zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Liouville-Green approximations via the Riccati transformation. (English) Zbl 0618.34011
Consider the scalar second order equation on (a,b) $u''=f(t)u$ with $f(t)=p(t)\sp 2+q(t)$ or $f(t)=-p(t)\sp 2+q(t),$ being p positive and piecewise differentiable and q piecewise continuous. Under suitable conditions on f, using a Riccati transformation, fundamental solutions are found and these are used to provide Liouville-Green approximations and explicit error bounds.
Reviewer: G.Caristi

MSC:
34A45Theoretical approximation of solutions of ODE
34B05Linear boundary value problems for ODE
WorldCat.org
Full Text: DOI
References:
[1] Atkinson, F. V.: The asymptotic solution of second order differential equations. Ann. mat. Pura appl. 37, 347-378 (1954) · Zbl 0056.08101
[2] Bellman, R.: Stability theory of differential equations. (1953) · Zbl 0053.24705
[3] Coppel, W. A.: Stability and asymptotic behavior of differential equations. (1965) · Zbl 0154.09301
[4] Coppel, W. A.: Dichotomies in stability theory. Lecture notes in mathematics 629 (1978) · Zbl 0376.34001
[5] Jr., W. A. Harris; Lutz, D. A.: On the asymptotic integration of linear differential systems. J. math. Anal. appl. 48, 1-16 (1974) · Zbl 0304.34043
[6] Jr., W. A. Harris; Lutz, D. A.: A unified theory of asymptotic integration. J. math. Anal. appl. 57, 571-586 (1977) · Zbl 0398.34012
[7] Hartman, P.; Wintner, A.: Asymptotic integrations of linear differential equations. Amer. J. Math. 77, 932 (1955) · Zbl 0064.08703
[8] Levinson, N.: The asymptotic nature of solutions of linear differential equations. Duke math. J. 15, 111-126 (1948) · Zbl 0040.19402
[9] Olver, F. W. J: Error bounds for the Liouville-Green (or WKB) approximation. Proc. Cambridge philos. Soc. 57, 790-810 (1961) · Zbl 0168.14003
[10] Olver, F. W. J: Asymptotics and special functions. (1974) · Zbl 0303.41035
[11] Perron, O.: Über stabilität und asymptotisches verhalten der integrale von differentialgleichungssystemen. Math. Z. 29, 129-160 (1929) · Zbl 54.0456.04
[12] Perron, O.: Über ein vermeintliches stabilitätskriterium. Math.-physik. Kl. fachgruppe 1 (1930) · Zbl 56.0380.03
[13] Smith, D. R.: Decoupling and order reduction via the Riccati transformation. (1984)
[14] Smith, D. R.: Singular perturbation theory. (1985) · Zbl 0567.34055
[15] Taylor, J. G.: Error bounds for the Liouville-Green approximation to initial-value problems. Z. angew. Math. mech. 58, 529-537 (1978) · Zbl 0399.34004
[16] Taylor, J. G.: Improved error bounds for the Liouville-Green (or WKB) approximation. J. math. Anal. appl. 85, 79-89 (1982) · Zbl 0499.34035