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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.

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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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