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Background. Jet streams of liquids and gases are used in various fields of technology as effective means of controlling the processes of heat and mass transfer, for intensifying and stabilizing various technological processes (for example, the stirring process, the combustion process), as a means of protecting various structures from the effects of thermal and other fields, for applying coatings, etc. Among the practically important objects of research, we also note burners, engine nozzles, jetvortex traces of aircraft. Jet streams are used in many branches of engineering and technology, which makes the problem of their study urgent. The aim of this work is to study the processes of heat and mass transfer in swirling jets. Materials and methods. To solve the problem, the asymptotic method is used, which involves the expansion of hydrodynamic functions (components of the velocity vector and pressure) and temperature, satisfying the system of Navier-Stokes equations for a viscous incompressible fluid, in series with respect to a small parameter. The solution of the obtained in the first approximation system of partial differential equations is sought in a self-similar form, which leads to the study of a system of ordinary differential equations for functions depending on the selfsimilar variable. Results. In a first approximation, the self-similar solution to the problem of the distribution of hydrodynamic (components of the velocity and pressure vector) and thermal (temperature) fields in an axisymmetric tangentially swirling stream of a viscous incompressible fluid is constructed. The material presented in the article supplements the previously known results by calculating the thermal field in the jet. Conclusions. Based on the obtained asymptotic differential equations and selfsimilar solutions of these equations, the fields of velocities, pressure and temperature are constructed in a tangentially swirling stream of a viscous incompressible fluid. It is shown that the longitudinal and tangential (rotational) velocity components influence the distribution of the thermal field in the jet as a first approximation. The method for clarifying the constructed solution is indicated.

A refinement of estimates of the convergence rate obtained earlier in the multidimensional central limit theorem for the sums of vectors generated by the sequences of random variables with mixing is close to optimal. This has been achieved by imposing an additional condition on the characteristic functions of these sums, more accurate estimates of the semi-invariants, and using asymptotic expansions for the characteristic functions of the sums of independent random vectors. The result has been obtained using the summation methods for weakly dependent random variables based on S.N. Bernstein's idea of partition of the sums of weakly dependent random variables into long and short partial sums, as a result of which the long sums are almost independent, and the contribution of short sums to the total distribution is small. To estimate the differences between the sum distributions, we have used the S.M. Sadikova's inequality connecting the difference between the characteristic functions of random vectors with the difference between the corresponding distributions. To estimate the contribution of short sums, Markov and Bernstein's inequalities have been used.

Исследован вопрос об асимптотике фундаментального решения линейного обыкновенного дифференциального уравнения дробного порядка с постоянными коэффициентами при больших значениях спектрального параметра ?.

Evolution equations are derived for the contrasting-structure-type solution of the gen-eralized Kolmogorov–Petrovskii–Piskunov (GKPP) equation with the small parameter with high order derivatives. The GKPP equation is a pseudoparabolic equation with third order derivatives. This equation describes numerous processes in physics, chemistry, biology, for example, magnetic field generation in a turbulent medium and the moving front for the carriers in semiconductors. The profile of the moving internal transitional layer (ITL) is found, and an expression for drift speed of the ITL is derived. An adaptive mesh (AM) algorithm for the numerical solution of the initial-boundary value problem for the GKPP equation is developed and rigorously substantiated. AM algorithm for the special point of the first kind is developed, in which drift speed of the ITL in the first order of the asymptotic expansion turns to zero. Sufficient conditions for ITL transitioning through the special point within finite time are formulated. AM algorithm for the special point of the second kind is developed, in which drift speed of the ITL in the first order formally turns to infinity. Substantiation of the AM method is given based on the method of differential inequalities. Upper and lower solutions are derived. The results of the numerical algorithm are presented.

Subject of Research. Creation of asymptotic expansions for solutions of partial differential equations with small parameter reduces, usually, to consequent solving of the Sturm-Liouville problems chain. To find some term of the series, the non-homogeneous Sturm-Liouville problem with the inhomogeneity depending on the previous term needs to be solved. At the same time, the corresponding homogeneous problem has a non-trivial solution. Hence, the solvability problem occures for the non-homogeneous Sturm-Liouville problem for functions or formal power series. The paper deals with creation of such asymptotic expansions. Method. To prove the necessary condition, we use conventional integration technique of the whole equation and boundary conditions. To prove the sufficient condition, we create an appropriate Cauchy problem (which is always solvable) and analyze its solution. We deal with the general case of power series and make no hypotheses about the series convergence. Main Result. Necessary and sufficient conditions of solvability for the non-homogeneous Sturm-Liouville problem in general case for formal power series are proved in the paper. As a particular case, the result is valid for functions instead of formal power series. Practical Relevance. The result is usable at creation of the solutions for partial differential equation in the form of power series. The result is general and is applicable to particular cases of such solutions, e.g., to asymptotic series or to functions (convergent power series).

This paper is devoted to the construction of a zero-order approximation of the solution of a three-time scale singular perturbed linear-quadratic optimal control problem with the help of the direct scheme method. The algorithm of the method consists in immediate substituting a postulated asymptotic expansion of solution into the problem condition and constructing a family of control problems to define the terms of the asymptotic expansion. Asymptotic approximation of the solution contains regular functions and four boundary ones of exponential type which are determined from the five linearquadratic optimal control problems. It is shown, that the system of equations for a zero-order approximation appeared from control optimality conditions of the initial perturbed problem corresponds to control optimality conditions appeared in respective five optimal control problems constructed for finding zero-order asymptotic approximation with the help of the direct scheme method. An illustrative example is given.

In a rectangular domain the first boundary value problem is considered for a singularly perturbed elliptic equationε 2∆u − ε αA(x, y) ∂u/∂y = F(u, x, y, ε)with a nonlinear on u function F. The complete asymptotic solution expansion uniform in a closed rectangle is constructed for α > 1. If 0 < α < 1, the uniform asymptotic approximation is constructed in zero and first approximations. The features of the asymptotic behavior are noted in the case α = 1 .

The paper deals with a formalism of constructing a zero order approximation of an asymptotic solution for a singularly perturbed linear-quadratic optimal control problem with discontinuous coefficients. This formalism is based on immediate substituting a postulated asymptotic expansion of boundary layer type for the solution into the problem condition and on defining four optimal control problems for finding asymptotics terms. The unique solvability of the problems, the solutions of which form the zero order approximation for the asymptotic solution, is proven. An illustrative example is given.