J Math Chem (2017) 55:1649–1668
An efﬁcient and computational effective method
for second order problems
· T. E. Simos
Received: 21 April 2017 / Accepted: 27 April 2017 / Published online: 16 May 2017
© Springer International Publishing Switzerland 2017
Abstract 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
, (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
. 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.
Highly Cited Researcher (http://isihighlycited.com/), Active Member of the European Academy of
Sciences and Arts. Active Member of the European Academy of Sciences. Corresponding Member of
European Academy of Arts, Sciences and Humanities.
Authors whose names appear on the submission have contributed sufﬁciently to the scientiﬁc work and
therefore share collective responsibility and accountability for the results.
Electronic supplementary material The online version of this article (doi:10.1007/s10910-017-0753-9)
contains supplementary material, which is available to authorized users.
T. E. Simos
School of Information Engineering, Chang’an University, Xi’an 710064,
People’s Republic of China
Laboratory of Computational Sciences, Department of Informatics and Telecommunications,
Faculty of Economy, Management and Informatics, University of Peloponnese, 221 00 Tripoli,
10 Konitsis Street, Amﬁthea - Paleon Faliron, 175 64 Athens, Greece