Finite difference methods for certain singular two-point boundary value problems (Q1919496)

The authors consider the boundary value problem (BVP) \[ Lu \equiv \sum^2_{k=0} (-1)^k {d^k \over dx^k} \left[ P_k(x)\;{d^k u\over dx^k}\right] = F(x,u),\quad x \in (0,1),\tag{1} \] \[ \sum^2_{k=0} A_{lk} \left(d^ku \over dx^k\right)_{x\to 0}+\sum^3_{k=0} B_{lk} \left( {{d^k u} \over {dx^k}} \right)_{x=1} = g_l, \quad l = 1,\dots, n, \] where one or more of the coefficients \(p_k(x)\) may be infinite at \(x =0\) and the singularities are of the first kind. At first, the BVP (1) is linearized by Newton's method. For each linear BVP the authors apply Gustafsson's method of series expansion at a small neighborhood of the point \(x = 0\) and a difference method in the rest of the interval. The consistency and stability of the method are proved and an error estimate is presented. In the conclusion numerical experiments and some comments are given.