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The problem of rectangular membrane deflection under alternating loads is solved in general terms by means of the method of fast expansions. The exact solution is represented by the finite expression borrowed from the theory of fast expansions as a sum of the boundary function and Fourier sine series with two Fourier coefficients taken into account. The obtained exact solution includes free parameters. Changing the values of these parameters, one can derive many new exact solutions. Obtaining of exact solutions to a problem of the rigidly fixed membrane under two types of loads (dome-shaped and sinusoidal) is shown as an example. Graphs of the dome-shaped and sinusoidal loads on the membrane and the curves of the corresponding deflections and stress components are presented in the paper. From the analysis of the exact solutions, it is obvious that only when a symmetrical alternating load is used, the membrane maximum deflection is attained in the center of the membrane, and the stresses reach the highest values in the middle of both long sides. In the case of a non-symmetrical load, the maximum stress occurs in the middle of either one of two long sides of the rectangular membrane, and the maximum deflection is found in the central region.

The fast decompositions method in analytical form was used to solve the elastic problem with mixed boundary conditions for stresses in a sharp wedge-shaped cutting tool of finite dimensions in the form of a truncated sector. A load is applied to the faces of the wedge and its nose, which rapidly decreases with distance from the nose. At a larger radius, the wedge is rigidly fixed. The resulting solution is valid for any angle of the wedge spread. If we take a small enough inner radius and the angle of the spread, then the shape of the proposed cutter will be close enough to real. The influence of the angle of the spread and the inner radius of the wedge on the magnitude and location of the highest stress σ was investigated. The critical dimensions of the wedge were found, but for any of its sizes the stresses are finite everywhere.

A semigroup of holomorphic self-mappings of a strip with hydrodynamic normalization is studied. A description of infinitesimal transformations of the semigroup is obtained. The differentiability with respect to the time parameter of normalized evolution families is established. An existence theorem is obtained for the evolution equation.

By the fast expansion method, the authors have obtained the approximate solution of the problem in analytical form, which holds true at all points of the cylinder up to the boundary. From an analysis of the solution, it follows that the broken thermal contact between annular regions gives rise to a weak thermal resistance. Beginning with four annular regions, the thermal resistance remains constant, in practice. This result was obtained for the first time. When no more than three terms in a Fourier series are used, the maximum residual of the differential heat-conduction equation is 10–2.

An analytical solution to the boundary value problem for the Poisson equation in a parallelepiped was obtained in general form using the fast expansions method. The construction of exact solutions of the heat conduction equation for the cases of an internal source depending on coordinates was shown in the work. The study of the influence of the variability of the internal source on the propagation of heat in the body at the same initial temperatures at the parallelepiped vertices was carried out. In this case, the heat in the body will be transferred only in those directions in which the internal source is changed.

The approximate analytic solution of the problem of temperature field in a rectangular plate with an internal temperature- dependent source is obtained by the method of fast expansions. The critical value of a parameter characterizing heat release, which fundamentally affects the analytic solution form, is found. The maximum solution error is shown to amount to 0.02 at the consideration of the first three terms of the Fourier series in fast expansion. Temperature fields are presented, and an analysis of the influence of the plate sizes and the heat release magnitude on their formation is given. Recommendations on the plate shape choice are given.

Using the fast expansion method, we obtained several exact solutions of the boundary value problem for the Poisson equation in a rectangular domain. We have given graphs of exact solutions corresponding to different boundary conditions and different types of the Poisson equation free term. We showed the influence of the type of boundary conditions and the right-hand side on the type of exact solution. We obtained a solution to the problem of membrane deflections under the action of variable load. We have given graphs of the stress components, from the analysis of which it follows that the greatest stress is in the middle of the rectangular membrane long sides.

This paper deals with the problems on the spacecraft motion in the Earth’s atmosphere taking into account the Earth’s atmospheric drag as well as on the extra-atmospheric part of the trajectory, where resistance is absent. An analytical solution of these problems by the method of fast expansions is proposed.

The paper is concerned with evolution families of conformal mappings of the unit disc to itself that fix an interior point and a boundary point. Conditions are obtained for the evolution families to be differentiable, and an existence and uniqueness theorem for an evolution equation is proved. A convergence theorem is established which describes the topology of locally uniform convergence of evolution families in terms of infinitesimal generating functions. The main result in this paper is the embedding theorem which shows that any conformal mapping of the unit disc to itself with two fixed points can be embedded into a differentiable evolution family of such mappings. This result extends the range of the parametric method in the theory of univalent functions. In this way the problem of the mutual change of the derivative at an interior point and the angular derivative at a fixed point on the boundary is solved for a class of mappings of the unit disc to itself. In particular, the rotation theorem is established for this class of mappings. Bibliography: 27 titles.

The stress problem for the elastic wedge-shaped cutter of finite dimensions with mixed boundary conditions is considered. The differential problem is reduced to the system of linear algebraic equations by applying twice the fast expansions with respect to the angular and radial coordinate. In order to determine the unknown coefficients of fast expansions, the pointwise method is utilized. The problem solution derived has explicit analytical form and it's valid for the entire domain including its boundary. The computed profiles of the displacements and stresses in a cross-section of the cutter are provided. The stress field is investigated for various values of opening angle and cusp's radius.

A two‐dimensional model of a recuperative heat exchanger which allows calculation of flows through a heat‐transfer wall and temperature at any point of it has been developed. The results of the calculations are in satisfactory agreement with experimental data.

The Koenigs function arises as the limit of an appropriately normalized sequence of iterates of holomorphic functions. On the other hand it is a solution of a certain functional equation and can be used for the definition of iterates of the original function. A description of the class of Koenigs functions corresponding to probability generating functions embeddable in a one-parameter group of fractional iterates is provided. The results obtained can be regarded as a test for the embeddability of a Galton-Watson process in a homogeneous Markov branching process.

The objects of this study are holomorphic functions, univalent in the upper half-plane, which are complex potentials of infinitely deep flows over a flat bottom with unperturbed flow at infinity. The semigroup of all such functions is determined and its infinitesimal description given.

We generalize Quine's result on the convergence of the extinction probability of the supercritical Galton-Watson process to one if the average number of offsprings tends to one without any restrictions on factorial moments.

Let SM, M > 1, be the class of functionsf(z) which are regular and univalent in the disk ¦z¦ < 1 and satisfy the conditionsf(0) = 0,f'(0) = 1, and ¦f(z)¦ < M. In the present note we will obtain an exact estimate for the argument of the derivative of a function of the class SM.