A new approach to constructing of explicit one-step methods of high order for singular initial value problems for nonlinear ordinary differential equations

A new approach to construction of one-step numerical methods of high order for the initial value problems on the interval [0,a] with a singularity of the first kind in the point x=0 is proposed. Using the substitution of the independent variable x=et, we reduce the original initial value problem to the one on the interval (−∞,ln⁡a]. On some finite irregular grid {tn∈(−∞,ln⁡a], n=0,1,...,N,tN=ln⁡a} Taylor series and Runge-Kutta methods for this problem have been developed. For finding of an approximate solution at the grid node t0, new one-step methods have been constructed. For finding of the solution at other grid nodes, the standard one-step methods have been used. An algorithm for the automatic generation of a grid which guarantees the prescribed accuracy is presented. The effectiveness of presented approach is illustrated by a set of numerical examples. The applicability of the constructed method to systems of singular differential equations is shown.