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A Class of Hybrid Methods for Direct Integration of Fourth-Order Ordinary Differential Equations

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Abstract

A class of hybrid methods for solving fourth-order ordinary differential equations (HMFD) is proposed and investigated. Using the theory of B-series, we study the order of convergence of the HMFD methods. The main result is a set of order conditions, analogous to those for two-step hybrid method, which offers a better alternative to the usual ad hoc Taylor expansions. Based on the algebraic order conditions, a one-stage and two-stage explicit HMFD methods are constructed. Results from numerical experiment suggest the superiority of the new methods in terms of accuracy and computational efficiency over hybrid methods for special second ODEs, Runge–Kutta methods recently proposed for solving special fourth-order ODEs directly and some linear multistep methods proposed for the same purpose in the literature.

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Authors and Affiliations

  1. Department of Mathematics, University Putra Malaysia, 43400, Serdang, Selangor, Malaysia

    Y. D. Jikantoro, F. Ismail, N. Senu & Z. B. Ibrahim

  2. Institute for Mathematical Research, University Putra Malaysia (UPM), 43400, Serdang, Selangor, Malaysia

    F. Ismail, N. Senu & Z. B. Ibrahim

  3. Department of Mathematics, Ibrahim Badamasi Babangida University, P.M.B 11, Lapai, Nigeria

    Y. D. Jikantoro

Authors
  1. Y. D. Jikantoro
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  2. F. Ismail
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  3. N. Senu
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  4. Z. B. Ibrahim
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Correspondence to Y. D. Jikantoro.

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Communicated by Ahmad Izani Md. Ismail.

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Jikantoro, Y.D., Ismail, F., Senu, N. et al. A Class of Hybrid Methods for Direct Integration of Fourth-Order Ordinary Differential Equations. Bull. Malays. Math. Sci. Soc. 41, 985–1010 (2018). https://doi.org/10.1007/s40840-017-0520-x

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  • DOI: https://doi.org/10.1007/s40840-017-0520-x

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