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

Authors:
Jiyong Li;Xinyuan Wu
Affiliations:
Department of Mathematics, Nanjing University, Nanjing, People's Republic of China 210093;Department of Mathematics, Nanjing University, Nanjing, People's Republic of China 210093
Venue:
Numerical Algorithms
Year:
2013

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Abstract

The classical Falkner methods (Falkner, Phil Mag S 7:621, 1936) are well-known for solving second-order initial-value problems u驴驴(t)驴=驴f(t, u(t), u驴(t)). In this paper, we propose the adapted Falkner-type methods for the systems of oscillatory second-order differential equations u驴驴(t)驴+驴Mu(t)驴=驴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.