Magomedyusuf Gasanov

In this paper, we consider a mathematical model of the dynamics of the behavior of a spherically symmetric Rayleigh – Plesset bubble in the van der Waals gas model. The analysis of the model takes into account various isoprocesses without the presence of condensation and a model that takes into account condensation in an isothermal process. In each case, various characteristics are searched for, such as oscillation frequency (linear/small oscillations), damping factor, relaxation time, decrement, and logarithmic decrement. Solutions are found in quadratures for various parameters of the equation. The theoretical results obtained are compared with the results of the numerical solution of the Cauchy problem for various isoprocesses.

This article introduces a mathematical model that utilizes a nonlinear differential equation to study a range of phenomena such as nonlinear wave processes, and beam deflections. Solving this equation is challenging due to the presence of moving singular points. The article addresses two main problems: first, it establishes the existence and uniqueness of the solution of the equation and, second, it provides precise criteria for determining the existence of a moving singular point. Additionally, the article presents estimates of the error in the analytical approximate solution and validates the results through a numerical experiment.

This paper considers a nonlinear fourth-order ordinary differential equation. The study of this class of equations is conducted using an analytical approximation method based on dividing the solution domain into two parts: the region of analyticity and the vicinity of a movable singular point. This work focuses on investigating the equation in the region of analyticity and solving two problems. The first problem is a classical problem in the theory of differential equations: proving the theorem of existence and uniqueness of a solution in the region of analyticity. The structure of the solution in this region takes the form of a power series. To transition from formal series to series converging in a neighborhood of the initial conditions, a modification of the majorant method is used, which is applied in the Cauchy – Kovalevskaya theorem. This method allows determining the domain of validity of the theorem. Within this domain, error estimates for the analytical approximate solution are obtained, enabling the solution to be found with any predefined accuracy. When leaving the domain of the theorem’s validity, analytical continuation is required. To do this, it is necessary to solve the second task of the study: to study the effect of perturbation of the initial data on the structure of the analytical approximate solution.