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Year 2020, Volume: 3 Issue: 1, 52 - 56, 10.06.2020
https://doi.org/10.33401/fujma.725913

Abstract

References

  • [1] H. Poincaré, Sur les int´egrales irr´eguli`eres, Acta Math., 8 (1) (1886), 295-344.
  • [2] E. J. Hinch, Perturbation Methods, Cambridge University Press, Cambridge, 1991.
  • [3] M. Van Dyke, Perturbation Methods in Fluid Mechanics, Academic Press, New York, 1964.
  • [4] C. M. Bender, 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.
  • [5] V. F. Zaitsev, A. D. Polyanin, Handbook of Exact Solutions for Ordinary Differential Equations, 2nd edition, Chapman & Hall/CRC Press Company, Boca Raton, FL, 2003.
  • [6] J. Cousteix, J. Mauss, Successive Complementary Expansion Method, Asymptotic Analysis and Boundary Layers, Springer, 2007, pp. 59-98.
  • [7] R. B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation, Academic Press, London-New York, 1973.
  • [8] M. V. Berry, Asymptotics, superasymptotics, hyperasymptotics, H. Segur, S. Tanveer, H. Levine (editors), Asymptotics Beyond All Orders, NATO Adv. Sci. Inst. Ser. B Phys., vol. 284, Springer Science & Business Media, Boston, 1991, 1-14.
  • [9] F. Say, Late-order terms of second order ODEs in terms of pre-factors, Hacet. J. Math. Stat., (in press).
  • [10] M. V. Berry, Uniform asymptotic smoothing of Stokes’s discontinuities, P. Roy. Soc. Lond. A Mat., 422 (1862) (1989), 7-21.
  • [11] M. V. Berry, Stokes’s phenomenon; smoothing a Victorian discontinuity, Inst. Hautes ´ Etudes Sci. Publ. Math., 68 (1) (1988), 211-221.
  • [12] B. L. J. Braaksma, G. K. Immink, M. Van der Put, J. Top (editors), Differential equations and the Stokes phenomenon, Proceedings of the workshop held at the University of Groningen, Groningen, May 28–30, 2001, World Scientific Publishing Co., Inc., River Edge, NJ, 2002, doi:10.1142/5107.
  • [13] P. J. Langman, When is a Stokes line not a Stokes line?, Ph.D. Thesis, University of Southampton, 2005.
  • [14] M. D. Kruskal, H. Segur, Asymptotics beyond all orders in a model of crystal growth, Stud. Appl. Math., 85 (2) (1991), 129-181.
  • [15] J. P. Boyd, The devil’s invention: Asymptotic, superasymptotic and hyperasymptotic series, Acta Appl. Math., 56 (1) (1999), 1-98.
  • [16] F. W. J. Olver, Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations, Methods Appl. Anal., 1 (1) (1994), 1-13.
  • [17] R. E. Meyer, A simple explanation of the Stokes phenomenon, SIAM Rev., 31 (3) (1989), 435-445.
  • [18] J. P. Boyd, A hyperasymptotic perturbative method for computing the radiation coefficient for weakly nonlocal solitary waves, J. Comput. Phys., 120 (1) (1995), 15-32.
  • [19] F. Say, Exponential asymptotics: multi-level asymptotics of model problems, Ph.D. Thesis, University of Nottingham, 2016.
  • [20] R. E. Meyer, Exponential asymptotics, SIAM Rev., 22 (2) (1980), 213-224.
  • [21] P. Hsieh, Y. Sibuya, On the asymptotic integration of second order linear ordinary differential equations with polynomial coefficients, J. Math. Anal. Appl., 16 (1) (1966), 84-103.
  • [22] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Reprint of the 1972 edition, Dover Publications, Inc., New York, 1992.
  • [23] E. C. Titchmarsh, The Theory of Functions, Oxford University Press, Oxford, 1939.

High-Order Coefficients of Second-Order ODEs in Relation to Pre-Factors for Complex Parameter

Year 2020, Volume: 3 Issue: 1, 52 - 56, 10.06.2020
https://doi.org/10.33401/fujma.725913

Abstract

In this study, we asymptotically reconsider the relations between the pre-factors of a general inhomogeneous second-order ordinary differential equation and the high-order coefficients of its asymptotic power series for complex values of the asymptotic parameter $ \epsilon_{1} $. The study provides a general formula for its generic high-order coefficients with the associated pre-factors for complex $ \epsilon_{1} $ based on the use of a well-known factorial divided by a power approach.

References

  • [1] H. Poincaré, Sur les int´egrales irr´eguli`eres, Acta Math., 8 (1) (1886), 295-344.
  • [2] E. J. Hinch, Perturbation Methods, Cambridge University Press, Cambridge, 1991.
  • [3] M. Van Dyke, Perturbation Methods in Fluid Mechanics, Academic Press, New York, 1964.
  • [4] C. M. Bender, 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.
  • [5] V. F. Zaitsev, A. D. Polyanin, Handbook of Exact Solutions for Ordinary Differential Equations, 2nd edition, Chapman & Hall/CRC Press Company, Boca Raton, FL, 2003.
  • [6] J. Cousteix, J. Mauss, Successive Complementary Expansion Method, Asymptotic Analysis and Boundary Layers, Springer, 2007, pp. 59-98.
  • [7] R. B. Dingle, Asymptotic Expansions: Their Derivation and Interpretation, Academic Press, London-New York, 1973.
  • [8] M. V. Berry, Asymptotics, superasymptotics, hyperasymptotics, H. Segur, S. Tanveer, H. Levine (editors), Asymptotics Beyond All Orders, NATO Adv. Sci. Inst. Ser. B Phys., vol. 284, Springer Science & Business Media, Boston, 1991, 1-14.
  • [9] F. Say, Late-order terms of second order ODEs in terms of pre-factors, Hacet. J. Math. Stat., (in press).
  • [10] M. V. Berry, Uniform asymptotic smoothing of Stokes’s discontinuities, P. Roy. Soc. Lond. A Mat., 422 (1862) (1989), 7-21.
  • [11] M. V. Berry, Stokes’s phenomenon; smoothing a Victorian discontinuity, Inst. Hautes ´ Etudes Sci. Publ. Math., 68 (1) (1988), 211-221.
  • [12] B. L. J. Braaksma, G. K. Immink, M. Van der Put, J. Top (editors), Differential equations and the Stokes phenomenon, Proceedings of the workshop held at the University of Groningen, Groningen, May 28–30, 2001, World Scientific Publishing Co., Inc., River Edge, NJ, 2002, doi:10.1142/5107.
  • [13] P. J. Langman, When is a Stokes line not a Stokes line?, Ph.D. Thesis, University of Southampton, 2005.
  • [14] M. D. Kruskal, H. Segur, Asymptotics beyond all orders in a model of crystal growth, Stud. Appl. Math., 85 (2) (1991), 129-181.
  • [15] J. P. Boyd, The devil’s invention: Asymptotic, superasymptotic and hyperasymptotic series, Acta Appl. Math., 56 (1) (1999), 1-98.
  • [16] F. W. J. Olver, Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations, Methods Appl. Anal., 1 (1) (1994), 1-13.
  • [17] R. E. Meyer, A simple explanation of the Stokes phenomenon, SIAM Rev., 31 (3) (1989), 435-445.
  • [18] J. P. Boyd, A hyperasymptotic perturbative method for computing the radiation coefficient for weakly nonlocal solitary waves, J. Comput. Phys., 120 (1) (1995), 15-32.
  • [19] F. Say, Exponential asymptotics: multi-level asymptotics of model problems, Ph.D. Thesis, University of Nottingham, 2016.
  • [20] R. E. Meyer, Exponential asymptotics, SIAM Rev., 22 (2) (1980), 213-224.
  • [21] P. Hsieh, Y. Sibuya, On the asymptotic integration of second order linear ordinary differential equations with polynomial coefficients, J. Math. Anal. Appl., 16 (1) (1966), 84-103.
  • [22] M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Reprint of the 1972 edition, Dover Publications, Inc., New York, 1992.
  • [23] E. C. Titchmarsh, The Theory of Functions, Oxford University Press, Oxford, 1939.
There are 23 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Fatih Say 0000-0002-4500-2830

Publication Date June 10, 2020
Submission Date January 23, 2020
Acceptance Date January 3, 2020
Published in Issue Year 2020 Volume: 3 Issue: 1

Cite

APA Say, F. (2020). High-Order Coefficients of Second-Order ODEs in Relation to Pre-Factors for Complex Parameter. Fundamental Journal of Mathematics and Applications, 3(1), 52-56. https://doi.org/10.33401/fujma.725913
AMA Say F. High-Order Coefficients of Second-Order ODEs in Relation to Pre-Factors for Complex Parameter. Fundam. J. Math. Appl. June 2020;3(1):52-56. doi:10.33401/fujma.725913
Chicago Say, Fatih. “High-Order Coefficients of Second-Order ODEs in Relation to Pre-Factors for Complex Parameter”. Fundamental Journal of Mathematics and Applications 3, no. 1 (June 2020): 52-56. https://doi.org/10.33401/fujma.725913.
EndNote Say F (June 1, 2020) High-Order Coefficients of Second-Order ODEs in Relation to Pre-Factors for Complex Parameter. Fundamental Journal of Mathematics and Applications 3 1 52–56.
IEEE F. Say, “High-Order Coefficients of Second-Order ODEs in Relation to Pre-Factors for Complex Parameter”, Fundam. J. Math. Appl., vol. 3, no. 1, pp. 52–56, 2020, doi: 10.33401/fujma.725913.
ISNAD Say, Fatih. “High-Order Coefficients of Second-Order ODEs in Relation to Pre-Factors for Complex Parameter”. Fundamental Journal of Mathematics and Applications 3/1 (June 2020), 52-56. https://doi.org/10.33401/fujma.725913.
JAMA Say F. High-Order Coefficients of Second-Order ODEs in Relation to Pre-Factors for Complex Parameter. Fundam. J. Math. Appl. 2020;3:52–56.
MLA Say, Fatih. “High-Order Coefficients of Second-Order ODEs in Relation to Pre-Factors for Complex Parameter”. Fundamental Journal of Mathematics and Applications, vol. 3, no. 1, 2020, pp. 52-56, doi:10.33401/fujma.725913.
Vancouver Say F. High-Order Coefficients of Second-Order ODEs in Relation to Pre-Factors for Complex Parameter. Fundam. J. Math. Appl. 2020;3(1):52-6.

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