A new coupled reduced alternating group explicit method for non-linear singular two point boundary value problems on a variable mesh

In this paper, we discuss a new coupled reduced alternating group explicit (CRAGE) and Newton-CRAGE iteration methods to solve non-linear singular two point boundary value problems $u''=f(r,u,u')$, $0<r<1$, subject to given natural boundary conditions $u(0)=A_1$, $u(1)=A_2$, where $A_1$ and $A_2$ are finite constants, along with a third order numerical method on a geometric mesh. The proposed method is applicable to singular and non-singular problems. We discuss the convergence of the CRAGE iteration method in detail. The results obtained from the proposed CRAGE iteration method are compared with the results of the corresponding two parameter alternating group explicit (TAGE) iteration methods to demonstrate computationally the efficiency of the proposed method.