A finite difference method with phase-lag and its derivatives equal to zero for problems in chemistry
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ORIGINAL PAPER
A finite difference method with phase-lag and its derivatives equal to zero for problems in chemistry Maxim A. Medvedev1,2 • T. E. Simos3,4,5,6,7 Received: 9 June 2020 / Accepted: 11 July 2020 Springer Nature Switzerland AG 2020
Abstract A new method with phase-lag and its derivatives up to order four equal to zero is introduced for the numerical solution of problems in quantum 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: dij ¼ 0 at x ¼ 0
ð2Þ
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. & T. E. Simos [email protected] Extended author information available on the last page of the article
123
Journal of Mathematical Chemistry
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 [60]). Categories of problems (see [60]): • Open Channels • Close Channels Asymptotic form (4)
ofð3Þfortheopenchannelsproblemðsee½60Þ
!
:
d N þ TW0 : where: Wij0
1=2 ki ¼ Kij kj
Nij ¼ ki xjli ðki xÞhij Tij ¼ ki xnli ðki xÞhij More details see in [65, 141–149] and references therein. Areas of application of the problems of the form (1) can also be found in [150–158] and references therein (Cybernetics, Neural Networks, Information Sciences, Neurocomputing, Social networks, Computational Intelligence, Simulation Modelling, etc.). [60] !: 2 d l0 ðl0 þ 1Þ Jjl 2 þ k dj0 l0 ðxÞ j0 j x2 dx2 2m X X 0 0 ¼ 2 \j l ; J j POTENT j j00 l00 ; J [ dJjl j00 l00 ðxÞ h j00 l00 where (j, l), ðj0 ; l0 Þ, J ¼ j þ l ¼ j0 þ l0 ! [60]. 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 ! [60] and references therein.
2 Production of the algorithm In this section we will present the development of the new proposed scheme.
123
Journal of Mathematical Chemistry
fghgn1 ¼ fghgn1 a0 h2 fbrgnþ1 2 fbrgn þ fbrgn1 2 a1 h2 fbrgn fghgn1 ¼ fghgn1 a2 h2 fbrgnþ1 2 fbrgn þ fbrgn1 2 a3 h2 fbrgn fghgn ¼ fghgn a4 h2 fbrgnþ1 2 fbrgn þ fbrgn1
ð4Þ
fghgnþ1 þ a5 fghgn þ fghgn1 ¼ h2 b1 fbrgnþ1 þ fbrgn1 þ b0 fbrgn where fbrgnþi ¼ fghg00 tnþi ; fghgnþi ; i ¼ 1ð1Þ1, fbrgn1 ¼ fghg00 tn1 ; fghgn1 , fbrgn1 ¼ fghg00 tn1 ; fghgn1 , fbrgn ¼ fghg00 tn ; fghgn and 1 • b0 ¼ 56, b1 ¼ 12 , • a5 ¼ 2 • aj ; j ¼ 0ð1Þ4, to be determined.
Figure 1 ! The Production of the new prop
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