The Influence of a Perturbation of a Moving Singular Point on the Structure of an Analytical Approximate Solution of a Class of Third-Order Nonlinear Differential Equations in a Complex Domain

Abstract

The authors prove the theorem of existence and uniqueness of the solution, construct an analytical approximate solution in the complex domain for one class of nonlinear differential equations of the third order, the solution of which are discontinuous functions. The solution of the listed mathematical problems is based on the classical approach. Since the existing methods allow obtaining moving singular points only approximately, it is necessary to investigate the effect of a perturbation of a moving singular point on the structure of an analytical approximate solution in the complex domain. Since the existing methods allow obtaining moving singular points only approximately, it is necessary to investigate the effect of a perturbation of a moving singular point on the structure of an analytical approximate solution in the complex domain. A theorem that allows us to determine a priori estimates of the error of the analytical approximate solution is proved. The study applies the classical approach to estimation and illustrates the application of series with fractional negative powers. The article presents the results of numerical experiment confirming the validity of the obtained theoretical position. The technique of optimization of a priori estimates of analytical approximate solution in the vicinity of perturbed value of moving singular point using a posteriori estimates is presented. The results allow expanding the classes of nonlinear differential equations used as a basis for mathematical models of processes and phenomena in various fields of human activity. In particular, the class of equations under consideration can be applied in the study of wave processes in elastic beams, which is confirmed by theoretical data

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

“…The first option is related to the solvability in quadratures, which is allowed only in special cases [17][18][19][20][21][22][23]. The second option is associated with the author's analytical approximate solution method, successfully tested on a number of classes of nonlinear differential equations [15,[24][25][26][27][28]. Therefore, with the exception of special cases, the existing method for finding moving singular points [28] allows only approximate values to be obtained.…”