Application of TAGE iterative algorithms to an efficient third order arithmetic average variable mesh discretization for two-point non-linear boundary value problems

In this article, we aim to discuss the application of two parameter alternating group explicit (TAGE) iterative method to an efficient third order numerical method for two point non-linear boundary value problems on a variable mesh. There is no need to use any special technique to handle singular problems. For the non-linear problems, the proposed method leads to a non-linear system whose Jacobian is tridiagonal. We also discuss Newton–TAGE algorithm for general non-linear differential equations. The proposed TAGE and Newton–TAGE iterative methods are suitable for use on parallel computers. The convergence theory is discussed in details. Numerical results are given to justify the utility of the proposed iterative methods.