A higher-order method for nonlinear singular two-point boundary value problems (Q1608177)

The authors propose a finite difference method to approximate the unique solution \(y\) of the nonlinear singular two-point boundary value problem \[ -\frac{1}{w(x)} (p(x) y'(x))'=g(x,y),\quad x\in (0,1), \] with mixed conditions \[ (py')(0^+)=y(1)=0, \] under some hypotheses on the data functions \(w\), \(p\) and \(g\) which guarantee that the problem is well-posed. Standard numerical methods exhibit a loss of accuracy or even a lack of convergence when applied to singular problems. There have been many numerical methods proposed for solving especial cases of this problem, namely \(w(x)=p(x)=x^\alpha\), and \(w(x)=p(x)\). Unlike the previous treatment, the method here is designed to work for general \(p\) and \(w\) which are not even required to be smooth. When \(w(x)=p(x)=x^\alpha\) with \(\alpha\geq 1\), the scheme coincides with that of \textit{M. M. Chawla, R. Subramanian} and \textit{H. L. Sathi} [BIT 28, No. 1, 88-97 (1988; Zbl 0636.65079)], and consequently is of fourth order. The convergence of the method for general \(w\) and \(p\) is proved and some numerical examples are presented in the final part of the paper.