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

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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′′(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.

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing, 210093, People’s Republic of China

    Jiyong Li & Xinyuan Wu

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  1. Jiyong Li
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  2. Xinyuan Wu
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Correspondence to Xinyuan Wu.

Additional information

The research was supported in part by the Specialized Research Foundation for the Doctoral Program of Higher Education under Grant 20100091110033, by A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions, by the 985 Project at Nanjing University under Grant 9112020301 and by the Natural Science Foundation of China under Grant 10771099.

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Li, J., Wu, X. Adapted Falkner-type methods solving oscillatory second-order differential equations. Numer Algor 62, 355–381 (2013). https://doi.org/10.1007/s11075-012-9583-9

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  • DOI: https://doi.org/10.1007/s11075-012-9583-9

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