Vol 65, No 2 (2025)

We study a three-level in time bilinear finite element method with weight for an initial-boundary value problem for the one-dimensional wave equation. We derive lower error estimates of orders (h + τ)2λ/3, 0 ⩽ λ ⩽ 3 in the L1 and W1,1 h norms. In them, each of the two initial functions or the free term in the equation belongs to Holder-type spaces of the corresponding orders of smoothness. They substantiate the accuracy in ¨ order of the corresponding known error estimates (from above) of the finite element method with a weight of the second-order approximation for second-order hyperbolic equations, as well as the impossibility of improving them with the maximum weakening of the degree of summability in the error norms and its maximum strengthening in the data norms. The derivation is based on the Fourier method.

An approximate numerical method for solving singular integro-differential equations of the generalized Prandtl equation type has been developed. The proposed approximate schemes are based on representing the solution and coefficients of the equation as an expansion over an orthogonal basis of Chebyshev polynomials. The use of known spectral relations has made it possible to obtain an analytical expression for the singular component of the equation. As a consequence, the proposed method demonstrates excellent accuracy and exponential rate of convergence of the approximate solution relative to the degree of interpolation polynomials. The computational qualities of the proposed method are demonstrated using a test example.

A two-dimensional medium in which the fields are described by the Helmholtz equation is considered. A linearized formulation of the problem is studied, which ultimately reduces to reconstructing the unknown right-hand side of the inhomogeneous Helmholtz equation in an infinite strip. The specified right-hand side in this work is taken as a sum of delta functions, which can be interpreted as the total conductivities of thin layers. The values of the solution of the Helmholtz equation and the normal derivative of the solution at the band boundary for several values of the parameter in the Helmholtz equation are used as information for solving the inverse problem. These data can be interpreted as the values of the electric and magnetic field strengths at the band boundary for a finite set of frequencies. Using the Fourier series expansion, an integral equation is obtained that relates the sought quantities to the data for solving the inverse problem. Using the Fourier transform, the conditions for the uniqueness of the solution of the inverse problem are established. Along with this, examples of the multivalued nature of the solution of the inverse problem in unexpected situations are given.

One-dimensional problems for the system of Maxwell equations cover a wide range of important applied problems. Among them are problems of photonics, plasmonics, microwave technology, etc. For such problems, we previously proposed a bicompact (two-point completely conservative) difference scheme. An exact solution to the corresponding system of grid equations is constructed. It is applicable to problems in piecewise homogeneous media with an arbitrary configuration of volume and surface currents. The constructed solution makes it possible to dramatically reduce the labor intensity of calculating such problems.

This paper considers an example of the combined use of the grid-characteristic method on regular structured computational grids and the discontinuous Galerkin method on tetrahedral grids to solve a three-dimensional direct problem of elastic wave propagation in a geological medium consisting of four layers represented as a linear-elastic medium with different parameters and arbitrary curvilinear boundaries. A special algorithm is used to stitch the numerical methods, taking into account the features of the transition from an irregular tetrahedral computational grid to a regular structured computational grid in three-dimensional space. A comparative analysis of the convergence of the resulting combined method with the grid-characteristic method on curvilinear structured computational grids is given depending on the change in the step in spatial directions. The wave field of the modulus of the disturbance propagation velocity from the source is obtained.

Determining the appropriate sample size is crucial for building effective machine learning models. Existing methods often either lack a rigorous theoretical basis or are tied to specific statistical hypotheses about the model parameters. In this paper, we present two new methods based on likelihood values on bootstrapped subsamples. We demonstrate the correctness of one of these methods in a linear regression model. Computational experiments with both synthetic and real datasets show that the proposed functions converge as the sample size increases, highlighting the practical usefulness of the approach.