The advection-diffusion problem in a thin curvilinear channel is considered. The multiscale finite element method is applied to solving the formulated model problem. It is shown that this method is efficient in the case of sufficiently large Peclet numbers. Numerical examples are presented and analysed.

In the paper, the possibility of the Appell hypergeometric function F4(1,2;2,2;z1,z2) approximation by a branched continued fraction of a special form is analysed. The correspondence of the constructed branched continued fraction to the Appell hypergeometric function F4 is proved. The convergence of the obtained branched continued fraction in some polycircular domain of two-dimensional complex space is established, and numerical experiments are carried out. The results of the calculations confirmed the efficiency of approximating the Appell hypergeometric function F4(1,2;2,2;z1,z2) by a branched continued fraction of special form and illustrated the hypothesis of the existence of a wider domain of convergence of the obtained expansion.

In the paper, using some recurrent relations, the expansion of the hypergeometric Appel function F4 (1,2;2,2; z1, z2 ) into a branched continued fraction of special form is constructed. Explicit formulas for the coefficients of constructed development are obtained. The structure of the obtained branched continued fraction is investigated. The values of the suitable fractions and the corresponding partial sums of the hypergeometric series at different points of the two-dimensional complex space are calculated. A comparative analysis of the obtained values is carried out, the results of which confirm the efficiency of using branched continued fractions to calculate the values of the hypergeometric function F4 (1,2;2,2; z1, z2 ) in space C2.

The paper deals with the mathematical model development of the kinematical behavior of the flexible transmission branch exemplified. A typical example of transmission with discrete-flexible connection can be considered as the movement of the drive elements of the chain. The use of chain gears as a drive for a wide range of technological machines with high requirements in order to ensure a certain law of motion of the executive bodies is the task of studying changes in its kinetic characteristics during operation. It is established that random deviations of the chain step from the nominal are the result of manufacturing inaccuracy elements of transmission and wear during operation. The mathematical model of motion gives an idea of the real interpretation of the kinematics of the chain transmission taking into account the uneven dimensions of individual links. The model makes it possible to present the components of deviations of the transmission movement from the given in two groups: deviations created by the accumulated error of the chain section, and movement. Calculations based on the developed mathematical model show that when the hinges of the sleeve-roller chain are worn by 2.5%, the value of the coefficient of non-uniformity of the transmission increases threefold.