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)
On higher-order Stokes phenomena of an inhomogeneous linear ordinary differential equation. (English) Zbl 1058.34123
The author considers the particular integral of the differential equation $d^2w/dz^2+u^2z^2w=e^{iu((1/2)z^2-4z)}$, where $u$ is a large parameter. He determines the complete Stokes geometry, i.e., the active and inactive Stokes curves and the higher-order Stokes curve. (A Stokes curve to be inactive or a Stokes multiplier to change its value are events typical for higher-order Stokes phenomena.) He presents numerical examples and suggests how the study can be extended to more complicated problems.
MSC:
 34M40 Stokes phenomena and connection problems (ODE in the complex domain) 34E05 Asymptotic expansions (ODE) 34E20 Asymptotic singular perturbations, turning point theory, WKB methods (ODE)
Full Text:
References:
 [1] T. Aoki, T. Kawai, Y. Takei, New turning points in the exact WKB analysis for higher-order ordinary differential equations, in: Analyse algébrique des perturbations singulières, I, Marseille-Luminy, 1991; Travaux en Cours 47 (1994) 69--84. [2] Aoki, T.; Kawai, T.; Takei, Y.: On the exact steepest descent methoda new method for the description of Stokes curves. J. math. Phys. 42, 3691-3713 (2001) · Zbl 1063.34047 [3] Berk, H. L.; Nevins, W. M.; Roberts, K. V.: New Stokes lines in WKB theory. J. math. Phys. 23, 988-1002 (1982) · Zbl 0488.34050 [4] C.J. Howls, P.J. Langman, A.B. Olde Daalhuis, On the higher-order Stokes phenomenon, Proc. Roy. Soc. London Ser. A, to appear. [5] C.J. Howls, A.B. Olde Daalhuis, Hyperasymptotic solutions of inhomogeneous linear differential equations with a singularity of rank one, Proc. Roy. Soc. London, Ser. A 459 (2003), no 2038, 2599--2612. · Zbl 1063.34084 [6] Daalhuis, A. B. Olde: Hyperasymptotic solutions of higher-order linear differential equations with a singularity of rank one. Proc. roy. Soc. London, ser. A 454, 1-29 (1998) · Zbl 0919.34012 [7] Daalhuis, A. B. Olde: On the computation of Stokes multipliers via hyperasymptotics. Sūrikaisekikenkyūsho kōkyūroku 1088, 68-78 (1999) · Zbl 0951.34501 [8] Paris, R. B.; Wood, A. D.: Stokes phenomenon demystified. Bull. inst. Math. appl. 31, 21-28 (1995) · Zbl 0827.34002 [9] Temme, N. M.: Special functions. An introduction to the classical functions of mathematical physics. (1996) · Zbl 0856.33001