Explicit numerical methods for solving singular initial value problems for systems of second-order nonlinear ODEs

Abstract

A new approach for numerical solving initial value problems for systems of second-order nonlinear ordinary differential equations with a singularity of the first kind at the start point $x=0$ is proposed. By substitution of the independent variable $x=e^t$, we reduce the original initial value problem on the interval [0, a] to the equivalent one on the interval $(-\infty ,\ln a]$. For solving this initial value problem at the grid node $t_0$ of finite grid $\{ t_{n}\in ( {-\infty , \ln a}], n =0,1,...,N, t_{N} = \ln a\}$, new fourth-order explicit Runge-Kutta-type methods have been constructed. For finding the solution in other nodes of the grid, we can apply any of the standard Runge-Kutta methods or linear multistep ones, using the solution at the point $t_0$, calculated by the constructed in this article methods, as an initial condition. For the proposed approach, a new effective numerical algorithm with a given tolerance has been developed.