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

摘要

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 = e(t), we reduce the original initial value problem to the one on the interval (-infinity, lna]. On some finite irregular grid {t(n) is an element of (-infinity, lna], n = 0,1,..., N, t(N) = lna} Taylor series and Runge-Kutta methods for this problem have been developed. For finding of an approximate solution at the grid node t(0), 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. (C) 2019 IMACS. Published by Elsevier B.V. All rights reserved.