New FD scheme with vanished phase-lag and its derivatives up to order six for problems in chemistry
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ORIGINAL PAPER
New FD scheme with vanished phase-lag and its derivatives up to order six for problems in chemistry Maxim A. Medvedev1,2 • T. E. Simos3,4,5,6,7 Received: 13 August 2020 / Accepted: 4 September 2020 Ó Springer Nature Switzerland AG 2020
Abstract A new FD (Finite Difference) method with zero phase-lag and its derivatives up to order six is proposed for the approximate solution of problems in Chemistry. Keywords Phase-lag Derivative of the phase-lag Initial value problems Oscillating solution Symmetric Hybrid Multistep Schro¨dinger equation
Mathematics Subject Classification 65L05
1 Introduction 1.1 The problem We discuss the solution of the following Systems of Differential Equations: 2 N X d li ðli þ 1Þ 2 þ k POTENT ¼ POTENTim dmj d ii ij i x2 dx2 m¼1
ð1Þ
where 1 i N and m 6¼ i. (3) ! BVP (¼ Boundary value problem) ! boundary conditions: Highly Cited Researcher (2001–2013 List, 2017 List and 2018 List), 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. Electronic supplementary material The online version of this article (https://doi.org/10.1007/s10910020-01168-5) contains supplementary material, which is available to authorized users. & T. E. Simos [email protected] Extended author information available on the last page of the article
123
Journal of Mathematical Chemistry
dij ¼0 at x ¼ 0
ð2Þ
1=2 ki dij ki xjli ðki xÞhij þ Kij ki x nli ðki xÞ kj
ð3Þ
where jl ðxÞ and nl ðxÞ ! spherical Bessel and Neumann functions respectively (see [62]). Categories of problems (see [62]): – Open Channels – Close Channels Asymptotic form (4) of (3) for the open channels problem (see [62]) !: d N þ TW0 : where: Wij0 ¼
1=2 ki Kij kj
Nij ¼ki xjli ðki xÞhij Tij ¼ki xnli ðki xÞhij More details see in [67, 143–151] and references therein. Areas of application of the problems of the form (1) can also be found in [152–160] and references therein (Neural Networks, impulsive delayed systems, delayed switched systems, Impulsive control method, impulsive functional differential equations with infinite delays, singularly perturbed nonlinear systems, etc) [62] !: 2 d l0 ðl0 þ 1Þ Jjl 2m X X 0 0 2 þ k ðxÞ ¼ \j l ; J j POTENT j j00 l00 ; J [ dJjl d 0 0 0 jj jl j00 l00 ðxÞ x2 dx2 h2 j00 l00 where (j, l), ðj0 ; l0 Þ, J ¼ j þ l ¼ j0 þ l0 ! [62]. and 2m h2 0 0 kj0 j ¼ 2 E þ fjðj þ 1Þ j ðj þ 1Þg : 2I h where for E, I and m ! [62] and references therein.
2 Introductiuon of the algorithm In this section we will present the development of the new proposed scheme.
123
Journal of Mathematical Chemistry
fbkgn1 ¼fbkgn1 a0 h2 fgrgnþ1 2 fgrgn þ fgrgn1 Þ 2 a1 h2 fgrgn fbkgn1 ¼fbkgn1 a2 h2 fgrgnþ1 2 fgrgn þ fgrgn1 2 a3 h2 fgrgn fbkgn ¼fbkgn a4 h2 fgrgnþ1 2 fgrgn þ fgrgn1
ð4Þ
fbkgnþ1 þ a5 fbkgn þ fbkgn1 ¼h2 b1 fgrgnþ1 þ fgrgn1 þ b0 fgrgn fgrgn1 ¼ fbkg
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