First-order general differential equation for multi-level asymptotics at higher levels and recurrence relationship of singulants

Fatih Say

doi:10.5802/crmath.264

Équations aux dérivées partielles, Physique mathématique

First-order general differential equation for multi-level asymptotics at higher levels and recurrence relationship of singulants

Fatih Say ¹

¹ Department of Mathematics, Faculty of Arts and Sciences, Ordu University, Ordu, Turkey

Comptes Rendus. Mathématique, Volume 359 (2021) no. 10, pp. 1267-1278.

Résumé

We construct a relation between the leading pre-factor function $A (z)$ and the singulants $u_{0} (z)$ , $u_{1} (z)$ , and recurrence relation of the singulants at higher levels for the solution of singularly-perturbed first-order ordinary general differential equation with a small parameter via the method of multi-level asymptotics. The particular equation is chosen due to its appearance at every level of multi-level asymptotic approach for the first-order differential equations. By the relations derived by the asymptotic analysis from the equation, Stokes and anti-Stokes lines can be extracted more quickly and so which exponentials of the expansions are actually contributed in each sector of the complex plane can be deduced faster. Multi-level asymptotic analysis of the first-order singular equations and the Stokes phenomenon may be done straightaway from the higher levels of the analysis.

Reçu le : 2021-04-19
Accepté le : 2021-08-23
Publié le : 2022-01-04

DOI : 10.5802/crmath.264

Classification : 34M40, 34E20

Affiliations des auteurs :

Fatih Say ¹

¹ Department of Mathematics, Faculty of Arts and Sciences, Ordu University, Ordu, Turkey

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2021__359_10_1267_0,
     author = {Fatih Say},
     title = {First-order general differential equation for multi-level asymptotics at higher levels and recurrence relationship of singulants},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1267--1278},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {10},
     year = {2021},
     doi = {10.5802/crmath.264},
     language = {en},
}

TY  - JOUR
AU  - Fatih Say
TI  - First-order general differential equation for multi-level asymptotics at higher levels and recurrence relationship of singulants
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 1267
EP  - 1278
VL  - 359
IS  - 10
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.264
LA  - en
ID  - CRMATH_2021__359_10_1267_0
ER  -

%0 Journal Article
%A Fatih Say
%T First-order general differential equation for multi-level asymptotics at higher levels and recurrence relationship of singulants
%J Comptes Rendus. Mathématique
%D 2021
%P 1267-1278
%V 359
%N 10
%I Académie des sciences, Paris
%R 10.5802/crmath.264
%G en
%F CRMATH_2021__359_10_1267_0

Fatih Say. First-order general differential equation for multi-level asymptotics at higher levels and recurrence relationship of singulants. Comptes Rendus. Mathématique, Volume 359 (2021) no. 10, pp. 1267-1278. doi : 10.5802/crmath.264. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.264/

Bibliographie
Cité par

[1] Handbook of mathematical functions with formulas, graphs, and mathematical tables (Milton Abramowitz; Irene A. Stegun, eds.), Dover Publications, 1992, xiv+1046 pages (Reprint of the 1972 edition) | MR

[2] Michael V. Berry Stokes’ phenomenon; smoothing a Victorian discontinuity, Publ. Math., Inst. Hautes Étud. Sci. (1988) no. 68, pp. 211-221 | DOI | Numdam | MR | Zbl

[3] Michael V. Berry Uniform asymptotic smoothing of Stokes’s discontinuities, Proc. R. Soc. Lond., Ser. A, Volume 422 (1989) no. 1862, pp. 7-21 | MR

[4] Michael V. Berry Waves near Stokes lines, Proc. R. Soc. Lond., Ser. A, Volume 427 (1990) no. 1873, pp. 265-280 | MR | Zbl

[5] Michael V. Berry Stokes’s phenomenon for superfactorial asymptotic series, Proc. R. Soc. Lond., Ser. A, Volume 435 (1991) no. 1894, pp. 437-444 | DOI | MR | Zbl

[6] Michael V. Berry; Christopher J. Howls Hyperasymptotics, Proc. R. Soc. Lond., Ser. A, Volume 430 (1990) no. 1880, pp. 653-668 | DOI | MR | Zbl

[7] Giles L. Body; John R. King; Richard H. Tew Exponential asymptotics of a fifth-order partial differential equation, Eur. J. Appl. Math., Volume 16 (2005) no. 5, pp. 647-681 | DOI | MR | Zbl

[8] John P. Boyd A comparison of numerical and analytical methods for the reduced wave equation with multiple spatial scales, Appl. Numer. Math., Volume 7 (1991) no. 6, pp. 453-479 | DOI | MR | Zbl

[9] John P. Boyd A hyperasymptotic perturbative method for computing the radiation coefficient for weakly nonlocal solitary waves, J. Comput. Phys., Volume 120 (1995) no. 1, pp. 15-32 | DOI | Zbl

[10] John P. Boyd The devil’s invention: asymptotic, superasymptotic and hyperasymptotic series, Acta Appl. Math., Volume 56 (1999) no. 1, pp. 1-98 | DOI | MR | Zbl

[11] John P. Boyd Hyperasymptotics and the linear boundary layer problem: why asymptotic series diverge, SIAM Rev., Volume 47 (2005) no. 3, pp. 553-575 | DOI | MR | Zbl

[12] Stephen J. Chapman On the non-universality of the error function in the smoothing of Stokes discontinuities, Proc. R. Soc. Lond., Ser. A, Volume 452 (1996) no. 1953, pp. 2225-2230 | DOI | MR | Zbl

[13] Steven J. Chapman; Christopher J. Howls; John R. King; Adri B. Olde Daalhuis Why is a shock not a caustic? The higher-order Stokes phenomenon and smoothed shock formation, Nonlinearity, Volume 20 (2007) no. 10, pp. 2425-2452 | DOI | MR | Zbl

[14] Steven J. Chapman; John R. King; K. L. Adams Exponential asymptotics and Stokes lines in nonlinear ordinary differential equations, Proc. R. Soc. Lond., Ser. A, Volume 454 (1998) no. 1978, pp. 2733-2755 | DOI | MR | Zbl

[15] Steven J. Chapman; James M. H. Lawry; John R. Ockendon; Richard H. Tew On the theory of complex rays, SIAM Rev., Volume 41 (1999) no. 3, pp. 417-509 | DOI | MR | Zbl

[16] Steven J. Chapman; David B. Mortimer Exponential asymptotics and Stokes lines in a partial differential equation, Proc. R. Soc. Lond., Ser. A, Volume 461 (2005) no. 2060, pp. 2385-2421 | DOI | MR | Zbl

[17] Steven J. Chapman; Philippe H. Trinh; Thomas P. Witelski Exponential asymptotics for thin film rupture, SIAM J. Appl. Math., Volume 73 (2013) no. 1, pp. 232-253 | DOI | MR | Zbl

[18] Gaston Darboux Mémoire sur l’approximation des fonctions de très-grands nombres, et sur une classe étendue de développements en série., J. Math. Pures Appl., Volume 4 (1878), pp. 5-56 | Zbl

[19] R. B. Dingle Asymptotic expansions: their derivation and interpretation, Academic Press Inc., 1973, xv+521 pages | MR

[20] John Heading The Stokes phenomenon and certain $n$ th order differential equations. II. The Stokes phenomenon, Proc. Camb. Philos. Soc., Volume 53 (1957), pp. 419-441 | DOI | MR | Zbl

[21] Edward J. Hinch Perturbation methods, Cambridge Texts in Applied Mathematics, Cambridge University Press, 1991, xii+160 pages | DOI | MR

[22] Christopher J. Howls; P. J. Langman; Adri B. Olde Daalhuis On the higher-order Stokes phenomenon, Proc. R. Soc. Lond., Ser. A, Volume 460 (2004) no. 2048, pp. 2285-2303 | DOI | MR | Zbl

[23] Christopher J. Howls; Adri B. Olde Daalhuis Exponentially accurate solution tracking for nonlinear ODEs, the higher order Stokes phenomenon and double transseries resummation, Nonlinearity, Volume 25 (2012) no. 6, pp. 1559-1584 | DOI | MR | Zbl

[24] Nalini Joshi; Christopher J. Lustri; S. Luu Nonlinear $q$ -Stokes phenomena for q-Painlevé I, J. Phys. A, Math. Theor., Volume 52 (2019) no. 6, 065204, 30 pages | Zbl

[25] John R. King; Steven J. Chapman Asymptotics beyond all orders and Stokes lines in nonlinear differential-difference equations, Eur. J. Appl. Math., Volume 12 (2001) no. 4, pp. 433-463 | DOI | MR | Zbl

[26] Christopher J. Lustri Exponential asymptotics in unsteady and three-dimensional flows, Ph. D. Thesis, University of Oxford (2013)

[27] John J. Mahony Error estimates for asymptotic representations of integrals, J. Aust. Math. Soc., Volume 13 (1972), pp. 395-410 | DOI | MR | Zbl

[28] J. Bryce McLeod Smoothing of Stokes discontinuities, Proc. R. Soc. Lond., Ser. A, Volume 437 (1992) no. 1900, pp. 343-354 | DOI | MR | Zbl

[29] Adri B. Olde Daalhuis; Steven J. Chapman; John R. King; John R. Ockendon; Richard H. Tew Stokes phenomenon and matched asymptotic expansions, SIAM J. Appl. Math., Volume 55 (1995) no. 6, pp. 1469-1483 | DOI | MR | Zbl

[30] Frank W. J. Olver Asymptotics and special functions, A K Peters, 1997, xviii+572 pages (Reprint of the 1974 original) | DOI | MR

[31] NIST Digital Library of Mathematical Functions (Frank W. J. Olver; Adri B. Olde Daalhuis; Daniel W. Lozier; Barry I. Schneider; Ronald F. Boisvert; Charles W. Clark; Bruce R. Miller; Bonita V. Saunders; Howard S. Cohl; Marjorie A. McClain, eds.) (http://dlmf.nist.gov/, Release 1.1.3 of 2021-09-15)

[32] Fatih Say Exponential asymptotics: multi-level asymptotics of model problems, Ph. D. Thesis, University of Nottingham (2016)

[33] Wolfgang P. Schleich E: Airy Function, Quantum Optics in Phase Space, Wiley Online Library, 2001, pp. 621-627

[34] George G. Stokes On the discontinuity of arbitrary constants that appear in divergent developments, Trans. Camb. Philos. Soc., Volume 10 (1864), pp. 106-128

Cité par Sources :

Commentaires - Politique