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High-order exponentially fitted and trigonometrically fitted explicit two-derivative Runge–Kutta-type methods for solving third-order oscillatory problems

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Abstract

Three stage sixth-order exponentially fitted and trigonometrically fitted explicit two-derivative Runge–Kutta-type methods are proposed for solving \(u^{'''}(t) = \, f(t,u(t),u'(t)).\) The idea of construction is based on linear composition of the set functions \(e^{\omega t}\) and \(e^{-\omega t}\) for exponentially fitted and \(e^{i\omega t}\) and \(e^{-i\omega t}\) for trigonometrically fitted with \(\omega \in \mathbb {R}\) to integrate initial value problems. The selected coefficients of two-derivative Runge–Kutta-type method are modified to depend on the principle frequency of the numerical problems to construct exponentially fitted and trigonometrically fitted Runge–Kutta-type direct methods, denoted as EFTDRKT6 and TFTDRKT6 methods. The numerical experiments illustrate competence of the new exponentially fitted and trigonometrically fitted method compared to existing methods for solving special type third-order ordinary differential equations with initial value problems.

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Authors and Affiliations

  1. Institute for Mathematical Research and Department of Mathematics and Statistics, Universiti Putra Malaysia, 43400, UPM Serdang, Selangor, Malaysia

    Norazak Senu & Siti Nur Iqmal Ibrahim

  2. Institute for Mathematical Research, Universiti Putra Malaysia, 43400, UPM Serdang, Selangor, Malaysia

    Khai Chien Lee

  3. Institute of Industry Revolution 4.0, The National University of Malaysia, 43600, UKM Bangi, Selangor, Malaysia

    Ali Ahmadian

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  1. Khai Chien Lee
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  2. Norazak Senu
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  3. Ali Ahmadian
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  4. Siti Nur Iqmal Ibrahim
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Correspondence to Khai Chien Lee.

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Appendix 1

Appendix 1

$$\begin{aligned} LTE(u) = \, &-\frac{5+\sqrt{5}}{216000}\left[ 9\sqrt{5}g_{u''u''}fg+9\sqrt{5}g_{uu''}u'g+6\sqrt{5}g_{uuu''}u'^{2}f\right. \nonumber \\&\left. +6\sqrt{5}g_{uu''u''}u'f^{2}-10g_{u'u'u'}u''^{3}-30g_{uu'}u''^{2}\right. \left. +12\sqrt{5}g_{uu'u''}u'u''f\right. \left. +6\sqrt{5}g_{uu'}u'f+7\sqrt{5}g_{u}g_{u''}u'-15g_{u''}^{2}f-10g_{u'}g\right. \left. +6\sqrt{5}g_{uu'}u''^{2} \right. \nonumber \\&\left. -25g_{u'u''}u''g-15g_{u'}g_{u''}u''-30g_{uu}u'u''-30g_{uu''}u''f\right. \nonumber \\&\left. -30g_{u'u'}u''f-30g_{u'u''u''}u''f^{2}\right. \nonumber \\&\left. -30g_{u'u'u''}u''^{2}f-30g_{uuu'}u'^{2}u''\right. \nonumber \\&\left. -30g_{uu'u'}u'u''^{2}+2\sqrt{5}g_{u'u'u'}u''^{3}-10g_{u''u''u''}f^{3}\right. \nonumber \\&\left. -30g_{uu'}u'f-30g_{uu''u''}u'f^{2}\right. \nonumber \\&\left. -30g_{uuu''}u'^{2}f+2\sqrt{5}g_{uuu}u'^{3}+2\sqrt{5}g_{u''u''u''}f^{3}\right. \nonumber \\&\left. +7\sqrt{5}g_{u''}^{2}f+2\sqrt{5}g_{u'}g\right. \nonumber \\&\left. +6\sqrt{5}g_{u'u''}f^{2}+2\sqrt{5}g_{u}f-25g_{uu''}u'g-10g_{u}f\right. \nonumber \\&\left. -30g_{u'u''}f^{2}+9\sqrt{5}g_{u'u''}u''g\right. \nonumber \\&\left. +6\sqrt{5}g_{u'u'u''}u''^{2}f+6\sqrt{5}g_{u'u''u''}u''f^{2}+6\sqrt{5}g_{uuu'}u'^{2}u''\right. \nonumber \\&\left. +6\sqrt{5}g_{uu'u'}u'u''^{2}-60g_{uu'u''}u'u''f\right. \nonumber \\&\left. +6\sqrt{5}g_{uu}u'u''+6\sqrt{5}g_{uu''}u''f+6\sqrt{5}g_{u'u'}u''f\right. \nonumber \\&\left. +7\sqrt{5}g_{u'}g_{u''}u''-25g_{u''u''}fg\right. \nonumber \\&\left. -10g_{uuu}u'^{3}-15g_{u}g_{u''}u'\right] v^{7}, \end{aligned}$$
(50)
$$\begin{aligned} LTE(u') = \, &-\frac{3+\sqrt{5}}{28800}\left[ 36\sqrt{5}g_{uuu'u''}u'^{2}u''f\right. \nonumber \\&\left. +36\sqrt{5}g_{uu'u''u''}u'u''f^{2}+36\sqrt{5}g_{uu'u'u''}u'u''^{2}f\right. \nonumber \\&\left. +36\sqrt{5}g_{uu'u''}u'u''g+36\sqrt{5}g_{u'u''u''}u''fg\right. \nonumber \\&\left. +36\sqrt{5}g_{uu'u'}u'u''f\right. \nonumber \\&\left. +36\sqrt{5}g_{uuu''}u'u''f\right. \nonumber \\&\left. -7g_{u'u'u'u'}u''^{4}-21g_{uu}u''^{2}\right. \nonumber \\&\left. -42g_{uu'u'}u''^{3}\right. \nonumber \\&\left. +36\sqrt{5}g_{uu''u''}u'fg\right. \nonumber \\&\left. +18\sqrt{5}g_{uuu''u''}u'^{2}f^{2}\right. \nonumber \\&\left. +36\sqrt{5}g_{uu'u''}u'u''g+36\sqrt{5}g_{uuu''}u'u''f\right. \nonumber \\&\left. -7g_{u'u'u'u'}u''^{4}\right. \nonumber \\&\left. -21g_{uu}u''^{2}-42g_{uu'u'}u''^{3}\right. \nonumber \\&\left. +36\sqrt{5}g_{uu''u''}u'fg+18\sqrt{5}g_{uuu''u''}u''^{2}f^{2}\right. \nonumber \\&\left. +36\sqrt{5}g_{uu'u''}u'f^{2}\right. \nonumber \\&\left. +12\sqrt{5}g_{uuuu''}u'^{3}f\right. \nonumber \\&\left. +18\sqrt{5}g_{uuu'}u'^{2}f+12\sqrt{5}g_{uu}u'f-70g_{u'u''}fg\right. \nonumber \\&\left. -42g_{u''u''u''}f^{2}g+30\sqrt{5}g_{u'u''}fg\right. \nonumber \\&\left. +12\sqrt{5}g_{uu'}u'g-18\sqrt{5}g_{u''u''u''}f^{2}g\right. \nonumber \\&\left. +18\sqrt{5}g_{uuu''}u'^{2}g\right. \nonumber \\&\left. +12\sqrt{5}g_{uu''u''u''}u'f^{3}\right. \nonumber \\&\left. -28g_{uuuu''}u'^{3}f-42g_{uuu'}u'^{2}f-28g_{uu}u'f\right. \nonumber \\&\left. +9\sqrt{5}g_{u''u''}g^{2}\right. \nonumber \\&\left. +3\sqrt{5}g_{uuuu}u'^{4}+3\sqrt{5}g_{u}g\right. \nonumber \\&\left. +18\sqrt{5}g_{u'u''u''}f^{3}+3\sqrt{5}g_{u''u''u''u''}f^{4}\right. \nonumber \\&\left. -28g_{uu''u''u''}u'f^{3}\right. \nonumber \\&\left. -42g_{uuu''u''}u'^{2}f^{2}\right. \nonumber \\&\left. -84g_{uu'u''}u'f^{2}-28g_{uu'}u'g-42g_{uuu''}u'^{2}g\right. \nonumber \\&\left. +9\sqrt{5}g_{u'u'}f^{2}+12\sqrt{5}g_{uu''}f^{2}-21g_{u'u'}f^{2}\right. \nonumber \\&\left. -28g_{uu''}f^{2}-21g_{u''u''}g^{2}-28g_{uu'u'u'}u'u''^{3}\right. \nonumber \\&\left. -70g_{uu'}u''f-84g_{u'u'u''}u''f^{2}\right. \nonumber \\&\left. -42g_{uuu}u'^{2}u''+3\sqrt{5}g_{u'u'u'u'}u''^{4}\right. \nonumber \\&\left. +9\sqrt{5}g_{uu}u''^{2}\right. \nonumber \\&\left. +18\sqrt{5}g_{uu'u'}u''^{3}-28g_{u'u''u''u''}u''f^{3}\right. \nonumber \\&\left. -42g_{u'u'u''u''}u''^{2}f^{2}-42g_{uuu'u'}u'^{2}u''^{2}\right. \nonumber \\&\left. -42g_{u'u'u'}u''^{2}f\right. \nonumber \\&\left. -42g_{uu''u''}u''f^{2}\right. \nonumber \\&\left. -28g_{u'u'u'u''}u''^{2}f-28g_{uuuu'}u'^{3}u''-42g_{uu''}u''g\right. \nonumber \\&\left. -28g_{u'u'}u''g-42g_{u'u'u''}u''^{2}g\right. \nonumber \\&\left. -84g_{uuu'}u'u''^{2}-84g_{uu'u''}u''^{2}f-7g_{u}g-42g_{u'u''u''}f^{3}\right. \nonumber \\&\left. -7g_{u''u''u''u''}f^{4}-7g_{uuuu}u'^{4}\right. \nonumber \\&\left. +36\sqrt{5}g_{u'u'u''}u''f^{2}+18\sqrt{5}g_{uuu}u'^{2}u''\right. \nonumber \\&\left. +36\sqrt{5}g_{uuu'}u'u''^{2}+18\sqrt{5}g_{u'u'u'}u''^{2}f\right. \nonumber \\&\left. +18\sqrt{5}g_{uu''u''}u''f^{2}+12\sqrt{5}g_{u'u'u'u''}u''^{3}f\right. \nonumber \\&\left. +12\sqrt{5}g_{uuuu'}u'^{3}u''+12\sqrt{5}g_{uu'u'u'}u'u''^{3}\right. \nonumber \\&\left. +30\sqrt{5}g_{uu'}u''f+36\sqrt{5}g_{uu'u''}u''^{2}f+12\sqrt{5}g_{u'u'}u''g\right. \nonumber \\&\left. +18\sqrt{5}g_{u'u'u''}u''^{2}g\right. \nonumber \\&\left. +18\sqrt{5}g_{uu''}u''g+12\sqrt{5}g_{u'u''u''u''}\right. \nonumber \\&\left. +18\sqrt{5}g_{u'u'u''u''}u''^{2}f^{2}\right. \nonumber \\&\left. +18\sqrt{5}g_{uuu'u'}u'^{2}u''^{2}\right. \nonumber \\&\left. -84g_{uuu'u''}u'^{2}u''f-84g_{uu'u''u''}u'u''f^{2}\right. \nonumber \\&\left. -84g_{uu'u'u''}u'u''^{2}f-84g_{u'u''u''}u''fg\right. \nonumber \\&\left. -84g_{uu'u''}u'u''g-84g_{uu'u'}u'u''f-84g_{uuu''}u'u''f\right. \nonumber \\&\left. -84g_{uu''u''}u'fg\right] v^{7}, \end{aligned}$$
(51)
(52)

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Lee, K.C., Senu, N., Ahmadian, A. et al. High-order exponentially fitted and trigonometrically fitted explicit two-derivative Runge–Kutta-type methods for solving third-order oscillatory problems. Math Sci 16, 281–297 (2022). https://doi.org/10.1007/s40096-021-00420-6

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