A new two stages tenth algebraic order symmetric six-step method with vanished phase-lag and its first and second derivatives for the solution of the radial Schrödinger equation and related problems

The presentation, development and analysis of a new two-stages tenth algebraic order symmetric six-step method is introduced, for the first time in the literature, in this paper. More specifically, we present the development of the new method (requesting the highest algebraic order and the elimination of the phase-lag and its first and second derivatives), the analysis (error analysis and stability and interval of periodicity analysis) and the evaluation of the new developed method comparing its efficiency with the efficiency of well known methods and very recently produced methods in the literature on the approximate solution of the resonance problem of the one dimensional (or radial) Schrödinger equation. From the developments achieved and the results presented, we prove that the new obtained method is most more effective than other well known or recently developed methods of the literature.     

New two stages high order symmetric six-step method with vanished phase–lag and its first,second and third derivatives for the numerical solution of the Schrödinger equation

In the present paper, we obtain and analyze, for the first time in the literature, a new two-stages high order symmetric six-step method. The specific characteristics of the new proposed method are the highest possible algebraic order, the elimination of the phase–lag and its first, second and third derivatives. Additionally, for the new method we give the analysis of the method (both error and stability and interval of periodicity analysis) and the comparison of the effectiveness of the new developed method with the effectiveness of well known methods and very recently produced methods in the literature. The comparison is based on the numerical solution of the Schrödinger equation. The theoretical achievements and the numerical results show the effectiveness of the new developed method in comparison with other well known or recently developed numerical methods.     

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.     

An explicit four-step method with vanished phase-lag and its first and second derivatives

In this paper we will develop an explicit fourth algebraic order four-step method with phase-lag and its first and second derivatives vanished. The comparative error and the stability analysis of the above mentioned paper is also presented. The new obtained method is applied on the resonance problem of the Schrödinger equationIn order in order to examine its efficiency. The theoretical and the computational results shown that the new obtained method is more efficient than other well known methods for the numerical solution of the Schrödinger equation and related initial-value or boundary-value problems with periodic and/or oscillating solutions.     

A new explicit hybrid four-step method with vanished phase-lag and its derivatives

A hybrid explicit sixth algebraic order four-step method with phase-lag and its first, second and third derivatives vanished is obtained in this paper. We present the development of the new method, its comparative error analysis and its stability analysis. The resonance problem of the Schrödinger equation, is used in order to study the efficiency of the new developed method. After the presentation of the theoretical and the computational results it is easy to see that the new constructed method is more efficient than other well known methods for the approximate solution of the Schrödinger equation and related initial-value or boundary-value problems with periodic and/or oscillating solutions.