Research of one class of nonlinear differential equations of third order for mathematical modelling the complex structures

Abstract

The study of many processes and phenomena in engineering, mechanics, biology, physics, medicine, the vitality of buildings and structures leads to mathematical modeling and the construction of a mathematical model Such models facilitate a rigorous justification of ongoing research and is the only objective option obtain reliable predictions based on analytical methods.Typically, the mathematical model is one of the aspects of differential equations The basis for more accurate models are nonlinear differential equations In the presence of mobile singular points in the General case it is a class of equations not solvable in quadratures. This circumstance is an obstacle to researchers when constructing mathematical models. This is the importance of the development of the mathematical theory of their decisions.This paper presents technology and results of solution of some problems of approximate analytical method of solution of nonlinear differential equations with movable singularities:1 In a complex domain for one class of nonlinear differential equations of third order, not solvable by quadratures, made evidence theorem of existence of solutions in the field of holomorphes.2 A constructive proof of existence theorems, in contrast to existing classical variants, allows to construct analytical approximate solution of the considered class of nonlinear differential equations, which are used in mathematical modeling of the complex structures.3 The influence of the perturbation of initial conditions on the obtained analytical approximate solution has been researched.The obtained results are accompanied by the computational experiments proving their adequacy Solved by the authors of the problem allows you to create mathematical models of complex structures and phenomena. The obtained results allow to carry out the analytical continuation of the approximate solution with a given accuracy In which case, a posteriori error allows obtaining significantly more accurate a priori error.