditions was numerically examined in our earlier work. In this paper we consider the stationary and nonstationary problems in large time interval with complex oscillating initial conditions. We use the implicit finite difference schemes and the method of lines realized with MATLAB. Two versions of gyrotron equation are investigated. We consider different methods for modelling new and old versions of the gyrotron equations. The main...