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A numerical method for evaluating zeros of solutions of second-order linear differential equations

Authors: Renato Spigler and Marco Vianello
Journal: Math. Comp. 55 (1990), 591-612
MSC: Primary 65L99; Secondary 65D15
MathSciNet review: 1035945
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Abstract: A numerical algorithm for computing real zeros of solutions of 2nd-order linear differential equations $ y''+ q(x)y = 0$ in the oscillatory case on a half line is studied. The method applies to the class $ q(x) = a + b/x + O({x^{ - p}})$, with $ a > 0$, $ b \in {\mathbf{R}}$, $ p > 1$.

This procedure is based on a certain nonlinear 3rd-order equation (Kummer's equation) which plays a role in the theory of transformations of 2nd-order differential equations into each other, and was earlier introduced by F. W. J. Olver in 1950 to compute zeros of cylinder functions. A rigorous asymptotic and numerical analysis is developed by combining Borůvka's approach to the study of Kummer's equation and Olver's original idea. Numerical examples are presented.

References [Enhancements On Off] (What's this?)

  • [1] M. Abramowitz and I. A. Stegun (eds.), Handbook of mathematical functions, Dover, New York, 1968.
  • [2] P. Appell, Sur la transformation des équations différentielles linéaires, C. R. Acad. Sci. Paris 91 (1880), 211-214.
  • [3] R. Bellman, On the linear differential equations whose solutions are the products of the solutions of two given linear differential equations, Boll. Un. Mat. Ital. (3) 12 (1957), 12-15. MR 0086960 (19:274e)
  • [4] O. Boruvka, Linear differential transformations of the second order, English Univ. Press, London, 1971. MR 0463539 (57:3484)
  • [5] P. Hartman, Ordinary differential equations, Wiley, New York, 1964. MR 0171038 (30:1270)
  • [6] E. Hille, Lectures on ordinary differential equations, Addison-Wesley, New York, 1968. MR 0249698 (40:2939)
  • [7] S. Lang, Analysis. II, Addison-Wesley, Reading, Mass., 1969.
  • [8] F. W. J. Olver, A new method for the evaluation of zeros of Bessel functions and of other solutions of second-order differential equations, Proc. Cambridge Philos. Soc. 46 (1950), 570-580. MR 0037609 (12:288b)
  • [9] -, (ed.), Bessel functions, Part III, Zeros and associated values, Roy. Soc. Math. Tables, vol. 7, Cambridge Univ. Press, Cambridge, 1960. MR 0119441 (22:10202)
  • [10] -, Asymptotics and special functions, Academic Press, New York, 1974. MR 0435697 (55:8655)
  • [11] R. Spigler, Alcuni risultati sugli zeri delle funzioni cilindriche e delle loro derivate, Rend. Sem. Mat. Univ. Politec. Torino 38 (1980), 67-85.
  • [12] -, The linear differential equations whose solutions are the products of solutions of two given differential equations, J. Math. Anal. Appl. 98 (1984), 130-147. MR 728521 (85b:34009)
  • [13] F. G. Tricomi, Funzioni ipergeometriche confluenti, Consiglio Nazionale delle Ricerche, Monografie Mat., no. 1, Edizioni Cremonese, Roma, 1954. MR 0076936 (17:967d)

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Keywords: Ordinary differential equations, zeros of functions, asymptotic and numerical approximation of zeros, special functions
Article copyright: © Copyright 1990 American Mathematical Society

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