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A new collocation formulation for the block Falkner-type methods with trigonometric coefficients for oscillatory second order ordinary differential equations

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Abstract

We consider a new class of modified block Falkner methods for the direct numerical integration of second-order initial value problems having periodic and oscillatory solutions. We will give a new collocation formulation different from that of Ramos et al. (J Comput Appl Math. https://doi.org/10.1016/j.cam.2015.12.018, 2016) for the coefficients of a modified block Falkner-type methods, which are frequency dependent. We give an example using our new approach to derive a practical method. Furthermore, the uniform general order conditions and the investigation of the stability properties are presented. Numerical experiments are carried out to illustrate the high effectiveness of the new methods compared with some recent methods in the literature.

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Authors and Affiliations

  1. College of Horticulture, Nanjing Agricultural University, Nanjing, 210095, China

    J. O. Ehigie

  2. Department of Mathematics, University of Lagos, Lagos, 23401, Nigeria

    J. O. Ehigie & S. A. Okunuga

Authors
  1. J. O. Ehigie
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  2. S. A. Okunuga
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Correspondence to J. O. Ehigie.

Appendix

Appendix

(61)
$$\begin{aligned} B=\left( \begin{array}{cccccccccccc} 0 &{}\quad 0 &{} \quad \ldots &{}\quad 0 &{}\quad 0 &{}\quad -z^2\beta _{1-k}(\nu ) &{}\quad 0 &{}\quad 0 &{}\quad \dots &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \ldots &{}\quad 0 &{}\quad 0 &{}\quad -z^2\beta _{1-k}^{(1)}(\nu ) &{}\quad 0 &{} \quad 0 &{} \quad \dots &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ \ldots &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} \\ 0 &{}\quad 0 &{}\quad \ldots &{}\quad 0 &{} \quad 0 &{}\quad -z^2\beta _{1-k}^{(k-3)}(\nu ) &{}\quad 0 &{} \quad 0 &{} \quad \dots &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \ldots &{}\quad 0 &{}\quad 0 &{}\quad -z^2\beta _{1-k}^{(k-2)}(\nu ) &{}\quad 0 &{} \quad 0 &{}\quad \dots &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \ldots &{} \quad 0 &{}\quad 0 &{}\quad -1-z^2\beta _{1-k}^{(k-1)}(\nu ) &{} \quad 0 &{}\quad 0 &{}\quad \dots &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad \ldots &{}\quad 0 &{}\quad 0 &{}\quad -z^2\gamma _{1-k}(\nu ) &{} \quad 0 &{}\quad 0 &{} \quad \dots &{}\quad 0 &{}\quad 0 &{} \quad 0 \\ 0 &{}\quad 0 &{} \quad \ldots &{}\quad 0 &{} \quad 0 &{} \quad -z^2\gamma _{1-k}^{(1)}(\nu ) &{} \quad 0 &{} \quad 0 &{} \quad \dots &{} \quad 0 &{}\quad 0 &{}\quad 0 \\ \ldots &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} &{} \\ 0 &{}\quad 0 &{}\quad \ldots &{}\quad 0 &{} \quad 0 &{}\quad -z^2\gamma _{1-k}^{(k-2)}(\nu ) &{} \quad 0 &{} \quad 0 &{} \quad \dots &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{} \quad \ldots &{}\quad 0 &{}\quad 0 &{} \quad -z^2\gamma _{1-k}^{(k-1)}(\nu ) &{}\quad 0 &{} \quad 0 &{} \quad \dots &{}\quad 0 &{}\quad 0 &{}\quad -1 \\ \end{array} \right) \end{aligned}$$
(62)

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Ehigie, J.O., Okunuga, S.A. A new collocation formulation for the block Falkner-type methods with trigonometric coefficients for oscillatory second order ordinary differential equations. Afr. Mat. 29, 531–555 (2018). https://doi.org/10.1007/s13370-018-0558-4

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