DISCRETE AND CONTINUOUS MODELS AND APPLIED COMPUTATIONAL SCIENCE

The article discusses a mathematical model and a finite-difference scheme for the heating process of an infinite plate. The disadvantages of using the classical parabolic heat equation for this case and the rationale for using the hyperbolic heat equation are given. The relationship between the hyperbolic thermal conductivity equation and the theory of equations with the retarded argument (delay equation) is shown. The considered mixed equation has 2 parts: parabolic and hyperbolic. Difference schemes use an integro-interpolation method to reduce errors. The problem with a nonlinear thermal conductivity coefficient was chosen as the initial boundaryvalue problem. The heat source in the parabolic part of the equation is equal to 0, and in the hyperbolic part of the equation sharp heating begins. The initial boundary-value problem with boundary conditions of the third kind in an infinite plate with nonlinear coefficients is formulated and numerically solved. An iterative method for solving the problem is described. A visual graph of the solution results is presented. A theoretical justification for the difference scheme is given. Also we consider the case of the nonlinear mixed equation of the fourth order.

There are two main approaches to the numerical solution of the eikonal equation: reducing it to a system of ODES (method of characteristics) and constructing specialized methods for the numerical solution of this equation in the form of a partial differential equation. The latter approach includes the FSM (Fast sweeping method) method. It is reasonable to assume that a specialized method should have greater versatility. The purpose of this work is to evaluate the applicability of the FSM method for constructing beams and fronts. The implementation of the FSM method in the Eikonal library of the Julia programming language was used. The method was used for numerical simulation of spherical lenses by Maxwell, Luneburg and Eaton. These lenses were chosen because their optical properties have been well studied. A special case of flat lenses was chosen as the easiest to visualize and interpret the results. The results of the calculations are presented in the form of images of fronts and rays for each of the lenses. From the analysis of the obtained images, it is concluded that the FSM method is well suited for constructing electromagnetic wave fronts. An attempt to visualize ray trajectories based on the results of his work encounters a number of difficulties and in some cases gives an incorrect visual picture.

In this paper, we study the effect of using the Metropolis–Hastings algorithm for sampling the integrand on the accuracy of calculating the value of the integral with the use of shallow neural network. In addition, a hybrid method for sampling the integrand is proposed, in which part of the training sample is generated by applying the Metropolis–Hastings algorithm, and the other part includes points of a uniform grid. Numerical experiments show that when integrating in high-dimensional domains, sampling of integrands both by the Metropolis–Hastings algorithm and by a hybrid method is more efficient with respect to the use of a uniform grid.

A number of initial boundary-value problems of classical mathematical physics is generally represented in the linear operator equation and its well-posedness and causality in a Hilbert space setting was established. If a problem has a unique solution and the solution continuously depends on given data, then the problem is called well-posed. The independence of the future behavior of a solution until a certain time indicates the causality of the solution. In this article, we established the well-posedness and causality of the solution of the evolutionary problems with a perturbation, which is defined by a quadratic form. As an example, we considered the coupled system of the heat and Maxwell’s equations (the microwave heating problem).

In this article, we propose fourth- and fifth-order two-step iterative methods for solving the systems of nonlinear equations in

Various approaches to calculating normal modes of a closed waveguide are considered. A review of the literature was given, a comparison of the two formulations of this problem was made. It is shown that using a self-adjoint formulation of the problem of normal waveguide modes eliminates the occurrence of artifacts associated with the appearance of a small imaginary additive to the eigenvalues. The implementation of this approach for a rectangular waveguide with rectangular inserts in the Sage computer algebra system is presented and tested on hybrid modes of layered waveguides. The tests showed that our program copes well with calculating the points of the dispersion curve corresponding to the hybrid modes of the waveguide.

The paper considers a single-line retrial queueing system with an unreliable server. Queuing systems are called unreliable if their servers may fail from time to time and require restoration (repair), only after which they can resume servicing customers. The input of the system is a simple Poisson flow of customers. The service time and uptime of the server are distributed exponentially. An incoming customer try to get service. The server can be free, busy or under repair. The customer is serviced immediately if the server is free. If it is busy or under repair, the customer goes into orbit. And after a random time it tries to get service again. The study is carried out by the method of asymptotically diffusion analysis under the condition of a large delay of requests in orbit. In this work, the transfer coefficient and diffusion coefficient were found and a diffusion approximation was constructed.

Integrated Access and Backhaul (IAB) technology facilitates the establishment of a compact network by utilizing repeater nodes rather than fully equipped base stations, which subsequently minimizes the expenses associated with the transition towards next-generation networks. The majority of studies focusing on IAB networks rely on simulation tools and the creation of discrete-time models. This paper introduces a mathematical model for the boundary node in an IAB network functioning in half-duplex mode. The proposed model is structured as a polling service system with a dual-queue setup, represented as a random process in continuous time, and is examined through the lens of queueing theory, integral transforms, and generating functions (GF). As a result, analytical expressions were obtained for the GF, marginal distribution, as well as the mean and variance of the number of requests in the queues, which correspond to packets pending transmission by the relay node via access and backhaul channels.