Asymptotic-numerical approximations for highly oscillatory second-order differential equations by the phase function method

Asymptotic approximations of “phase functions” for linear second-order differential equations, whose solutions are highly oscillatory, can be obtained using Borůvka's theory of linear differential transformations coupled to Liouville-Green (WKB) asymptotics. A numerical method, very effective in case of asymptotically polynomial coefficients, is extended to other cases of rapidly growing coefficients. Zeros of solutions can be computed without prior evaluation of the solutions themselves, but the method can also be applied to Initial- and Boundary-Value problems, as well as to the case of forced oscillations. Numerical examples are given to illustrate the performance of the algorithm. In all cases, the error turns out to be of the order of that made approximating the phase functions.