Exact boundaries of the scope of the approximate solution for a certain class of nonlinear differential equations in the complex domain

Abstract

Introduction. A nonlinear differential equation of the third order with a polynomial right part of the seventh degree describing wave processes in beams is considered. When studying this class of equations, generally unsolvable in quadrature, the authors use a method allowing to obtain an analytical approximate solution. This kind of research is based on the solution of several mathematical problems. The paper generalizes previously obtained results of the study of one class of nonlinear differential equations with moving singularities to the complex domain.Materials and methods. The problem of finding the exact limits of application of the approximate solution of a nonlinear differential equation in the neighborhood of a moving singular point is considered. Previously, the boundaries of the application area of the approximate solution were determined on the basis of the theorem of unique existence, but further, the obtained area was reduced due to the perturbation of the moving singular point. Applying elements of differential calculus to estimate the error of the solution, in this work, it is possible to expand the scope of the approximate solution and bring it closer to the initially obtained area.Results. The exact boundaries of the area of application of the approximate solution are obtained. Theoretical provisions are confirmed by numerical calculations, which characterizes their reliability. Two numerical experiments are considered. In the first one, a point falling under the previous and new area obtained in this paper is taken. In the second, the point falling only under the new theorem is taken.Conclusions. The author’s approach of the method of analytical approximate solution is further developed on the example of the considered class of nonlinear equations. The paper summarizes the results obtained earlier in the research of the exact limits of application of the approximate solution of the considered class of equations in the vicinity of a moving singular point in the complex domain. The presented studies are confirmed by numerical experiments.

Keywords

• nonlinearity aspect
• perturbation of a moving singular point
• exact boundaries
• a priori estimation
• cauchy problem