An efficient and computational effective method for second order problems

An efficient and computational effective algorithm is introduced, for the first time in the literature, in the present paper. The main properties of the scheme are: (1) the algorithm is a two-step scheme, (2) the algorithm is symmetric one, (3) it is a hight algebraic order scheme (i.e of eight algebraic order), (4) it is a three-stages algorithm, (5) the first layer of the new method is based on an approximation to the point (x_{n-1}), (6) the scheme has vanished phase-lag and its first, second and third derivatives, (7) the new proposed algorithm has an interval of periodicity equal to (left( 0, 9.8 right) ). For the present new scheme we study: (1) its construction, (2) its error analysis (3) its stability analysis. Finally, the investigation of the effectiveness of the new algorithm leads to its application to systems of differential equations arising from the Schrödinger equation.