Explicit high-order one-step methods are constructed for singular initial value problems (IVP) of the following type: \begin{gather*} \frac{1}{x^\lambda} \frac{d}{dx} \left (x^\lambda k(x) \frac{du}{dx} \right) = -f(x,u),\quad x \in(0,a],\\ \quad u(0)=u_0, \; \frac{du(0)}{dx} = 0, \end{gather*} where \(0 < c \le k(x),\; f (x, u)\) are given functions, and \(\lambda = 1, 2\).
The singular IVP is reduced, using the substitution \(x = e^t\), to an IVP on an infinite interval \((-\infty, \ln a]\). Since all derivatives of the function \(w(x) = k(x) \frac{du}{dx}\) are singular at the point \(x = 0\), their limits are calculated as \(x\rightarrow 0\) and the Taylor expansion of the solution of the singular IVP is obtained around \(x = 0\).
Based on this transformation, new three-stage Runge-Kutta methods, which are consistent of order \(4\) with the obtained Taylor expansion, are developed . The approximate solution at the node \(t_0\) is found on some finite irregular grid \({t_n \in (-\infty, \ln a], n = 0, 1, \dots, N,\; t_N =\ln a}\). Using standard fourth-order Runge-Kutta methods, the solution at other grid nodes is calculated, whereas the classical fourth-order of convergence is retained. Because the methods are explicit, unlike the implicit Runge-Kutta methods, they do not require additional iterative procedures on each step. The effectiveness and convergence of the constructed method are illustrated for three different singular IVPs.
The applicability of the method to a coupled system of two singular differential equations is proved by numerical experiments.