卷 63, 编号 8 (2023)

The use of spectral methods for solution of boundary value problems is very effective but involves great technical difficulties associated with the implementation of the boundary conditions. There exist several methods of such an implementation, but they are either very cumbersome or require a preliminary analysis of the problem and its reduction to an integral form. We propose a universal means of implementation of the boundary conditions for linear differential operators on a finite interval, which is very simple in its realization. The use of the rational arithmetic allows to assess the effectiveness of this method without interference of the round-off errors. We apply this approach for computation of rational approximations for some fundamental constants. We obtained approximations that in a number of cases are better than those that are given by convergents of regular continued fractions of these constants.

The study of nonlinear problems associated with heat transfer in substance is important for practice. Earlier, the authors proposed an efficient algorithm for determining the thermal conductivity from experimental observations of the dynamics of the temperature field in an object. In this work, we explore the possibility of extending the algorithm to the numerical solution of the problem of simultaneous identification of the temperature-dependent volume heat capacity and the thermal conductivity of the substance under study. The consideration is based on the Dirichlet boundary value problem for the one-dimensional nonstationary heat equation. The coefficient inverse problem in question is reduced to a variational problem, which is solved by applying gradient methods based on the fast automatic differentiation technique. The uniqueness of the solution to the inverse problem is analyzed.

In this work, Improved Bernoulli Sub-Equation Function (IBSEF) method is proposed to seek solitary solutions of nonlinear differential equations. Chaffee–Infante equations are chosen to illustrate the effectiveness and convenience of the suggested method. Abundant new and more general exact solutions are obtained of these equations. As a result, by selecting the suitable parameters, two and three dimensional surfaces and contour plots of the results are drawn with the help of the software program.

In weighted Hölder spaces, classes of smooth arcs and piecewise smooth contours are introduced that are invariant under power mappings. The boundary properties of conformal mappings are described in terms of these classes by analogy with Kellogg’s classical theorem.

The problem of calculating the processes of electron emission from metal surfaces in strong electromagnetic fields is considered with allowance for relativistic effects. One of the methods of simulation in these processes is the particle method combined with grid calculation of fields on the basis of Maxwell’s equations. Similar techniques have been developed since the 1960s to the present. However, existing approaches have certain limitations. In this work, for an axisymmetric geometry of the generating system, a new numerical technique simulating the processes of electron emission from metal cathode surfaces is presented. The technique uses the representation of large smoothed Gaussian particles and implements the calculation of electromagnetic fields on Cartesian spatial grids. The software implementation is oriented to parallel computing. The aim of numerical experiments was to determine the parameters of electron emission. Diode and triode cylindrical systems were chosen as test problems. In numerical calculations, the spatiotemporal characteristics of relativistic electron beams generated by emission processes are obtained, including the reproduction of the Child–Langmuir current. The numerical technique developed has confirmed its correctness and efficiency.

In the current paper, numerical simulation has been performed for the two dimensional time dependent Pennes’ heat transfer model which is solved for irregular diseased tumor cells. An elliptic cryoprobe of varying sizes is taken at the center of the computational domain in such a manner that the location of the probe is fixed throughout the computation. The phase transition occurs due to the effect of probe with infusion of different nanoparticles Au, Al2O3, Fe3O4. The cooling performance of these nanoparticles injected at very low temperature, has been studied by implementing hybrid FEM/EFGM method in which the whole domain is decomposed into two subdomains. The results are shown in terms of temperature profile inside the computational domain. Rate of cooling is obtained for various nanoparticles and it is observed that infusion of Au nanoparticles is very much efficient on increasing the heating rate than other nanoparticles. Such numerical scheme has direct applications where the domain is irregular.

The topic of stable matchings (marriages) in bipartite graphs gained popularity beginning from the appearance of the classical Gale and Shapley work. In this paper, a detailed review of selected and other related statements in this field that describe structured, polyhedral, and algorithmic properties of such objects and their sets accompanied by short proofs is given.