Research Article
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Year 2021, Volume: 50 Issue: 2, 342 - 350, 11.04.2021
https://doi.org/10.15672/hujms.657267

Abstract

References

  • [1] M. Abramowitz and I.A. Stegun, (editors), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Reprint of the 1972 edition, Dover Publications, Inc., New York, 1992.
  • [2] K.L. Adams, Exponential asymptotics, Ph.D. Thesis, University of Nottingham, Nottingham, UK, 1997.
  • [3] C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Reprint of the 1978 original, Springer-Verlag, New York, 1999.
  • [4] M.V. Berry, Uniform asymptotic smoothing of Stokes’s discontinuities, Proc. Roy. Soc. London Ser. A 422 (1862), 7-21, 1989.
  • [5] M.V. Berry and C.J. Howls, Unfolding the high orders of asymptotic expansions with coalescing saddles: singularity theory, crossover and duality, Proc. Roy. Soc. London Ser. A 443 (1917), 107126, 1993.
  • [6] M.V. Berry, Asymptotics, superasymptotics, hyperasymptotics..., in: Asymptotics Beyond All Orders, NATO Adv. Sci. Inst. Ser. B Phys. 284, 1-14, H. Segur, S. Tanveer, H. Levine (editors), Springer Science & Business Media, Boston, 1991.
  • [7] J.P. Boyd, The devil’s invention: asymptotic, superasymptotic and hyperasymptotic series, Acta Appl. Math. 56 (1), 1-98, 1999.
  • [8] J.P. Boyd, Hyperasymptotics and the linear boundary layer problem: why asymptotic series diverge, SIAM Rev. 47 (3), 553575, 2005.
  • [9] S.J. Chapman and D.B. Mortimer, Exponential asymptotics and Stokes lines in a partial differential equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2060), 2385-2421, 2005.
  • [10] G. Darboux, Mémoire sur l’approximation des fonctions de très-grands nombres, et sur une classe étendue de développements en série, (French), J. Math. Pures Appl. 4, 556, 1878.
  • [11] R.B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation, Academic Press, London-New York, 1973.
  • [12] P. Henrici, Applied and Computational Complex Analysis: Special Functions - Integral Transforms - Asymptotics - Continued Fractions, 2, Reprint of the 1977 original, John Wiley & Sons, Inc., New York, 1991.
  • [13] E.J. Hinch, Perturbation Methods, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1991.
  • [14] P.J. Langman, When is a Stokes line not a Stokes line?, Ph.D. Thesis, University of Southampton, Southampton, UK, 2005.
  • [15] F.W.J. Olver, Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations, Methods Appl. Anal. 1 (1), 1-13, 1994.
  • [16] F.W.J. Olver, Asymptotics and Special Functions, Reprint of the 1974 original, A K Peters, Ltd., Wellesley, MA, 1997.
  • [17] H. Poincaré, Sur les intégrales irrégulières: Des équations linéaires (French), Acta Math. 8 (1), 295-344, 1886.
  • [18] F. Say, Exponential asymptotics: multi-level asymptotics of model problems, Ph.D. Thesis, University of Nottingham, Nottingham, UK, 2016.
  • [19] S. Tanveer, Viscous displacement in a Hele-Shaw cell, in: Asymptotics Beyond All Orders, NATO Adv. Sci. Inst. Ser. B Phys. 284, 131-153, H. Segur, S. Tanveer, H. Levine (editors), Springer Science & Business Media, Boston, MA, 1991.
  • [20] W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Reprint of the 1965 edition, Robert E. Krieger Publishing Co., Huntington, NY, 1976.

Late-order terms of second order ODEs in terms of pre-factors

Year 2021, Volume: 50 Issue: 2, 342 - 350, 11.04.2021
https://doi.org/10.15672/hujms.657267

Abstract

Factorial over a power approach is one of the fundamental techniques for deriving the late-order terms in the asymptotic approximation of integrals and differential equations. To our knowledge, although many differential equations depending on small or large parameters are addressed thoroughly and intensively by this approach in the literature to date, no explicit formula of the general representation of singularly-perturbed second order inhomogeneous ODEs in the form of this paper has yet been discussed generally in terms of their pre-factors. In this paper, we obtain a leading order asymptotic formula of the general asymptotic expansions suitable for the particular type of ODE by its pre-factors.

References

  • [1] M. Abramowitz and I.A. Stegun, (editors), Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Reprint of the 1972 edition, Dover Publications, Inc., New York, 1992.
  • [2] K.L. Adams, Exponential asymptotics, Ph.D. Thesis, University of Nottingham, Nottingham, UK, 1997.
  • [3] C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Reprint of the 1978 original, Springer-Verlag, New York, 1999.
  • [4] M.V. Berry, Uniform asymptotic smoothing of Stokes’s discontinuities, Proc. Roy. Soc. London Ser. A 422 (1862), 7-21, 1989.
  • [5] M.V. Berry and C.J. Howls, Unfolding the high orders of asymptotic expansions with coalescing saddles: singularity theory, crossover and duality, Proc. Roy. Soc. London Ser. A 443 (1917), 107126, 1993.
  • [6] M.V. Berry, Asymptotics, superasymptotics, hyperasymptotics..., in: Asymptotics Beyond All Orders, NATO Adv. Sci. Inst. Ser. B Phys. 284, 1-14, H. Segur, S. Tanveer, H. Levine (editors), Springer Science & Business Media, Boston, 1991.
  • [7] J.P. Boyd, The devil’s invention: asymptotic, superasymptotic and hyperasymptotic series, Acta Appl. Math. 56 (1), 1-98, 1999.
  • [8] J.P. Boyd, Hyperasymptotics and the linear boundary layer problem: why asymptotic series diverge, SIAM Rev. 47 (3), 553575, 2005.
  • [9] S.J. Chapman and D.B. Mortimer, Exponential asymptotics and Stokes lines in a partial differential equation, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2060), 2385-2421, 2005.
  • [10] G. Darboux, Mémoire sur l’approximation des fonctions de très-grands nombres, et sur une classe étendue de développements en série, (French), J. Math. Pures Appl. 4, 556, 1878.
  • [11] R.B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation, Academic Press, London-New York, 1973.
  • [12] P. Henrici, Applied and Computational Complex Analysis: Special Functions - Integral Transforms - Asymptotics - Continued Fractions, 2, Reprint of the 1977 original, John Wiley & Sons, Inc., New York, 1991.
  • [13] E.J. Hinch, Perturbation Methods, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1991.
  • [14] P.J. Langman, When is a Stokes line not a Stokes line?, Ph.D. Thesis, University of Southampton, Southampton, UK, 2005.
  • [15] F.W.J. Olver, Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations, Methods Appl. Anal. 1 (1), 1-13, 1994.
  • [16] F.W.J. Olver, Asymptotics and Special Functions, Reprint of the 1974 original, A K Peters, Ltd., Wellesley, MA, 1997.
  • [17] H. Poincaré, Sur les intégrales irrégulières: Des équations linéaires (French), Acta Math. 8 (1), 295-344, 1886.
  • [18] F. Say, Exponential asymptotics: multi-level asymptotics of model problems, Ph.D. Thesis, University of Nottingham, Nottingham, UK, 2016.
  • [19] S. Tanveer, Viscous displacement in a Hele-Shaw cell, in: Asymptotics Beyond All Orders, NATO Adv. Sci. Inst. Ser. B Phys. 284, 131-153, H. Segur, S. Tanveer, H. Levine (editors), Springer Science & Business Media, Boston, MA, 1991.
  • [20] W. Wasow, Asymptotic Expansions for Ordinary Differential Equations, Reprint of the 1965 edition, Robert E. Krieger Publishing Co., Huntington, NY, 1976.
There are 20 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Fatih Say 0000-0002-4500-2830

Publication Date April 11, 2021
Published in Issue Year 2021 Volume: 50 Issue: 2

Cite

APA Say, F. (2021). Late-order terms of second order ODEs in terms of pre-factors. Hacettepe Journal of Mathematics and Statistics, 50(2), 342-350. https://doi.org/10.15672/hujms.657267
AMA Say F. Late-order terms of second order ODEs in terms of pre-factors. Hacettepe Journal of Mathematics and Statistics. April 2021;50(2):342-350. doi:10.15672/hujms.657267
Chicago Say, Fatih. “Late-Order Terms of Second Order ODEs in Terms of Pre-Factors”. Hacettepe Journal of Mathematics and Statistics 50, no. 2 (April 2021): 342-50. https://doi.org/10.15672/hujms.657267.
EndNote Say F (April 1, 2021) Late-order terms of second order ODEs in terms of pre-factors. Hacettepe Journal of Mathematics and Statistics 50 2 342–350.
IEEE F. Say, “Late-order terms of second order ODEs in terms of pre-factors”, Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 2, pp. 342–350, 2021, doi: 10.15672/hujms.657267.
ISNAD Say, Fatih. “Late-Order Terms of Second Order ODEs in Terms of Pre-Factors”. Hacettepe Journal of Mathematics and Statistics 50/2 (April 2021), 342-350. https://doi.org/10.15672/hujms.657267.
JAMA Say F. Late-order terms of second order ODEs in terms of pre-factors. Hacettepe Journal of Mathematics and Statistics. 2021;50:342–350.
MLA Say, Fatih. “Late-Order Terms of Second Order ODEs in Terms of Pre-Factors”. Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 2, 2021, pp. 342-50, doi:10.15672/hujms.657267.
Vancouver Say F. Late-order terms of second order ODEs in terms of pre-factors. Hacettepe Journal of Mathematics and Statistics. 2021;50(2):342-50.

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