Myroslava Vovk

A simple cell population growth model is proposed, where cells are assumed to have a physiological structure (e.g., a model describing cancer cell maturation, where cells are structured by maturation stage, size, or mass). The main question is whether we can guarantee, using the death rate as a control mechanism, that the total number of cells or the total cell biomass has prescribed dynamics, which may be applied to modeling the effect of chemotherapeutic agents on malignant cells. Such types of models are usually described by partial differential equations (PDE). The population dynamics are modeled by an inverse problem for PDE in our paper. The main idea is to reduce this model to a simplified integral equation that can be more easily studied by various analytical and numerical methods. Our results were obtained using the characteristics method.

This paper analyzes the influence of kinetic and physical-mechanical parameters of systems on the characteristics of dynamic processes in moving one-dimensional nonlinear-elastic systems. Improved convenient calculation formulas have been derived that describe the laws of changing the amplitude-frequency characteristics of systems for both a non-resonant case and a resonant one. An important issue of studying the influence of the speed of movement of elements of mechanisms on the oscillations of one-dimensional nonlinear-elastic systems has not been considered in detail until now in the scientific literature. This issue relates to the vibrations of shafts in gears, pipe strings when drilling oil and gas wells, the oscillations of turbine blades and rotating turbine discs, the longitudinal vibrations of the beam as an element of structures. The main reason for this in the analytical study of dynamic processes were the shortcomings of the mathematical apparatus for solving the corresponding nonlinear differential equations that describe the laws of motion of those systems. It was found that in the case of longitudinal oscillations in the moving beam with an increase in the longitudinal speed of the medium to 10 m/s, the amplitude of the oscillation also increases by 13.5 %. However, when the longitudinal velocity of the beam is 5 m/s, the amplitude will increase by only 3 %. It is established that with the growth of the amplitude, the frequency of longitudinal oscillations decreases sharply, and if the system moves at a higher speed, for example, 20 m/s, it reduces the frequency of oscillation by about 13 %. The results reported here make it possible to assess the effect of kinetic and physical-mechanical parameters on the frequency and amplitude of oscillations. The research that involved the asymptotic method makes it possible to predict resonant phenomena and obtain engineering solutions to improve the efficiency of technological equipment.