Adapted Falkner-type methods solving oscillatory second-order differential equations

Abstract

The classical Falkner methods (Falkner, Phil Mag S 7:621, 1936) are well-known for solving second-order initial-value problems u??(t)?=?f(t, u(t), u?(t)). In this paper, we propose the adapted Falkner-type methods for the systems of oscillatory second-order differential equations u??(t)?+?Mu(t)?=?g(t, u(t)) and make a rigorous error analysis. The error bounds for the global errors on the solution and the derivative are presented. In particular, the error bound for the global error of the solution is shown to be independent of ||M||. We also give a stability analysis and plot the regions of stability for our new methods. Numerical examples are included to show that our new methods are very competitive compared with the reformed Falkner methods in the scientific literature.