Investigation of asymptotics of solutions of second-order differential equations with meromorphic coefficients

Shiyao Cheng

Abstract


This paper investigates the asymptotic expansions of solutions to second-order ordinary differential equations with meromorphic coefficients in the vicinity of irregular singular points. Unlike regular singular points, irregular singular points are characterized by the presence of exponential terms in the solutions, which is related to the fact that the leading coefficient of the differential operator tends to zero faster than allowed for regular singularities. The primary focus is on computing all the coefficients in these expansions, both in cases where the roots of the principal symbol of the differential operator are simple and where they are multiple. Additionally, particular attention is given to the case where the equation contains multiple roots, as such situations significantly complicate the structure of the asymptotic expansions of solutions. The study proves that for any equation with multiple roots, there always exists a transformation that reduces it to an equivalent equation with simple roots while simultaneously decreasing the order of degeneracy of the differential operator. Furthermore, it is shown that in some cases, such a transformation can convert the equation into a Fuchsian-type equation, i.e., a regular one. The results obtained extend asymptotic analysis methods to a broader class of singular differential equations, including those with a high degree of degeneracy.

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References


Kats D.S. Computation of the asymptotics of solutions for equations with polynomial degeneration of the coefficients // Differential Equations. – 2015. – Т. 51. № 12. – с. 1589-1594.

Poincaré H. Sur les intégrales irrégulières: Des équations linéaires //Acta Mathematica. – 1886. – Т. 8. №. 1. – с. 295-344.

Poincaré H. Analysis of the mathematical and natural works of Henri Poincaré // Selected Works: Mathematics; Theoretical Physics. – 1974. – Vol. 3.

Ince E. L. Ordinary differential equations. – Dover Publications, 1956. – 576 pp.

Coddington E., Levinson N. Theory of ordinary differential equations. – New York : McGraw-Hill, 1955. – 444 pp.

L. Cesari. Asymptotic behavior and stability problems in ordinary differential equations. – Springer Science & Business Media, 2012. – T. 16. – 274 pp.

Olver F. Asymptotics and Special Functions. – New York : Academic Press, 1974. – 572 pp.

E ́calle J. Resurgent functions // Vol. I-III Mathematical Publications of Orsay. 1981. С. 81–05.

E ́calle J. Les Fonctions Resurgentes Vol. I-III // Publ. Math. Orsay. 1981. Т. 15.

E ́calle J. The acceleration operators and their applications // Proc. In. 1990.

Korovina M. V., Shatalov V. E. Differential equations with degeneration and resurgent analysis // Differential Equations. – 2010. –

Т. 46. № 9. – с. 1267–1286.

Korovina M. V. Existence of resurgent solutions for equations with higher-order degeneration // Differential Equations. – 2011. – Т. 47. № 3. – с. 346–354.

Korovina M. V. Asymptotics solutions of equations with higher-order degeneracies // Doklady Mathematics. – 2011. – Т. 83. № 2. – с. 182–184.

Korovina M. V. Asymptotics of solutions of equations with higher degenerations // Differential Equations. – 2012. – Т. 48. № 5. – с. 717–729.

Korovina M. V., Matevossian H.A. Uniform Asymptotics of Solutions of Second-Order Differential Equations with Meromorphic Coefficients in a Neighborhood of Singular Points and Their Applications // Mathematics. – 2022. – Vol. 10. № 14. – p. 2465.

Dorigoni D. An introduction to resurgence, trans-series and alien calculus //Annals of Physics. – 2019. – Vol. 409. – p. 167914.


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