New explicit high-order one-step methods for singular initial value problems

New explicit one-step high-order numerical methods (Taylor series and Runge-Kutta methods) for singular initial value problems for second-order nonlinear ordinary differential equations are constructed. These methods allow to calculate an approximate solution in the neighborhood of a singular point x=0 with a high-order of accuracy. By the substitution of the independent variable, we transform the original singular initial value problem into a nonsingular one at an infinite interval. To solve this nonsingular problem, standard numerical methods with initial conditions obtained near a singular point can be used. The effectiveness of the presented approach is confirmed by numerical experiments.