# Vol 30, No 2 (2022)

**Year:**2022**Articles:**6**URL:**https://journals.rudn.ru/miph/issue/view/1543**DOI:**https://doi.org/10.22363/2658-4670-2022-30-2

## Full Issue

### Numerical solution of Cauchy problems with multiple poles of integer order

#### Abstract

We consider Cauchy problem for ordinary differential equation with solution possessing a sequence of multiple poles. We propose the generalized reciprocal function method. It reduces calculation of a multiple pole to retrieval of a simple zero of accordingly chosen function. Advantages of this approach are illustrated by numerical examples. We propose two representative test problems which constitute interest for verification of other numerical methods for problems with poles.

**Discrete and Continuous Models and Applied Computational Science**. 2022;30(2):105-114

### Optimization of an isotropic metasurface on a substrate

#### Abstract

Mathematical statement of one-wavelength antireflective coating based on two-dimensional metamaterial is formulated for the first time. The constraints on geometric parameters of the structure are found. We propose a penalty function, which ensures the applicability of physical model and provides the uniqueness of the desired minimum. As an example, we consider the optimization of metasurface composed of PbTe spheres located on germanium substrate. It is shown that the accuracy of the minimization with properly chosen penalty term is the same as for the objective function without it.

**Discrete and Continuous Models and Applied Computational Science**. 2022;30(2):115-126

### Multistage pseudo-spectral method (method of collocations) for the approximate solution of an ordinary differential equation of the first order

#### Abstract

The classical pseudospectral collocation method based on the expansion of the solution in a basis of Chebyshev polynomials is considered. A new approach to constructing systems of linear algebraic equations for solving ordinary differential equations with variable coefficients and with initial (and/or boundary) conditions makes possible a significant simplification of the structure of matrices, reducing it to a diagonal form. The solution of the system is reduced to multiplying the matrix of values of the Chebyshev polynomials on the selected collocation grid by the vector of values of the function describing the given derivative at the collocation points. The subsequent multiplication of the obtained vector by the two-diagonal spectral matrix, ‘inverse’ with respect to the Chebyshev differentiation matrix, yields all the expansion coefficients of the sought solution except for the first one. This first coefficient is determined at the second stage based on a given initial (and/or boundary) condition. The novelty of the approach is to first select a class (set) of functions that satisfy the differential equation, using a stable and computationally simple method of interpolation (collocation) of the derivative of the future solution. Then the coefficients (except for the first one) of the expansion of the future solution are determined in terms of the calculated expansion coefficients of the derivative using the integration matrix. Finally, from this set of solutions only those that correspond to the given initial conditions are selected.

**Discrete and Continuous Models and Applied Computational Science**. 2022;30(2):127-138

### Complex eigenvalues in Kuryshkin-Wodkiewicz quantum mechanics

#### Abstract

One of the possible versions of quantum mechanics, known as Kuryshkin-Wodkiewicz quantum mechanics, is considered. In this version, the quantum distribution function is positive, but, as a retribution for this, the von Neumann quantization rule is replaced by a more complicated rule, in which an observed value $A$ is associated with a pseudodifferential operator $\hat{O}\left(A\right)$. This version is an example of a dissipative quantum system and, therefore, it was expected that the eigenvalues of the Hamiltonian should have imaginary parts. However, the discrete spectrum of the Hamiltonian of a hydrogen-like atom in this theory turned out to be real-valued. In this paper, we propose the following explanation for this paradox. It is traditionally assumed that in some state $\psi $ the quantity $A$ is equal to $\lambda $ if $\psi $ is an eigenfunction of the operator $\hat{O}\left(A\right)$. In this case, the variance $\hat{O}\left(\right(A-\lambda \left)2\right)\psi $ is zero in the standard version of quantum mechanics, but nonzero in Kuryshkin’s mechanics. Therefore, it is possible to consider such a range of values and states corresponding to them for which the variance $\hat{O}\left(\right(A-\lambda \left)2\right)$ is zero. The spectrum of the quadratic pencil $\hat{O}\left(A2\right)-2\hat{O}\left(A\right)\lambda +\lambda 2\hat{E}$ is studied by the methods of perturbation theory under the assumption of small variance $\hat{D}\left(A\right)=\hat{O}\left(A2\right)-\hat{O}\left(A\right)2$ of the observable $A$. It is shown that in the neighborhood of the real eigenvalue $\lambda $ of the operator $\hat{O}\left(A\right)$, there are two eigenvalues of the operator pencil, which differ in the first order of perturbation theory by $\pm i\sqrt{\u27e8\hat{D}\u27e9}$.

**Discrete and Continuous Models and Applied Computational Science**. 2022;30(2):139-148

### Investigation of adiabatic waveguide modes model for smoothly irregular integrated optical waveguides

#### Abstract

The model of adiabatic waveguide modes (AWMs) in a smoothly irregular integrated optical waveguide is studied. The model explicitly takes into account the dependence on the rapidly varying transverse coordinate and on the slowly varying horizontal coordinates. Equations are formulated for the strengths of the AWM fields in the approximations of zero and first order of smallness. The contributions of the first order of smallness introduce depolarization and complex values characteristic of leaky modes into the expressions of the AWM electromagnetic fields. A stable method is proposed for calculating the vertical distribution of the electromagnetic field of guided modes in regular multilayer waveguides, including those with a variable number of layers. A stable method for solving a nonlinear equation in partial derivatives of the first order (dispersion equation) for the thickness profile of a smoothly irregular integrated optical waveguide in models of adiabatic waveguide modes of zero and first orders of smallness is described. Stable regularized methods for calculating the AWM field strengths depending on vertical and horizontal coordinates are described. Within the framework of the listed matrix models, the same methods and algorithms for the approximate solution of problems arising in these models are used. Verification of approximate solutions of models of adiabatic waveguide modes of the first and zero orders is proposed; we compare them with the results obtained by other authors in the study of more crude models.

**Discrete and Continuous Models and Applied Computational Science**. 2022;30(2):149-159

### Analysis of queuing systems with threshold renovation mechanism and inverse service discipline

#### Abstract

The paper presents a study of three queuing systems with a threshold renovation mechanism and an inverse service discipline. In the model of the first type, the threshold value is only responsible for activating the renovation mechanism (the mechanism for probabilistic reset of claims). In the second model, the threshold value not only turns on the renovation mechanism, but also determines the boundaries of the area in the queue from which claims that have entered the system cannot be dropped. In the model of the third type (generalizing the previous two models), two threshold values are used: one to activate the mechanism for dropping requests, the second - to set a safe zone in the queue. Based on the results obtained earlier, the main time-probabilistic characteristics of these models are presented. With the help of simulation modeling, the analysis and comparison of the behavior of the considered models were carried out.

**Discrete and Continuous Models and Applied Computational Science**. 2022;30(2):160-182