Research of a third-order nonlinear differential equation in the vicinity of a moving singular point for a complex plane

2023

Axioms

This paper considers a theoretical substantiation of the influence of a perturbation of a moving singular point on the analytical approximate solution to the Van der Pol equation obtained earlier by the author. A priori estimates of the error of the analytical approximate solution are obtained, which allows the solving of the inverse problem of the theory of error: what should the structure of the analytical approximate solution be in order to obtain a result with a given accuracy? Thanks to a new approach for obtaining a priori evaluations of errors, based on elements of differential calculus, the domain, used to obtain an analytical approximate solution, was substantially expanded. A variant of optimizing a priori estimates using a posteriori estimates is illustrated. The results of a numerical experiment are also presented.

Section: Introductionmentioning

confidence: 99%

Dependence of the Analytical Approximate Solution to the Van der Pol Equation on the Perturbation of a Moving Singular Point in the Complex Domain

2023

Axioms

Exact Criteria for the Existence of a Moving Singular Point in a Complex Domain for a Nonlinear Differential Third-Degree Equation with a Polynomial Seventh-Degree Right-Hand Side

Gasanov

2022

Axioms

Earlier, the authors formulated and proved interval and point criteria for the existence of moving singular points of a third-degree nonlinear differential equation with a polynomial seventh-degree right-hand side for a real domain. For the complex domain, these criteria are associated with specificity of transition to phase spaces. Necessary as well as necessary and sufficient conditions for the existence of moving singular points are obtained. The results of these studies are the basis of the algorithm for obtaining moving singular points of nonlinear differential equations in the complex domain. The results of this paper allow us to expand the classes of the studied nonlinear differential equations with moving peculiarities.

Moving Singular Points and the Van der Pol Equation, as Well as the Uniqueness of Its Solution

2023

Mathematics

The article considers the Van der Pol equation nonlinearity aspect related to a moving singular point. The fact of the existence of moving singular points and the uniqueness of their solution for complex domains have been proved. An answer to the question about the existence of moving singular points in the real domain was obtained. The proof of existence and uniqueness is based on an author’s modification of the technology of the classical Cauchy theorem. A priori estimates of the analytical approximate solution in the vicinity of a moving singular point are obtained. Calculations of a numerical experiment are presented.