The Borel transform and its use in the summation of asymptotic expansions

Studies in Applied Mathematics

Volumn 103, Issue 4, 1999, Pages 339-369

  Byatt Smith, J G ^a  

^a UNIVERSITY OF EDINBURGH   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EXPANSION;

ASYMPTOTIC EXPANSION; BOREL TRANSFORM; INTEGRAL REPRESENTATION; REAL LINE; ZERO EXPANSION;

ASYMPTOTIC ANALYSIS;

EID: 0008105511     PISSN: 00222526     EISSN: None     Source Type: Journal    
DOI: 10.1111/1467-9590.1034136     Document Type: Article

References (22)

1. M. D. KRUSKAL and H. SEGUR, Asymptotics beyond all orders in a model of crystal growth, Stud. App. Math. 85:129-181 (1991).
2. F. W. J. OLVER, Asymptotics and Special Functions, Academic Press, Chap. 3, Section 6. Examples; Cases of Failure, 1970.
3. J. G. BYATT-SMITH, Formulation and summation of hyperasymptotic expansions obtained from integrals, Eur. J. Appl. Math. 9:159-185 (1998).
4. J. ÉCALLE, Les fonctions résurgentes. Tome I (French) [Resurgent functions. Vol. I] Les algébres de fonctions résurgentes, [The algebras of resurgent functions] With an English foreword, Publications Mathématiques d'Orsay [Mathematical Publications of Orsay], 1981.
5. J. ÉCALLE, Les fonctions résurgentes, Tome II (French) [Resurgent functions. Vol. II], Les fonctions résurgentes appliquées à l'itération [Resurgent functions applied to iteration], Publications Mathématiques d'Orsay [Mathematical Publications of Orsay], 1981.
6. O. COSTIN and M. D. KRUSKAL, On optimal truncation of divergent series solutions of nonlinear differential systems; Berry smoothing. Preprint University of Chicago, (1997), submitted to Proc. Roy. Soc. London A (1998).
7. M. V. BERRY, Uniform asymptotic smoothing of Stokes' discontinuities, Proc. Roy. Soc. London A 422:7-21 (1989).
8. M. V. BERRY and C. J. HOWLS, Hyperasymptotics. Proc. Roy. Soc. London A 430:654-668 (1990).
9. S. J. CHAPMAN, J. R. KING, and K. L. ADAMS, Exponential asymptotics and Stokes lines in nonlinear ordinary differential equations. Preprint University of Oxford, (1996), submitted to Proc. Roy. Soc. London A (1998).
10. J. B. MCLEOD, Smoothing of Stokes discontinuities. Proc. Roy. Soc. London A 437:343-354 (1992).
11. Y. POMMEAU, A. RAMANI, and B. GRAMMATICOS, Singular perturbations of solitons, Physica D 31:127-143 (1988).
12. V. HAKIM, Asymptotics Beyond All Orders, H. Segur, S. Tanveer, and H. Levine, Eds. NATO ASI series B, 284, pp. 15-28, Plenum, New York, 1991.
13. J. G. BYATT-SMITH, The asymptotic solutions of a connection problem of a second order ordinary differential equation, Stud. Appl. Math. 80:109-135 (1989).
14. J. G. BYATT-SMITH, On the existence of homoclinic and heteroclinic orbits for differential equations with a small parameter, Eur. J. Appl. Math. 2:133-158 (1991).
15. J. G. BYATT-SMITH, On solutions of ordinary differential equations arising from a model of crystal growth, Stud. Appl. Math. 89:167-187 (1993).
16. R. GRIMSHAW and N. JOSHI, Nonlocal solitary waves, SIAM. J. Appl. Math. 55:124-135 (1995).
17. T. R. AKYLAS and T. S. YANG, On short-scale oscillatory tails of long wave disturbances, Stud. Appl. Math. 94:1-20 (1995).
18. E. BOREL, Memoire sur les séries divergentes, Ann. Sci. Ecole Normale Supevieure Paris 16:1-136 (1899).
19. E. BOREL, Leçons sur les Séries Divergentes, Gauthier-Villars, Paris, 1901, or 2nd. ed., Jacques Gabay, Paris, 1988.
20. R. B. DINGLE, Asymptotic expansions and converging factors 1. General theory and basic converging factors, Proc. Roy. Soc. London A. 224:456-475 (1958).
21. R. B. DINGLE, Asymptotic Expansions: Their Derivations and Interpretation, Academic Press, 1973.
22. J. G. BYATT-SMITH and A. M. DAVIE, A rigorous proof of an exponentially small estimate for a boundary value arising from an ordinary differential equation, Proc. Roy. Soc. Edinburgh 114A:243-268 (1990).