A variant of the complex Liouville-Green approximation theorem

Renato Spigler; Marco Vianello

Archivum Mathematicum (2000)

  • Volume: 036, Issue: 3, page 213-218
  • ISSN: 0044-8753

Abstract

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We propose a variant of the classical Liouville-Green approximation theorem for linear complex differential equations of the second order. We obtain rigorous error bounds for the asymptotics at infinity, in the spirit of F. W. J. Olver’s formulation, by using rather arbitrary ξ -progressive paths. This approach can provide higher flexibility in practical applications of the method.

How to cite

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Spigler, Renato, and Vianello, Marco. "A variant of the complex Liouville-Green approximation theorem." Archivum Mathematicum 036.3 (2000): 213-218. <http://eudml.org/doc/248526>.

@article{Spigler2000,
abstract = {We propose a variant of the classical Liouville-Green approximation theorem for linear complex differential equations of the second order. We obtain rigorous error bounds for the asymptotics at infinity, in the spirit of F. W. J. Olver’s formulation, by using rather arbitrary $\xi $-progressive paths. This approach can provide higher flexibility in practical applications of the method.},
author = {Spigler, Renato, Vianello, Marco},
journal = {Archivum Mathematicum},
keywords = {complex Liouville-Green; WKB; asymptotic approximations; complex Liouville-Green; WKB; asymptotic approximations},
language = {eng},
number = {3},
pages = {213-218},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {A variant of the complex Liouville-Green approximation theorem},
url = {http://eudml.org/doc/248526},
volume = {036},
year = {2000},
}

TY - JOUR
AU - Spigler, Renato
AU - Vianello, Marco
TI - A variant of the complex Liouville-Green approximation theorem
JO - Archivum Mathematicum
PY - 2000
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 036
IS - 3
SP - 213
EP - 218
AB - We propose a variant of the classical Liouville-Green approximation theorem for linear complex differential equations of the second order. We obtain rigorous error bounds for the asymptotics at infinity, in the spirit of F. W. J. Olver’s formulation, by using rather arbitrary $\xi $-progressive paths. This approach can provide higher flexibility in practical applications of the method.
LA - eng
KW - complex Liouville-Green; WKB; asymptotic approximations; complex Liouville-Green; WKB; asymptotic approximations
UR - http://eudml.org/doc/248526
ER -

References

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  1. Olver F. W. J., Asymptotics and Special Functions, Academic Press, New York, 1974; reprinted by A. K. Peters, Wellesley, MA, 1997. (1974) Zbl0308.41023MR0435697
  2. Spigler R., Vianello M., A numerical method for evaluating zeros of solutions of second-order linear differential equations, Math. Comp. 55 (1990), 591–612. (1990) Zbl0676.65041MR1035945
  3. Spigler R., Vianello M., On the complex differential equation Y ' ' + G ( z ) Y = 0 in Banach algebras, Stud. Appl. Math. 102 (1999), 291–308. (1999) Zbl1001.34051MR1669480
  4. Thorne R. C., Asymptotic formulae for solutions of second-order differential equations with a large parameter, J. Austral. Math. Soc. 1 (1960), 439–464. (1960) MR0123766
  5. Vianello M., Extensions and numerical applications of the Liouville-Green approximation, Ph. D. Thesis (in Italian), University of Padova, 1992 (advisor: R. Spigler). (1992) 

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