A fourth-order finite-difference method based on non-uniform mesh for a class of singular two-point boundary value problems (Q5948587)

This paper is concerned with the construction of finite difference schemes on non-uniform meshes for the singular two-point boundary value problem NEWLINE\[NEWLINE( x^{\alpha} y'(x))' = f(x, y(x)), \quad x \in (0,1], \qquad y(0)= A, \quad y(1) = B,NEWLINE\]NEWLINE where \(A\) and \(B\) are given constants, \( \alpha \in [0,1)\) and \( f\) is assumed to be sufficiently smooth. By using a three-point identity relating \( y(x_{k-1}), y(x_k)\) and \( y(x_{k+1})\) considered by \textit{M. M. Chawla} and \textit{C. P. Katti} [Numer. Math. 39, 341-350 (1982; Zbl 0489.65055)] and taking a non uniform grid on \( [0,1]\) defined by \( x_k = ( k / N)^{(1/( 1 - \alpha))}\), \( k=0,1, \ldots ,N\) the authors derive a fourth order scheme for the above problem. Finally two test problems (linear and non linear) are solved with the proposed scheme to check numerically the order of convergence of the method.