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

Abstract

A numerical algorithm for computing real zeros of solutions of 2nd-order linear differential equations y + q ( x ) y = 0 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 ) q(x) = a + b/x + O({x^{ - p}}) , with a > 0 a > 0 , b ∈ R b \in {\mathbf {R}} , p > 1 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.