Analytical theory of differential equations

• \begin{thebibliography}{9}
• \Bibitem{1} \by Orlov V, Gasanov M. \paper Technology for Obtaining the Approximate Value of Moving Singular Points for a Class of Nonlinear Differential Equations in a Complex Domain. \paperinfo In previous studies, the authors formulated precise criteria for finding moving singular points of one class of nonlinear differential equations with a second degree polynomial right-hand side for a real domain. In this paper, the authors generalize these exact criteria to a complex one by using phase spaces. The proposed technology for obtaining an approximate value of moving singular points is necessary for developing PC programs. This technology has been tested in a manual version based on a numerical experiment. \jour Mathematics \yr 2022 \vol 10 \issue 21 \pages 7
• \Bibitem{2} \by Orlov V, Gasanov M. \paper Existence and Uniqueness Theorem for a Solution to a Class of a Third-Order Nonlinear Differential Equation in the Domain of Analyticity. \paperinfo The paper considers the specifics of nonlinear differential equations that have applications in different areas. Earlier, the authors proved the existence and uniqueness theorem for a solution to a class of non-linear differential equations in a neighborhood of a moving singular point. In this paper, we consider the first problem of studying a third-order nonlinear differential equation in the domain of analyticity. An analytical approximate solution is built, taking into account the solution search area. A priori estimates of the analytical approximate solution are obtained, and the technology of their optimization using a posteriori ones is illustrated. The result of a numerical experiment is presented. The presented results allow to expand the class of nonlinear differential equations for describing various phenomena and processes. \jour Axioms \yr 2022 \vol 11 \issue 5 \pages 6
• \Bibitem{3} \by Orlov V.N., Gasanov M.V. \paper 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. \paperinfo 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. \jour Herald of the Bauman Moscow State Technical University, Series Natural Scie

• Moscow State University of Civil Engineering