A sixth order accurate AGE iterative method for non-linear singular two point boundary value problems

Abstract

We report alternating group explicit (AGE) iteration method to solve the tri-diagonal linear system derived from a new finite difference discretization of sixth order accuracy of the two point singular boundary value problem ${\mathit{u}}^{″}+\frac{\mathit{\alpha }}{\mathit{r}}{\mathit{u}}^{\prime }=\mathit{f}\mathrm{(}\mathit{r}\mathrm{)}$ , $0\mathrm{<}\mathit{r}\mathrm{<}1$ , $\mathit{\alpha }=1$ and 2 subject to boundary conditions $\mathit{u}\mathrm{(}0\mathrm{)}=\mathit{A}$ , $\mathit{u}\mathrm{(}1\mathrm{)}=\mathit{B}$ , where A and B are finite constants. We also discuss Newton-AGE iteration method for the sixth order numerical solution of two point non-linear boundary value problem. The proof for the convergence of the AGE iteration method when the coefficient matrix is real and unsymmetric is discussed. Numerical results are presented to illustrate the proposed iterative methods.