The generalization of the incomplete Galerkin method to the description of the transitions of the “horn” type between open planar waveguides is discussed. The obtained result is characterized by a high degree of analyticity of the derived equations and this is expected to enhance the efficiency of the eigenwave modeling in open irregular waveguides.

To implement the method of adiabatic waveguide modes for modeling the propagation of polarized monochromatic electromagnetic radiation in irregular integrated optics structures it is necessary to expand the desired solution in basic adiabatic waveguide modes. This expansion requires the use of the scalar product in the space of waveguide vector fields of integrated optics waveguide. This work solves the first stage of this problem – the construction of the scalar product in the space of vector solutions of the eigenmode problem (classical and generalized) waveguide modes of an open planar waveguide. In constructing the mentioned sesquilinear form, we used the Lorentz reciprocity principle of waveguide modes and tensor form of the Ostrogradsky-Gauss theorem.

Presentation of the probability as an intrinsic property of the nature leads researchers to switch from deterministic to stochastic description of the phenomena. The procedure of stochastization of one-step process was formulated. It allows to write down the master equation based on the type of of the kinetic equations and assumptions about the nature of the process. The kinetics of the interaction has recently attracted attention because it often occurs in the physical, chemical, technical, biological, environmental, economic, and sociological systems. However, there are no general methods for the direct study of this equation. Leaving in the expansion terms up to the second order we can get the Fokker-Planck equation, and thus the Langevin equation. It should be clearly understood that these equations are approximate recording of the master equation. However, this does not eliminate the need for the study of the master equation. Moreover, the power series produced during the master equation decomposition may be divergent (for example, in spatial models). This makes it impossible to apply the classical perturbation theory. It is proposed to use quantum field perturbation theory for the statistical systems (the so-called Doi method). This work is a methodological material that describes the principles of master equation solution based on quantum field perturbation theory methods. The characteristic property of the work is that it is intelligible for non-specialists in quantum field theory. As an example the Verhulst model is used because of its simplicity and clarity (the first order equation is independent of the spatial variables, however, contains non-linearity). We show the full equivalence of the operator and combinatorial methods of obtaining and study of the one-step process master equation.

As a result of the application of a technique of multistep processes stochastic models construction the range of models, implemented as a self-consistent differential equations, was obtained. These are partial differential equations (master equation, the Fokker--Planck equation) and stochastic differential equations (Langevin equation). However, analytical methods do not always allow to research these equations adequately. It is proposed to use the combined analytical and numerical approach studying these equations. For this purpose the numerical part is realized within the framework of symbolic computation. It is recommended to apply stochastic Runge--Kutta methods for numerical study of stochastic differential equations in the form of the Langevin. Under this approach, a program complex on the basis of analytical calculations metasystem Sage is developed. For model verification logarithmic walks and Black--Scholes two-dimensional model are used. To illustrate the stochastic "predator--prey" type model is used. The utility of the combined numerical-analytical approach is demonstrated.

The paper describes the relationship between the solutions of Maxwell’s equations which can be considered at least locally as plane waves and the curvilinear coordinates of geometrical optics; it generalizes the results achieved by Lüneburg, concerning the evolution of surfaces of electromagnetic fields discontinuities. If vectors and are orthogonal to each other and their directions do not change with time t, but may vary from point to point in the domain G, then under some conditions there is an orthogonal coordinate system in which -lines represent rays of geometrical optics, -lines point out -direction and, -lines point out -direction. This coordinate system will be called phase-ray coordinate system. In the article, it will be proved that field under study can be represented by two scalar functions. The article will also specify the necessary and sufficient conditions for the existence of a coordinate system, generated by the solution of Maxwell’s equations with the holonomic field of the Poynting vector. It is shown that the class of solutions of Maxwell’s equations, as described in this work, includes monochromatic polarized waves, and the Hilbert–Courant solutions and their generalizations.

The mathematical model of light propagation in a planar gradient optical waveguide consists of the Maxwell’s equations supplemented by the matter equations and boundary conditions. In the coordinates adapted to the waveguide geometry, the Maxwell’s equations are separated into two independent sets for the TE and TM polarizations. For each there are three types of waveguide modes in a regular planar optical waveguide: guided modes, substrate radiation modes, and cover radiation modes. We implemented in our work the numerical-analytical calculation of typical representatives of all the classes of waveguide modes. In this paper we consider the case of a linear profile of planar gradient waveguide, which allows for the most complete analytical description of the solution for the electromagnetic field of the waveguide modes. Namely, in each layer we are looking for a solution by expansion in the fundamental system of solutions of the reduced equations for the particular polarizations and subsequent matching them at the boundaries of the waveguide layer. The problem on eigenvalues (discrete spectrum) and eigenvectors is solved in the way that first we numerically calculate (approximately, with double precision) eigenvalues, then numerically and analytically—eigenvectors. Our modelling method for the radiation modes consists in reducing the initial potential scattering problem (in the case of the continuous spectrum) to the equivalent ones for the Jost functions: the Jost solution from the left for the substrate radiation modes and the Jost solution from the right for the cover radiation modes.

Background. By the means of the method of stochastization of one-step processes we get the simplified mathematical model of the original stochastic system. We can explore these models by standard methods, as opposed to the original system. The process of stochastization depends on the type of the system under study. Purpose. We want to get a unified abstract formalism for stochastization of one-step processes. This formalism should be equivalent to the previously introduced. Methods. To unify the methods of construction of the master equation, we propose to use the diagram technique. Results. We get a diagram technique, which allows to unify getting master equation for the system under study. We demonstrate the equivalence of the occupation number representation and the state vectors representation by using a Verhulst model. Conclusions. We have suggested a convenient diagram formalism for unified construction of stochastic systems.

This paper studies program implementation problem of pseudo-random number generators in OpenModelica. We give an overview of generators of pseudo-random uniform distributed numbers. They are used as a basis for construction of generators of normal and Poisson distributions. The last step is the creation of Wiener and Poisson stochastic processes generators. We also describe the algorithm to call external C-functions from programs written in Modelica. This allows us to use random number generators implemented in the C language.

This paper studies program implementation problem of pseudo-random number generators in OpenModelica. We give an overview of generators of pseudo-random uniform distributed numbers. They are used as a basis for construction of generators of normal and Poisson distributions. The last step is the creation of Wiener and Poisson stochastic processes generators. We also describe the algorithm to call external C-functions from programs written in Modelica. This allows us to use random number generators implemented in the C language.

Representing a probability density function (PDF) and other quantities describing a solution of stochastic differential equations by a functional integral is considered in this paper. Methods for the approximate evaluation of the arising functional integrals are presented. Onsager–Machlup functionals are used to represent PDF by a functional integral. Using these functionals the expression for PDF on a small time interval Δt can be written. This expression is true up to terms having an order higher than one relative to Δt. A method for the approximate evaluation of the arising functional integrals is considered. This method is based on expanding the action along the classical path. As an example the application of the proposed method to evaluate some quantities to solve the equation for the Cox–Ingersol–Ross type model is considered.

The problem of diffraction of electromagnetic TE-polarized monochromatic radiation on a three-dimensional thickening of a waveguide layer of a regular planar three-layer dielectric waveguide forming a thin-film waveguide lens is considered in this paper. An approximate mathematical model with an open waveguide considered inside an auxiliary closed waveguide is presented. It leads to a correct mathematical formulation of the diffraction problem. It is shown that the parameters of the guided modes of the open waveguide are stable to the shifts of the boundaries of the enclosing closed waveguide. Consequently, the proposed approach adequately describes the propagation of polarized light in an open smoothly irregular waveguide. Due to the local thickening of the waveguide layer, the effect of radiation depolarization arises, which requires consideration of the vector nature of the propagating electromagnetic radiation. In the paper, the diffraction problem is solved in the adiabatic approximation with respect to a small parameter corresponding to the irregularity. Carrying out numerical experiments made it possible to show that with decreasing small parameter the matrix of reflection coefficients tends to zero, and the matrix of transmission coefficients tends to a unit matrix. Moreover, the exchange contributions to which the off-diagonal elements of the matrices correspond tend to zero much faster than the diagonal terms. So, the depolarization effects in the considered configuration can be neglected.

The paper deals with a numerical solution of the problem of waveguide propagation of polarized light in smoothly-irregular transition between closed regular waveguides using the incomplete Galerkin method. This method consists in replacement of variables in the problem of reduction of the Helmholtz equation to the system of differential equations by the Kantorovich method and in formulation of the boundary conditions for the resulting system. The formulation of the boundary problem for the ODE system is realized in computer algebra system Maple. The stated boundary problem is solved using Maples libraries of numerical methods.