On extension of the domain for analytical approximate solution of one class of nonlinear differential equations of the second order in a complex domain

In previous research the authors have implemented the investigation of one class of nonlinear differential equations of the second order in the neighborhood of variable exceptional point. The authors have proven the following: the existence of variable exceptional point, theorem of the existence and uniqueness of solution in the neighborhood of variable exceptional point. The analytical approximated solution in the neighborhood of variable exceptional point was built. The authors researched the influence of disturbance of variable exceptional point on an approximated solution. The results obtained for the real domain have been extended to the complex domain $|z|<|\tilde z^*|\leqslant |z^*|$, where $z^*$ is precise value of variable exceptional point, $\tilde z^*$ is approximate value of variable exceptional point. In the present paper, the authors have carried out the investigation of analytical approximate solution of the influence of disturbance of variable exceptional point in the domain $|z|>|\tilde z^*|\geqslant |z^*|$, giving special attention to change of direction of movement along the beam towards the origin of coordinates of a complex domain. These researches are actual due to the variable exceptional point pattern (even fractional degree of critical pole). The received results are accompanied by the numerical experiment and complete the investigation of analytical approximated solution of the considered class of nonlinear differential equations in the neighborhood of variable exceptional point depending on the direction of movement along the beam in a complex domain.

movable singular point, nonlinear differential equation, analytical approximate solution, neighborhood of variable exceptional point, complex domain, a posteriori estimate