Mixing processes are found in the chemical, pharmaceutical and food industries. Fluid mixing is one of the fundamental scientific problems associated with modern concepts of regular and chaotic dynamics. The paper considers the problem of mathematical modeling of the quasi-stationary process of mixing a viscous mixture. This problem consists of two sub-problems: determination of the velocity field in the flow region (Eulerian formalism) and investigation of the trajectories of individual fluid particles (Lagrange formalism). To solve the first subproblem, it is proposed to jointly use the principle of superposition, the structural method (method of R-functions) and the Ritz variational method. The methods of nonlinear dynamics and qualitative theory of differential equations are used to solve the second subproblem. A plane quasi-steady flow is considered in a rectangular region and it is assumed that the side walls are at rest, and the upper and lower walls move alternately according to the given laws. According to the method of R-functions, the structures of the solutions were built and the use of the Ritz variational method for the approximation of the uncertain components of the structures was justified. The operation of the proposed method is illustrated by the results of a computational experiment, which was conducted for different modes of wall motion. The practical interest of the considered regimes is due to the fact that they lead to the emergence of chaotic behavior when mixing occurs most efficiently. Using the methods of nonlinear dynamics, the location of periodic (hyperbolic and elliptical) points was investigated and the Poincaré section was constructed. Further research with the help of the method proposed in the work can be related to the consideration of flows in more geometrically complex regions and more complex mixing regimes, as well as in the application to the calculation of industrial problems.

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The first boundary value problem for a one-dimensional nonlinear heat equation is considered, where the heat conductivity coefficient and the power function of heat sources have a power-law dependence on temperature. For a numerical analysis of this problem, it is proposed to use the method of two-sided approximations based on the method of Green’s functions. After replacing the unknown function, the boundary value problem is reduced to the Hammerstein integral equation, which is considered as a nonlinear operator equation in a semi-ordered Banach space. The conditions for the existence of a single positive solution of the problem and the conditions for two-sided convergence of successive approximations to it are obtained. The developed method is programmatically implemented and researched in solving test problems. The results of the computational experiment are illustrated by graphical and tabular information. The conducted experiments confirmed the efficiency and effectiveness of the developed method that allowed recommending its practical use for solving problems of system analysis and mathematical modeling of nonlinear processes.

Context. The question of constructing a method of two-sided approximations for finding a positive solution of the Dirichlet problem for a semilinear elliptic equation based on the use of the Green’s functions method is considered. The object of research is the first boundary value problem (the Dirichlet problem) for a second-order semilinear elliptic equation.  Objective. The purpose of the research is to develop a method of two-sided approximations for solving the Dirichlet problem for second-order semilinear elliptic equations based on the use of the Green’s functions method and to study its work in solving test problems. Method. Using the Green’s functions method, the initial first boundary value problem for a semilinear elliptic equation is replaced by the equivalent Hammerstein integral equation. The integral equation is represented in the form of a nonlinear operator equation with a heterotone operator and is considered in the space of continuous functions, which is semi-ordered using the cone of nonnegative functions. As a solution (generalized) of the boundary value problem, it was taken the solution of the equivalent integral equation. For a heterotone operator, a strongly invariant cone segment is found, the ends of which are the initial approximations for two iteration sequences. The first of these iterative sequences is monotonically increasing and approximates the desired solution to the boundary value problem from below, and the second is monotonically decreasing and approximates it from above. Conditions for the existence of a unique positive solution of the considered Dirichlet problem and two-sided convergence of successive approximations to it are given. General guidelines for constructing a strongly invariant cone segment are also given. The method developed has a simple computational implementation and a posteriori error estimate that is convenient for use in practice.  Results. The method developed was programmed and studied when solving test problems. The results of the computational experiment are illustrated with graphical and tabular informations. Conclusions. The experiments carried out have confirmed the efficiency and effectiveness of the developed method and make it possible to recommend it for practical use in solving problems of mathematical modeling of nonlinear processes. Prospects for further research may consist the development of two-sided methods for solving problems for systems of partial differential equations, partial differential equations of higher orders and nonstationary multidimensional problems, using semi-discrete methods (for example, the Rothe’s method of lines).