**8. Conclusions**

A modified method of the parameter continuation (MMPC) is proposed. This method enables simplification of the calculations both at the stage of constructing the model, and also within its continued use due to precise values of the Taylor coefficients for the solution of the degree not exceeding the number of approximation.

The expression to calculate approximations by the MMPC in the general case and with the nonlinearity type of products and squares of the desired functions is presented.

The application of fractional-rational transformation for the polynomial approximation in the form of the 1-D and 2-D PAs used for increasing the degree of convergence and for the analytical continuation of the approximation in the region of its meromorphy was analyzed. It was concluded that such a transformation is justified if it is applied to polynomials which depend on the variable of integration. We used 2-D PAs for the independent variable and for the artificial parameter applying the scheme proposed by V. Vavilov. In this paper it is shown that this transformation provides a satisfactory quality for the approximation behavior and minimizes its error, in spite of the fact that the use of 2-D PAs requires a further theoretical justification.

The estimation of stability using MMPC approximation is also proposed. A study of numerical results was conducted by applying the methods for three model examples which were perturbed with a natural small parameter. It is shown that the application of PAs provides them with sufficient accuracy in the studied area. This paper shows the advantage of approximations which were obtained based on the MMPC.

Calculations of nonlinear deformation and stability of elastic flexible circular cylindrical shell under uniform external pressures and of free oscillations of simply supported stringer shell demonstrated the efficiency and accuracy of the proposed method.

The methodology and results of a holographic experiment with thin low-conical shells having oval ends are presented. They show good agreement with calculation results.
