An Approximate Analytical Solution for a Class of Fourth-Order Nonlinear Differential Equations in the Domain of Analyticity

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.