Extension and improvement to the Krylov–Bogoliubov methods using elliptic functions

Abstract It is shown that the Krylov–Bogoliubov methods that give the approximate oscillatory solution of the equation in terms of Jacobi elliptic functions are applicable not only when c 1>0 and c 3>0, but also when c 1>0 and c 3 0. In particular, the most precise of these methods, the Christopher-Brocklehurst method, is discussed in detail. Its accuracy has been improved by utilizing the transformation properties of elliptic functions with respect to their parameters.