In this article we continue our previously conducted research on the construction of a mathematical model for obtaining medicinal nanoforms using cryochemical synthesis methods. In connection with the need to increase their therapeutic effectiveness, it is necessary to take into account the particles size, structure and shape. Thus, to reduce side effects and a toxicity we can reduce the particle size of drugs to nanoscales. It allows us to obtain highly effective drugs and to use its smaller doses. One of the most powerful new methods for obtaining nanoforms of drugs is its cryochemical synthesis. This method is a leading-edge process for producing drugs in nanoparticle form. The procedure involves vaporizing the raw drug material in a vacuum and then channeling this vapor into a stream of gas. The gas stream, now carrying the drug molecules, is directed onto an extremely cold surface where the molecules instantly condense and form nanoscale structures. The first stage of mathematical modeling of cryochemical synthesis processes was the calculation of the temperature field in the carrier gas flow interacting with the cooled surface. At this stage, taking into account the previously obtained results, we study the change in pressure and supersaturation, determining the coordinate of the formation of the first embryo and its critical size, which will allow us to describe the process of embryo growth at the next stage of constructing a mathematical model, and determine their molecular mass as they reach the cooling surface.

We discuss the problem of asymptotic behavior of all positive non-extensible (so-called “blow-up”) solutions to higher-order ordinary differential equations with a power-law nonlinearity. We are interested in typicality and a-typicality of the power-law asymptotic behavior of its singular solutions.

We consider an extremum problem associated with a mathematical model of the temperature control. It is based on a one-dimensional non-self-adjoint parabolic equation of general form. Determining the optimal control as a function minimizing the weighted quadratic functional, we prove the existence of a solution to the problem of the double minimum by control and weight functions. We also obtain upper estimates for the norm of the control function in terms of the value of the functional. These estimates are used to prove the existence of the minimizing function for unbounded sets of control functions.

We consider Riccati's equation on the real axis with continuous coefficients and non-negative discriminant of the right-hand side. We study the extensibility of its solutions to unbounded intervals. We obtain asymptotic formulae for its solutions in their dependence on the initial values and the properties of the functions representing roots of the right-hand side of the equation. We obtain results on the asymptotical behaviour of solutions defined near $\pm\infty$. We study the structure of the set of bounded solutions in the case when the roots of the right-hand side of the equation are $C^1$-functions which are different on the whole of their domain and tend monotonically to some limits as $x\to\pm\infty$. We extend, improve, or refine some well-known results. Bibliography: 47 titles.

The work is aimed at creating a mathematical model of cryochemical synthesis of nanoforms of pharmaceutical substances. The therapeutic efficacy of pharmaceutical substances largely depends on the size and morphology of the particles. Reducing the particle size of pharmaceutical substances to nanoscale makes it possible to obtain highly effective drugs, which makes it possible to use smaller doses of drugs and, thus, reduce side effects and toxicity. Cryochemical synthesis is one of the most powerful methods for obtaining nanoforms of medicament. The method, which is completely new, is based on sublimation or evaporation of the initial pharmaceutical substance under high vacuum conditions and the introduction of the resulting vapors into an inert gas stream, followed by low-temperature condensation of the flow of molecules of the substance from the gas phase on the cooled surface. The first step in the mathematical modeling of cryochemical synthesis processes is the calculation of the temperature field in the carrier gas flow interacting with the cooled surface. For this purpose, a stationary equation of thermal conductivity with mass transfer is used for the one-dimensional case. We prove existence and uniqueness theorems of the solution. Analytical solutions of the equation for Dirichlet, Neumann and Robin boundary conditions are found.

This paper is devoted to the problem of asymptotic equivalence of n-th order differential equations with exponentially equivalent right-hand sides. With the help of the obtained result asymptotic behavior of solutions to perturbed differential equations is described.

The existence of unbounded solutions and their asymptotic behavior is studied for higher order differential equations considered as perturbations of certain linear differential equations. In particular, the existence of solutions with polynomial-like or noninteger power-law asymptotic behavior is proved. These results give a relation between solutions to nonlinear and corresponding linear equations, which can be interpreted, roughly speaking, as an asymptotic proximity between the linear case and the nonlinear one. Our approach is based on the induction method, an iterative process and suitable estimates for solutions to the linear equation.

In this paper we consider a control problem with pointwise observation for a one-dimensional parabolic equation which arises in a mathematical model of climate control in industrial greenhouses. We study a general equation with variable diffusion coefficient, convection coefficient, and depletion potential. For the extremum problem of minimizing an integral weighted quadratic cost functional, we establish the existence and uniqueness of a minimizing function. We also study exact controllability and dense controllability of the problem. Necessary conditions for an extremum are obtained, and qualitative properties of the minimizing function are studied.

For a minimization problem with pointwise observation governed by a one-dimensional parabolic equation with a free convection term and a depletion potential we consider a weight quadratic cost functional. We state a problem of double minimization to our functional obtain by finding first minimum of the functional in some class of control functions and iterated minimum by weight. For such problem we prove the existence of a pair of minimizers. To prove these results we establish the special form of maximum principle. Also we use some results on *-weakly closed sets and weak convergence of functionals.

We consider a control problem associated with a mathematical model of temperature control in industrial greenhouses. It is based on a one-dimensional non-self-adjoint parabolic equation with variable coefficients. Defining the optimal control as the minimizer quadratic functional with point observation, we prove the existence of the minimizer and obtain necessary conditions.

Асташова И.В., Барабанов Е.А., Боровских А.В., Бутузов В.Ф., Быков В.В., Ветохин А.Н., Глызин С.Д., Горицкий А.Ю., Денисова Н.В., Изобов Н.А., Ильин А.В., Ильяшенко Ю.С., Капустина Т.О., Кигурадзе И.Т., Козлов В.В., Колесов А.Ю., Коньков А.А., Ломов И.С., Моисеев Е.И., Палин В.В. и др.//Дифференциальные уравнения, 2021

For the minimization problem with pointwise observation governed by a one-dimensional parabolic equation with a free convection term and a depletion potential, we formulate a result on the existence and uniqueness of a minimizer from a prescribed set. We use a weight quadratic cost functional showing the temperature deviation. We obtain estimates for the norm of control functions in terms of the value of the quality functional in different functional spaces. It gives us a possibility to estimate the required internal energy of the system. To prove these results we establish the positivity principle.

The paper studies the asymptotic behaviour of solutions to a second-order non-linear discrete equation of Emden–Fowler type \begin{document}$ \Delta^2 u(k) \pm k^\alpha u^m(k) = 0 $\end{document} where \begin{document}$ u\colon \{k_0, k_0+1, \dots\}\to \mathbb{R} $\end{document} is an unknown solution, \begin{document}$ \Delta^2 u(k) $\end{document} is its second-order forward difference, \begin{document}$ k_0 $\end{document} is a fixed integer and \begin{document}$ \alpha $\end{document}, \begin{document}$ m $\end{document} are real numbers, \begin{document}$ m\not = 0, 1 $\end{document}.

We consider a temperature control problem based on the heat equation with a convective term and a quadratic quality functional. We examine the structure of the set of attainable functions and establish controllability of the problem on various sets of admissible controls.