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On the existence of nonoscillatory phase functions for second order ordinary differential equations in the high-frequency regime. (English) Zbl 1349.34037

Summary: We observe that solutions of a large class of highly oscillatory second order linear ordinary differential equations can be approximated using nonoscillatory phase functions. In addition, we describe numerical experiments which illustrate several implications of this fact. For example, that many special functions of great interest – such as the Bessel functions \(J_{\nu}\) and \(Y_{\nu}\) – can be evaluated accurately using a number of operations which is \(O(1)\) in the order \(\nu\). The present paper is devoted to the development of an analytical apparatus. Numerical aspects of this work will be reported at a later date.

MSC:

34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
33E30 Other functions coming from differential, difference and integral equations
33F05 Numerical approximation and evaluation of special functions
65L99 Numerical methods for ordinary differential equations

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References:

[1] Andrews, G.; Askey, R.; Roy, R., Special functions, (1999), Cambridge University Press
[2] Bellman, R., Stability theory of differential equations, (1953), Dover Publications Mineola, NY · Zbl 0052.31505
[3] Borůvka, O., Linear differential transformations of the second order, (1971), The English University Press London
[4] Coddington, E.; Levinson, N., Theory of ordinary differential equations, (1984), Krieger Publishing Company Malabar, FL
[5] Corless, R.; Gonnet, G.; Hare, D.; Jeffrey, D.; Knuth, D., On the Lambert W function, Adv. Comput. Math., 5, 329-359, (1996) · Zbl 0863.65008
[6] Goldstein, M.; Thaler, R. M., Bessel functions for large arguments, Math. Tables Other Aids Comput., 12, 18-26, (1958) · Zbl 0084.06802
[7] Grafakos, L., Classical Fourier analysis, (2009), Springer
[8] Z. Heitman, J. Bremer, V. Rokhlin, B. Vioreanu, On the asymptotics of Bessel functions in the Fresnel regime, preprint, 2014. · Zbl 1321.33007
[9] Kummer, E., De generali quadam aequatione differentiali tertti ordinis, (Progr. Evang. Köngil. Stadtgymnasium Liegnitz, (1834)) · JFM 18.0297.01
[10] Neuman, F., Global properties of linear ordinary differential equations, (1991), Kluwer Academic Publishers Dordrecht, The Netherlands · Zbl 0784.34009
[11] Olver, F.; Lozier, D.; Boisvert, R.; Clark, C., NIST handbook of mathematical functions, (2010), Cambridge University Press · Zbl 1198.00002
[12] Olver, F. W., Asymptotics and special functions, (1997), A.K. Peters Natick, MA · Zbl 0982.41018
[13] Rudin, W., Principles of mathematical analysis, (1976), McGraw-Hill · Zbl 0148.02903
[14] Segura, J., Bounds for the ratios of modified Bessel functions and associated Turán-type inequalities, J. Math. Anal. Appl., 374, 516-528, (2011) · Zbl 1207.33009
[15] Spigler, R.; Vianello, M., A numerical method for evaluating the zeros of solutions of second-order linear differential equations, Math. Comput., 55, 591-612, (1990) · Zbl 0676.65041
[16] Spigler, R.; Vianello, M., The phase function method to solve second-order asymptotically polynomial differential equations, Numer. Math., 121, 565-586, (2012) · Zbl 1256.65080
[17] Stein, E.; Weiss, G., Introduction to Fourier analysis on Euclidean spaces, (1971), Princeton University Press · Zbl 0232.42007
[18] Zeidler, E., Nonlinear functional analysis and its applications, vol. I: fixed-point theorems, (1986), Springer-Verlag New York
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