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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

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 | References | Similar Articles | Additional Information

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] Richard Bellman, On the linear differential equations whose solutions are the products of solutions of two given linear differential equations, Boll. Un. Mat. Ital. (3) 12 (1957), 12–15. MR 0086960 (19,274e)
  • [4] Otakar Boruvka, Linear differential transformations of the second order, The English Universities Press, Ltd., London, 1971. Translated from the German by F. M. Arscott. MR 0463539 (57 #3484)
  • [5] Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038 (30 #1270)
  • [6] Einar Hille, Lectures on ordinary differential equations, Addison-Wesley Publ. Co., Reading, Mass.-London-Don Mills, Ont., 1969. 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] Bessel functions. Part III: Zeros and associated values, Royal Society Mathematical Tables, Vol. 7. Prepared under the direction of the Bessel Functions Panel of the Mathematical Tables Committee, Cambridge University Press, New York, 1960. MR 0119441 (22 #10202)
  • [10] F. W. J. Olver, Asymptotics and special functions, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1974. Computer Science and Applied Mathematics. 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] Renato Spigler, The linear differential equation whose solutions are the products of solutions of two given differential equations, J. Math. Anal. Appl. 98 (1984), no. 1, 130–147. MR 728521 (85b:34009), http://dx.doi.org/10.1016/0022-247X(84)90282-8
  • [13] Francesco G. Tricomi, Funzioni ipergeometriche confluenti, Edizioni Cremonese, Roma, 1954 (Italian). MR 0076936 (17,967d)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1990-1035945-7
PII: S 0025-5718(1990)1035945-7
Keywords: Ordinary differential equations, zeros of functions, asymptotic and numerical approximation of zeros, special functions
Article copyright: © Copyright 1990 American Mathematical Society