A Compact Alternating Direction Implicit Scheme for Two-Dimensional Fractional Oldroyd-B Fluids

[Vasileva Daniela; Василева Даниела]; [Bazhlekov Ivan; Бажлеков Иван]; [Bazhlekova Emilia; Бажлекова Емилия]The two-dimensional Rayleigh-Stokes problem for a generalized Oldroyd-B fluid is considered in the present work. The fractional time derivatives are discretized using L1 and L2 approximations. A fourth order compact approximation is implemented for the space derivatives and two variants of an alternating direction implicit finite difference scheme are numerically investigated. 2010 Mathematics Subject Classification: 26A33, 35R11, 65M06, 65M22, 74D05

Modeling Static Electric Field Effect on Nematic Liquid Crystal Director Orientation in Side-Electrode Cell

A two-dimensional model of Fredericks effect was used for the investigation of the static electric field influence on nematic liquid crystal director orientation in the side-electrode cell. The solutions of the equations describing the model were obtained by finite-difference methods. Fredericks transition threshold for the central part of the cell, as well as dependencies of the distribution of the director orientation patterns on the electric field and location, were obtained. The numerical results are found to agree qualitatively with the experiment. Further investigations are needed to elucidate completely the Fredericks effect

Approximate Calculation of Functional Integrals Generated by Nonrelativistic and Relativistic Hamiltonians

The discussion revolves around the most recent outcomes in the realm of approximating functional integrals through calculations. Review of works devoted to the application of functional integrals in quantum mechanics and quantum field theory, nuclear physics and in other areas is presented. Methods obtained by the authors for approximate calculation of functional integrals generated by nonrelativistic Hamiltonians are given. One of the methods is based on the expansion in eigenfunctions of the Hamiltonian. In an alternate approach, the functional integrals are tackled using the semiclassical approximation. Methods for approximate evaluation of functional integrals generated by relativistic Hamiltonians are presented. These are the methods using functional polynomial approximation (analogue of formulas of a given degree of accuracy) and methods based on the expansion in eigenfunctions of the Hamiltonian, generating a functional integral

Quasi-Vector Model of Propagation of Polarized Light in a Thin-Film Waveguide Lens

Maxwell equations describe the propagation with diffraction of waveguide modes through a thin-film waveguide lens. If the radius of the thin-film lens is large, then the thickness of the lens varies slowly in the yz plane. For this case we propose the model, which is based on the assumption of a small change in the electromagnetic field in a direction y. Under this assumption the vector diffraction problem is reduced to a number of scalar diffraction problems. The solutions demonstrate the vector nature of the electromagnetic field, which allows us to call the proposed model a quasi-vector model

Functional Integral Approach to the Solution of a System of Stochastic Differential Equations

A new method for the evaluation of the characteristics of the solution of a system of stochastic differential equations is presented. This method is based on the representation of a probability density function p through a functional integral. The functional integral representation is obtained by means of the Onsager-Machlup functional technique for a special case when the diffusion matrix for the SDE system defines a Riemannian space with zero curvature

Functional Integral Approach to the Solution of a System of Stochastic Differential Equations

A new method for the evaluation of the characteristics of the solution of a system of stochastic differential equations is presented. This method is based on the representation of a probability density function p through a functional integral. The functional integral representation is obtained by means of the Onsager-Machlup functional technique for a special case when the diffusion matrix for the SDE system defines a Riemannian space with zero curvature

Modelling Leaky Waves in Planar Dielectric Waveguides

Experimentally observed leaky modes of a dielectric waveguide are characterised by a weak tunnelling of the light through the waveguide and its long-time propagation along the waveguide. Traditional mathematical models of leaky waveguide modes meet some contradictions resolved using additional considerations. We propose a model of leaky modes in a waveguide free from the above contradictions, akin to the quantum mechanical model of the “pseudo-stable” Gamow-Siegert states. By separating variables, from the complete problem for plane inhomogeneous waves we obtain a non-self-adjoint Sturm-Liouville problem to determine the complex coefficient of the phase delay of the studied mode. The solution of the complete wave problem determines the propagation cone for the leaky mode of the waveguide, inside which there are no contradictions. Thus, solution is in qualitative agreement with experimental data

Quasi-Vector Model of Propagation of Polarized Light in a Thin-Film Waveguide Lens

Maxwell equations describe the propagation with diffraction of waveguide modes through a thin-film waveguide lens. If the radius of the thin-film lens is large, then the thickness of the lens varies slowly in the yz plane. For this case we propose the model, which is based on the assumption of a small change in the electromagnetic field in a direction y. Under this assumption the vector diffraction problem is reduced to a number of scalar diffraction problems. The solutions demonstrate the vector nature of the electromagnetic field, which allows us to call the proposed model a quasi-vector model