Information on the thesis

The methods of Teylor series as well as Runge-Kutta method of the four order
accuracy have been developed for the numerical solving the singular initial value prob-
lems for the nonlinear ordinary differential equations of the order 2 in this dissertation
thesis. It has been constructed and justified the exact three-point difference scheme
(ETDS) for solving the boundary problems of nonlinear stationary equations in the
cylindrical and spherical coordinate systems on the non-uniform grid. An effective
algorithmic realization of ETDS has been implemented via the truncated three-point
difference schemes (TDS) of the rank ¯m = 2[(m + 1)/2], where m is a natural num-
ber and [ · ] is an integer part. The enough conditions of existence and uniqueness of
the truncated TDS solution have been obtained. It has been proved that the TDS of
rank ¯m has the accuracy order of ¯m both for the solution of u(x) and for the flow of
k(x)du/dx. The estimation of iteration method of serial approximation for the solv-
ing the nonlinear differential schemes has been obtained and the convergence of this
method has been proved as well. The theoretical conclusions have been verified by
the results of numerical experiments.
Key words: singular ordinary differential equation, initial value problem, boundary
value problem, three-point difference scheme, iterative method, order of accuracy .