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New hybrid two-step method with optimized phase and stability characteristics

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Abstract

In this paper and for the first time in the literature, we develop a new Runge–Kutta type symmetric two-step finite difference pair with the following characteristics:

  • the new algorithm is of symmetric type,

  • the new algorithm is of two-step,

  • the new algorithm is of five-stages,

  • the new algorithm is of twelfth-algebraic order,

  • the new algorithm is based on the following approximations:

    1. 1.

      the first layer on the point \(x_{n-1}\),

    2. 2.

      the second layer on the point \(x_{n-1}\),

    3. 3.

      the third layer on the point \(x_{n-1}\),

    4. 4.

      the fourth layer on the point \(x_{n}\) and finally,

    5. 5.

      the fifth (final) layer on the point \(x_{n+1}\),

  • the new algorithm has vanished the phase-lag and its first, second, third and fourth derivatives,

  • the new algorithm has improved stability characteristics for the general problems,

  • the new algorithm is of P-stable type since it has an interval of periodicity equal to \(\left( 0, \infty \right) \).

For the new developed algorithm we present a detailed numerical analysis (local truncation error and stability analysis). The effectiveness of the new developed algorithm is evaluated with the approximate solution of coupled differential equations arising from the Schrödinger type.

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Acknowledgements

The research was funded by a Grant of the Russian Foundation for Basic Research (RFBR) for the Project No. 16-38-60114

Author information

Authors and Affiliations

  1. Group of Numerical and Applied Mathematics on Urgent Problems of Energy and Power Engineering, Faculty of Power Engineering, Ulyanovsk State Technical University, Severny Venets Street 32, Ulyanovsk, Russian Federation, 432027

    V. N. Kovalnogov & R. V. Fedorov

  2. Ulyanovsk Institute of Civil Aviation named after Chief Marshal of Aviation BP Bugaev State Technical University, Mozhaiskogo str. 8/8, Ulyanovsk, Russian Federation, 432071

    A. A. Bondarenko

  3. Department of Mathematics, College of Sciences, King Saud University, P. O. Box 2455, Riyadh, 11451, Saudi Arabia

    T. E. Simos

  4. Group of Modern Computational Methods, Ural Federal University, 19 Mira Street, Ekaterinburg, Russian Federation, 620002

    T. E. Simos

  5. Department of Automation Engineering, TEI of Sterea Hellas, 34400, Psachna Campus, Psachna, Greece

    T. E. Simos

  6. Section of Mathematics, Department of Civil Engineering, Democritus University of Thrace, Xanthi, Greece

    T. E. Simos

  7. 10 Konitsis Street, Amphithea - Paleon Phaleron, 175 64, Athens, Greece

    T. E. Simos

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  1. V. N. Kovalnogov
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  2. R. V. Fedorov
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  4. T. E. Simos
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Highly Cited Researcher (http://isihighlycited.com/), Active Member of the European Academy of Sciences and Arts. Active Member of the European Academy of Sciences. Corresponding Member of European Academy of Arts, Sciences and Humanities.

Appendices

Appendix A

$$\begin{aligned} T_{0} \left( v \right)= & {} 5\, \left( \cos \left( v \right) \right) ^{2}{v}^{6}+30\,\cos \left( v \right) \sin \left( v \right) {v}^{5}\\&+ 65\,\cos \left( v \right) {v}^{6}-123\, \left( \cos \left( v \right) \right) ^{2}{v}^{4}\\&+ 150\,{v}^{5}\sin \left( v \right) +80\,{v}^{6}-267\,\cos \left( v \right) {v}^{3}\sin \left( v \right) \\&+699\,\cos \left( v \right) {v}^{4}-144\, \left( \cos \left( v \right) \right) ^{2}{v}^{2}-885\,{v}^{3}\sin \left( v \right) \\&- 618\,{v}^{4}+4608\, \left( \cos \left( v \right) \right) ^{3}-2448\,\cos \left( v \right) v\sin \left( v \right) \\&- 144\,{v}^{2}\cos \left( v \right) -9216\, \left( \cos \left( v \right) \right) ^{2}+2448\,v\sin \left( v \right) \\&+ 288\,{v}^{2}+4608\,\cos \left( v \right) \\ T_{1} \left( v \right)= & {} \left( \cos \left( v \right) \right) ^{2}{v}^{6}+8\,\cos \left( v \right) \sin \left( v \right) {v}^{5}\\&+ 13\,\cos \left( v \right) {v}^{6}-30\, \left( \cos \left( v \right) \right) ^{2}{v}^{4}\\&+ 40\,{v}^{5}\sin \left( v \right) +16\,{v}^{6}-75\,\cos \left( v \right) {v}^{3}\sin \left( v \right) \\&- 138\,\cos \left( v \right) {v}^{4}-9\, \left( \cos \left( v \right) \right) ^{2}{v}^{2}-249\,{v}^{3}\sin \left( v \right) \\&- 102\,{v}^{4}+1152\, \left( \cos \left( v \right) \right) ^{3}-585\,\cos \left( v \right) v\sin \left( v \right) -9\,{v}^{2}\cos \left( v \right) \\&- 2304\, \left( \cos \left( v \right) \right) ^{2}+585\,v\sin \left( v \right) \\&+ 18\,{v}^{2} + 1152\,\cos \left( v \right) \\ T_{2} \left( v \right)= & {} \left( \cos \left( v \right) \right) ^{2}{v}^{6}+10\,\cos \left( v \right) \sin \left( v \right) {v}^{5}\\ \end{aligned}$$
$$\begin{aligned}&+ 13\,\cos \left( v \right) {v}^{6}-43\, \left( \cos \left( v \right) \right) ^{2}{v}^{4}\\&+ 50\,{v}^{5}\sin \left( v \right) +16\,{v}^{6}-99\,\cos \left( v \right) {v}^{3}\sin \left( v \right) \\&- 139\,\cos \left( v \right) {v}^{4}+72\, \left( \cos \left( v \right) \right) ^{2}{v}^{2}-285\,{v}^{3}\sin \left( v \right) \\&- 58\,{v}^{4}+1536\, \left( \cos \left( v \right) \right) ^{3}-696\,\cos \left( v \right) v\sin \left( v \right) +72\,{v}^{2}\cos \left( v \right) \\&- 3072\, \left( \cos \left( v \right) \right) ^{2}+696\,v\sin \left( v \right) \\&- 144\,{v}^{2} + 1536\,\cos \left( v \right) \\ T_{3} \left( v \right)= & {} \left( \cos \left( v \right) \right) ^{2}{v}^{6}+12\,\cos \left( v \right) \sin \left( v \right) {v}^{5}\\&+ 13\,\cos \left( v \right) {v}^{6}-63\, \left( \cos \left( v \right) \right) ^{2}{v}^{4}\\&+ 60\,{v}^{5}\sin \left( v \right) +16\,{v}^{6}-165\,\cos \left( v \right) {v}^{3}\sin \left( v \right) \\&- 135\,\cos \left( v \right) {v}^{4}+162\, \left( \cos \left( v \right) \right) ^{2}{v}^{2}-195\,{v}^{3}\sin \left( v \right) \\&+ 18\,{v}^{4}+2304\, \left( \cos \left( v \right) \right) ^{3}-414\,\cos \left( v \right) v\sin \left( v \right) \\&+ 162\,{v}^{2}\cos \left( v \right) -4608\, \left( \cos \left( v \right) \right) ^{2}\\&+ 414\,v\sin \left( v \right) -324\,{v}^{2} + 2304\,\cos \left( v \right) \\ T_{4} \left( v \right)= & {} \left( \cos \left( v \right) \right) ^{2}{v}^{6}+70\,\cos \left( v \right) \sin \left( v \right) {v}^{5}\\&+ 65\,\cos \left( v \right) {v}^{6}-435\, \left( \cos \left( v \right) \right) ^{2}{v}^{4}\\ \end{aligned}$$
$$\begin{aligned}&+ 350\,{v}^{5}\sin \left( v \right) +80\,{v}^{6}-1395\,\cos \left( v \right) {v}^{3}\sin \left( v \right) \\&- 435\,\cos \left( v \right) {v}^{4}+1920\, \left( \cos \left( v \right) \right) ^{2}{v}^{2}\\&+ 1395\,{v}^{3}\sin \left( v \right) +870\,{v}^{4}+23040\, \left( \cos \left( v \right) \right) ^{3}\\&- 3840\,{v}^{2}\cos \left( v \right) -69120\, \left( \cos \left( v \right) \right) ^{2}\\&+ 1920\,{v}^{2} + 69120\,\cos \left( v \right) -23040 \\ T_{denom0} \left( v \right)= & {} \left( \cos \left( v \right) \right) ^{2}{v}^{6}+8\,\cos \left( v \right) \sin \left( v \right) {v}^{5}\\&+ 13\,\cos \left( v \right) {v}^{6}-30\, \left( \cos \left( v \right) \right) ^{2}{v}^{4}+40\,{v}^{5}\sin \left( v \right) \\&+ 16\,{v}^{6}-75\,\cos \left( v \right) {v}^{3}\sin \left( v \right) -138\,\cos \left( v \right) {v}^{4}\\&- 9\, \left( \cos \left( v \right) \right) ^{2}{v}^{2}-249\,{v}^{3}\sin \left( v \right) -102\,{v}^{4}\\&+ 1152\, \left( \cos \left( v \right) \right) ^{3}-585\,\cos \left( v \right) v\sin \left( v \right) -9\,{v}^{2}\cos \left( v \right) \\&- 2304\, \left( \cos \left( v \right) \right) ^{2}+585\,v\sin \left( v \right) +18\,{v}^{2}+1152\,\cos \left( v \right) \\ T_{denom1} \left( v \right)= & {} \left( \cos \left( v \right) \right) ^{2}{v}^{6}+10\,\cos \left( v \right) \sin \left( v \right) {v}^{5}\\ \end{aligned}$$
$$\begin{aligned}&+ 13\,\cos \left( v \right) {v}^{6}-43\, \left( \cos \left( v \right) \right) ^{2}{v}^{4}+50\,{v}^{5}\sin \left( v \right) \\&+ 16\,{v}^{6}-99\,\cos \left( v \right) {v}^{3}\sin \left( v \right) -139\,\cos \left( v \right) {v}^{4}\\&+ 72\, \left( \cos \left( v \right) \right) ^{2}{v}^{2}-285\,{v}^{3}\sin \left( v \right) -58\,{v}^{4}\\&+ 1536\, \left( \cos \left( v \right) \right) ^{3}-696\,\cos \left( v \right) v\sin \left( v \right) \\&+ 72\,{v}^{2}\cos \left( v \right) -3072\, \left( \cos \left( v \right) \right) ^{2}\\&+ 696\,v\sin \left( v \right) -144\,{v}^{2}+1536\,\cos \left( v \right) \\ T_{denom2} \left( v \right)= & {} \left( \cos \left( v \right) \right) ^{2}{v}^{6}+12\,\cos \left( v \right) \sin \left( v \right) {v}^{5}\\&+ 13\,\cos \left( v \right) {v}^{6}-63\, \left( \cos \left( v \right) \right) ^{2}{v}^{4}+60\,{v}^{5}\sin \left( v \right) \\&+ 16\,{v}^{6}-165\,\cos \left( v \right) {v}^{3}\sin \left( v \right) -135\,\cos \left( v \right) {v}^{4}\\&+ 162\, \left( \cos \left( v \right) \right) ^{2}{v}^{2}-195\,{v}^{3}\sin \left( v \right) +18\,{v}^{4}\\&+ 2304\, \left( \cos \left( v \right) \right) ^{3}-414\,\cos \left( v \right) v\sin \left( v \right) \\&+ 162\,{v}^{2}\cos \left( v \right) -4608\, \left( \cos \left( v \right) \right) ^{2}\\&+ 414\,v\sin \left( v \right) -324\,{v}^{2} + 2304\,\cos \left( v \right) \\ T_{denom3} \left( v \right)= & {} \left( \cos \left( v \right) \right) ^{2}{v}^{4}+13\,\cos \left( v \right) {v}^{4}\\&+ 14\,\cos \left( v \right) {v}^{3}\sin \left( v \right) +16\,{v}^{4}+70\,{v}^{3}\sin \left( v \right) \\&- 87\, \left( \cos \left( v \right) \right) ^{2}{v}^{2} - 87\,{v}^{2}\cos \left( v \right) \\&- 279\,\cos \left( v \right) v\sin \left( v \right) +384\, \left( \cos \left( v \right) \right) ^{3}\\&+ 174\,{v}^{2}+279\,v\sin \left( v \right) -768\, \left( \cos \left( v \right) \right) ^{2}+384\,\cos \left( v \right) \end{aligned}$$

Appendix B

$$\begin{aligned} LTE_{CL}= & {} LTE_{NM2S5SDVD} = LTE_{NM2S5SDVD2} = LTE_{NM2S5SDVD3} \\= & {} LTE_{NM2S5SDVD4} =\,\approx h^{14} \, \vartheta _{0}\\= & {} h^{14} \Biggl [ -{\frac{351719\,g \left( x \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\varphi \left( x \right) \right) \left( {\frac{{\mathrm{d}}^{6}}{{\mathrm{d}}{x}^{6}}}g \left( x \right) \right) {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) }{8491392000}}\\&- {\frac{253597\,g \left( x \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\varphi \left( x \right) \right) \left( {\frac{{\mathrm{d}}^{5}}{{\mathrm{d}}{x}^{5}}}g \left( x \right) \right) {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}g \left( x \right) }{2830464000}}\\&- {\frac{54589\,g \left( x \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\varphi \left( x \right) \right) \left( {\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}g \left( x \right) \right) {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}g \left( x \right) }{424569600}}\\&-{\frac{1292861\,g \left( x \right) y \left( x \right) \left( {\frac{{\mathrm{d}}^{7}}{{\mathrm{d}}{x}^{7}}}g \left( x \right) \right) {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) }{59439744000}}\\&- {\frac{1989389\,g \left( x \right) y \left( x \right) \left( {\frac{{\mathrm{d}}^{6}}{{\mathrm{d}}{x}^{6}}}g \left( x \right) \right) {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}g \left( x \right) }{39626496000}}\\&-{\frac{2405371\,g \left( x \right) y \left( x \right) \left( {\frac{{\mathrm{d}}^{5}}{{\mathrm{d}}{x}^{5}}}g \left( x \right) \right) {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}g \left( x \right) }{29719872000}}\\&- {\frac{434639\, \left( g \left( x \right) \right) ^{2}y \left( x \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) \right) ^{2}{\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}g \left( x \right) }{4245696000}}\\ \end{aligned}$$
$$\begin{aligned}&-{\frac{691\, \left( g \left( x \right) \right) ^{3} \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\varphi \left( x \right) \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) \right) {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}g \left( x \right) }{35380800}}\\&- {\frac{78083\, \left( g \left( x \right) \right) ^{4}y \left( x \right) {\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}g \left( x \right) }{16982784000}}\\&-{\frac{15893\,g \left( x \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\varphi \left( x \right) \right) {\frac{{\mathrm{d}}^{9}}{{\mathrm{d}}{x}^{9}}}g \left( x \right) }{11887948800}}\\&- {\frac{691\, \left( g \left( x \right) \right) ^{4} \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\varphi \left( x \right) \right) {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}g \left( x \right) }{283046400}}\\&-{\frac{691\,g \left( x \right) y \left( x \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) \right) ^{4}}{32659200}}\\&- {\frac{373831\, \left( g \left( x \right) \right) ^{2}y \left( x \right) {\frac{{\mathrm{d}}^{8}}{{\mathrm{d}}{x}^{8}}}g \left( x \right) }{118879488000}}\\&-{\frac{803633\, \left( g \left( x \right) \right) ^{3}y \left( x \right) {\frac{{\mathrm{d}}^{6}}{{\mathrm{d}}{x}^{6}}}g \left( x \right) }{118879488000}}\\&- {\frac{330989\, \left( g \left( x \right) \right) ^{2}y \left( x \right) \left( {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}g \left( x \right) \right) ^{2}}{4953312000}}\\&-{\frac{187261\,g \left( x \right) y \left( x \right) \left( {\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}g \left( x \right) \right) ^{2}}{3962649600}}\\ \end{aligned}$$
$$\begin{aligned}&- {\frac{54589\, \left( g \left( x \right) \right) ^{3} \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\varphi \left( x \right) \right) {\frac{{\mathrm{d}}^{5}}{{\mathrm{d}}{x}^{5}}}g \left( x \right) }{8491392000}}\\&-{\frac{691\, \left( g \left( x \right) \right) ^{5} \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\varphi \left( x \right) \right) {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) }{2830464000}}\\&- {\frac{71173\,g \left( x \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\varphi \left( x \right) \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) \right) \left( {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}g \left( x \right) \right) ^{2}}{404352000}}\\&-{\frac{691\, \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\varphi \left( x \right) \right) \left( {\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}g \left( x \right) \right) {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}g \left( x \right) }{2808000}}\\&- {\frac{7601\, \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) \right) ^{3} \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\varphi \left( x \right) \right) {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}g \left( x \right) }{88452000}}\\&-{\frac{129217\, \left( {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}g \left( x \right) \right) y \left( x \right) {\frac{{\mathrm{d}}^{8}}{{\mathrm{d}}{x}^{8}}}g \left( x \right) }{39626496000}}\\ \end{aligned}$$
$$\begin{aligned}&- {\frac{46297\,g \left( x \right) y \left( x \right) {\frac{{\mathrm{d}}^{10}}{{\mathrm{d}}{x}^{10}}}g \left( x \right) }{118879488000}}\\&-{\frac{691\, \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) \right) ^{3}y \left( x \right) {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}g \left( x \right) }{9144576}}\\&- {\frac{567311\, \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) \right) y \left( x \right) \left( {\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}g \left( x \right) \right) {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}g \left( x \right) }{3962649600}}\\&-{\frac{444313\, \left( g \left( x \right) \right) ^{3}y \left( x \right) \left( {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}g \left( x \right) \right) ^{2}}{16982784000}}\\&- {\frac{3363097\,g \left( x \right) y \left( x \right) \left( {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}g \left( x \right) \right) ^{3}}{39626496000}}\\&-{\frac{32477\, \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) \right) y \left( x \right) \left( {\frac{{\mathrm{d}}^{5}}{{\mathrm{d}}{x}^{5}}}g \left( x \right) \right) {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}g \left( x \right) }{309582000}}\\&- {\frac{42151\, \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\varphi \left( x \right) \right) {\frac{{\mathrm{d}}^{8}}{{\mathrm{d}}{x}^{8}}}g \left( x \right) }{6604416000}}\\&-{\frac{7601\, \left( {\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}g \left( x \right) \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\varphi \left( x \right) \right) {\frac{{\mathrm{d}}^{5}}{{\mathrm{d}}{x}^{5}}}g \left( x \right) }{188697600}}\\ \end{aligned}$$
$$\begin{aligned}&- {\frac{691\, \left( g \left( x \right) \right) ^{2} \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\varphi \left( x \right) \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) \right) ^{3}}{47174400}}\\&-{\frac{326843\, \left( {\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}g \left( x \right) \right) y \left( x \right) {\frac{{\mathrm{d}}^{6}}{{\mathrm{d}}{x}^{6}}}g \left( x \right) }{39626496000}}\\&- {\frac{691\, \left( {\frac{{\mathrm{d}}^{12}}{{\mathrm{d}}{x}^{12}}}g \left( x \right) \right) y \left( x \right) }{118879488000}}\\&-{\frac{691\, \left( {\frac{{\mathrm{d}}^{11}}{{\mathrm{d}}{x}^{11}}}g \left( x \right) \right) {\frac{\mathrm{d}}{{\mathrm{d}}x}}y \left( x \right) }{9906624000}}\\&- {\frac{15893\, \left( g \left( x \right) \right) ^{5}y \left( x \right) {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}g \left( x \right) }{16982784000}}\\&-{\frac{277091\, \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) \right) ^{2}y \left( x \right) {\frac{{\mathrm{d}}^{6}}{{\mathrm{d}}{x}^{6}}}g \left( x \right) }{9906624000}}\\&- {\frac{691\, \left( g \left( x \right) \right) ^{7}y \left( x \right) }{118879488000}}\\&-{\frac{7601\, \left( {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}g \left( x \right) \right) ^{2} \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\varphi \left( x \right) \right) {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}g \left( x \right) }{40435200}}\\&- {\frac{3200021\, \left( {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}g \left( x \right) \right) ^{2}y \left( x \right) {\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}g \left( x \right) }{39626496000}}\\ \end{aligned}$$
$$\begin{aligned}&-{\frac{20039\, \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) \right) y \left( x \right) {\frac{{\mathrm{d}}^{9}}{{\mathrm{d}}{x}^{9}}}g \left( x \right) }{14859936000}}\\&- {\frac{21421\,g \left( x \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\varphi \left( x \right) \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) \right) ^{2}{\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}g \left( x \right) }{157248000}}\\&-{\frac{691\,g \left( x \right) y \left( x \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) \right) ^{2}{\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}g \left( x \right) }{5080320}}\\&- {\frac{691\, \left( g \left( x \right) \right) ^{2} \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\varphi \left( x \right) \right) \left( {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}g \left( x \right) \right) {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}g \left( x \right) }{7581600}}\\&-{\frac{241159\, \left( g \left( x \right) \right) ^{2} \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\varphi \left( x \right) \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) \right) {\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}g \left( x \right) }{4245696000}}\\&- {\frac{6514057\, \left( g \left( x \right) \right) ^{2}y \left( x \right) \left( {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}g \left( x \right) \right) {\frac{{\mathrm{d}}^{4}}{{\mathrm{d}}{x}^{4}}}g \left( x \right) }{59439744000}}\\&-{\frac{1761359\, \left( g \left( x \right) \right) ^{2}y \left( x \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) \right) {\frac{{\mathrm{d}}^{5}}{{\mathrm{d}}{x}^{5}}}g \left( x \right) }{29719872000}}\\&- {\frac{174823\, \left( {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}g \left( x \right) \right) y \left( x \right) {\frac{{\mathrm{d}}^{7}}{{\mathrm{d}}{x}^{7}}}g \left( x \right) }{29719872000}}\\&-{\frac{15893\, \left( g \left( x \right) \right) ^{2} \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\varphi \left( x \right) \right) {\frac{{\mathrm{d}}^{7}}{{\mathrm{d}}{x}^{7}}}g \left( x \right) }{2971987200}}\\ \end{aligned}$$
$$\begin{aligned}&- {\frac{691\, \left( g \left( x \right) \right) ^{4}y \left( x \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) \right) ^{2}}{212284800}}\\&-{\frac{315787\, \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\varphi \left( x \right) \right) \left( {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}g \left( x \right) \right) ^{2}}{2122848000}}\\&- {\frac{7601\, \left( {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}g \left( x \right) \right) y \left( x \right) \left( {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}g \left( x \right) \right) ^{2}}{81648000}}\\&-{\frac{83611\, \left( {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}g \left( x \right) \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\varphi \left( x \right) \right) {\frac{{\mathrm{d}}^{7}}{{\mathrm{d}}{x}^{7}}}g \left( x \right) }{4953312000}}\\&- {\frac{1348141\, \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) \right) ^{2}y \left( x \right) \left( {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}g \left( x \right) \right) ^{2}}{9906624000}}\\&-{\frac{129217\, \left( {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}g \left( x \right) \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\varphi \left( x \right) \right) {\frac{{\mathrm{d}}^{6}}{{\mathrm{d}}{x}^{6}}}g \left( x \right) }{4245696000}}\\&- {\frac{15893\, \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) \right) ^{2} \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}\varphi \left( x \right) \right) {\frac{{\mathrm{d}}^{5}}{{\mathrm{d}}{x}^{5}}}g \left( x \right) }{235872000}}\\&-{\frac{7601\, \left( {\frac{{\mathrm{d}}^{5}}{{\mathrm{d}}{x}^{5}}}g \left( x \right) \right) ^{2}y \left( x \right) }{1651104000}}\\&- {\frac{49061\, \left( g \left( x \right) \right) ^{3}y \left( x \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) \right) {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}g \left( x \right) }{1213056000}}\\&-{\frac{8162783\,g \left( x \right) y \left( x \right) \left( {\frac{\mathrm{d}}{{\mathrm{d}}x}}g \left( x \right) \right) \left( {\frac{{\mathrm{d}}^{2}}{{\mathrm{d}}{x}^{2}}}g \left( x \right) \right) {\frac{{\mathrm{d}}^{3}}{{\mathrm{d}}{x}^{3}}}g \left( x \right) }{19813248000}} \Biggr ] \end{aligned}$$

where \(\varphi \left( x \right) = \varphi _{n}\).

Appendix C

$$\begin{aligned} T_{5}\left( s,v \right)= & {} -8280\,\sin \left( v \right) {s}^{4}{v}^{7}-23040\,\cos \left( v \right) {s}^{8}{v}^{2}\\&+46080\,\cos \left( v \right) {s}^{6}{v}^{4}\\&- 46080\,\cos \left( v \right) {s}^{4}{v}^{6}\\&+69120\,\cos \left( v \right) {s}^{2}{v}^{8}-192\,{v}^{14}+46080\, \left( \cos \left( v \right) \right) ^{2}{s}^{8}{v}^{2}\\&- 92160\, \left( \cos \left( v \right) \right) ^{2}{s}^{6}{v}^{4}+92160\, \left( \cos \left( v \right) \right) ^{2}{s}^{4}{v}^{6}\\&-69120\, \left( \cos \left( v \right) \right) ^{2}{s}^{2}{v}^{8}\\&- 2088\,{v}^{12}-123\, \left( \cos \left( v \right) \right) ^{2}{s}^{10}{v}^{4}+600\, \left( \cos \left( v \right) \right) ^{2}{s}^{8}{v}^{6}\\&-1290\, \left( \cos \left( v \right) \right) ^{2}{s}^{6}{v}^{8}\\&+ 1260\, \left( \cos \left( v \right) \right) ^{2}{s}^{4}{v}^{10}-435\, \left( \cos \left( v \right) \right) ^{2}{s}^{2}{v}^{12}\\&+150\,\sin \left( v \right) {s}^{10}{v}^{5}\\&- 800\,\sin \left( v \right) {s}^{8}{v}^{7}+1500\,\sin \left( v \right) {s}^{6}{v}^{9}\\&- 1200\,\sin \left( v \right) {s}^{4}{v}^{11}+350\,\sin \left( v \right) {s}^{2}{v}^{13}\\&-168\,\cos \left( v \right) \sin \left( v \right) {v}^{13}\\ \end{aligned}$$
$$\begin{aligned}&- 699\,\cos \left( v \right) {s}^{10}{v}^{4}+2760\,\cos \left( v \right) {s}^{8}{v}^{6}-4170\,\cos \left( v \right) {s}^{6}{v}^{8}\\&+2700\,\cos \left( v \right) {s}^{4}{v}^{10}\\&- 435\,\cos \left( v \right) {s}^{2}{v}^{12}-144\, \left( \cos \left( v \right) \right) ^{2}{s}^{10}{v}^{2}+2448\,\sin \left( v \right) {s}^{10}v\\&-11700\,\sin \left( v \right) {s}^{8}{v}^{3}\\&+ 20880\,\sin \left( v \right) {s}^{6}{v}^{5}+1500\,\cos \left( v \right) \sin \left( v \right) {s}^{8}{v}^{5}\\&- 2970\,\cos \left( v \right) \sin \left( v \right) {s}^{6}{v}^{7}+3300\,\cos \left( v \right) \sin \left( v \right) {s}^{4}{v}^{9}\\&- 1395\,\cos \left( v \right) \sin \left( v \right) {s}^{2}{v}^{11}-2448\,\cos \left( v \right) \sin \left( v \right) {s}^{10}v\\&+11700\,\cos \left( v \right) \sin \left( v \right) {s}^{8}{v}^{3}\\&- 20880\,\cos \left( v \right) \sin \left( v \right) {s}^{6}{v}^{5}+8280\,\cos \left( v \right) \sin \left( v \right) {s}^{4}{v}^{7}\\&+5\, \left( \cos \left( v \right) \right) ^{2}{s}^{10}{v}^{6}\\&- 20\, \left( \cos \left( v \right) \right) ^{2}{s}^{8}{v}^{8}+30\, \left( \cos \left( v \right) \right) ^{2}{s}^{6}{v}^{10}\\&-20\, \left( \cos \left( v \right) \right) ^{2}{s}^{4}{v}^{12}\\&+ 5\, \left( \cos \left( v \right) \right) ^{2}{s}^{2}{v}^{14}+65\,\cos \left( v \right) {s}^{10}{v}^{6}-260\,\cos \left( v \right) {s}^{8}{v}^{8}\\ \end{aligned}$$
$$\begin{aligned}&+390\,\cos \left( v \right) {s}^{6}{v}^{10}\\&- 260\,\cos \left( v \right) {s}^{4}{v}^{12}+65\,\cos \left( v \right) {s}^{2}{v}^{14}-4320\,{s}^{6}{v}^{6}+6480\,{s}^{4}{v}^{8}\\&+1920\,{s}^{2}{v}^{10} -23040\,{s}^{2}{v}^{8}\\&- 12\, \left( \cos \left( v \right) \right) ^{2}{v}^{14}-156\,\cos \left( v \right) {v}^{14}\\&+1044\, \left( \cos \left( v \right) \right) ^{2}{v}^{12}-840\,\sin \left( v \right) {v}^{13}\\&+ 4608\, \left( \cos \left( v \right) \right) ^{3}{s}^{10}-4608\, \left( \cos \left( v \right) \right) ^{3}{v}^{10}\\&+1044\,\cos \left( v \right) {v}^{12}-9216\, \left( \cos \left( v \right) \right) ^{2}{s}^{10}\\&+ 9216\, \left( \cos \left( v \right) \right) ^{2}{v}^{10}-3348\,\sin \left( v \right) {v}^{11}+4608\,\cos \left( v \right) {s}^{10}\\&-4608\,\cos \left( v \right) {v}^{10}\\&+ 30\,\cos \left( v \right) \sin \left( v \right) {s}^{10}{v}^{5}-160\,\cos \left( v \right) \sin \left( v \right) {s}^{8}{v}^{7}\\&+ 300\,\cos \left( v \right) \sin \left( v \right) {s}^{6}{v}^{9}-240\,\cos \left( v \right) \sin \left( v \right) {s}^{4}{v}^{11}\\&+ 70\,\cos \left( v \right) \sin \left( v \right) {s}^{2}{v}^{13}-267\,\cos \left( v \right) \sin \left( v \right) {s}^{10}{v}^{3}\\&+180\, \left( \cos \left( v \right) \right) ^{2}{s}^{8}{v}^{4}\\&+ 2160\, \left( \cos \left( v \right) \right) ^{2}{s}^{6}{v}^{6}-3240\, \left( \cos \left( v \right) \right) ^{2}{s}^{4}{v}^{8}\\ \end{aligned}$$
$$\begin{aligned}&+1920\, \left( \cos \left( v \right) \right) ^{2}{s}^{2}{v}^{10}\\&- 885\,\sin \left( v \right) {s}^{10}{v}^{3}+4980\,\sin \left( v \right) {s}^{8}{v}^{5}\\&-8550\,\sin \left( v \right) {s}^{6}{v}^{7}+3900\,\sin \left( v \right) {s}^{4}{v}^{9}\\&+ 1395\,\sin \left( v \right) {s}^{2}{v}^{11}-23040\, \left( \cos \left( v \right) \right) ^{3}{s}^{8}{v}^{2}\\&+46080\, \left( \cos \left( v \right) \right) ^{3}{s}^{6}{v}^{4}\\&- 46080\, \left( \cos \left( v \right) \right) ^{3}{s}^{4}{v}^{6}+23040\, \left( \cos \left( v \right) \right) ^{3}{s}^{2}{v}^{8}\\&+3348\,\cos \left( v \right) \sin \left( v \right) {v}^{11}\\&- 144\,\cos \left( v \right) {s}^{10}{v}^{2}+180\,\cos \left( v \right) {s}^{8}{v}^{4}\\&+2160\,\cos \left( v \right) {s}^{6}{v}^{6}\\&- 3240\,\cos \left( v \right) {s}^{4}{v}^{8}-3840\,\cos \left( v \right) {s}^{2}{v}^{10}\\&+80\,{s}^{10}{v}^{6}-320\,{s}^{8}{v}^{8}+480\,{s}^{6}{v}^{10}-320\,{s}^{4}{v}^{12}\\&+80\,{s}^{2}{v}^{14}\\&-618\,{s}^{10}{v}^{4}+2040\,{s}^{8}{v}^{6}\\&- 1740\,{s}^{6}{v}^{8}-360\,{s}^{4}{v}^{10}+870\,{s}^{2}{v}^{12}\\&+288\,{s}^{10}{v}^{2}-360\,{s}^{8}{v}^{4}\\ T_{6}\left( s,v \right)= & {} -8280\,\sin \left( v \right) {s}^{4}{v}^{7}-23040\,\cos \left( v \right) {s}^{8}{v}^{2}\\&+46080\,\cos \left( v \right) {s}^{6}{v}^{4}\\&- 46080\,\cos \left( v \right) {s}^{4}{v}^{6}-192\,{v}^{14}+46080\, \left( \cos \left( v \right) \right) ^{2}{s}^{8}{v}^{2}\\&-92160\, \left( \cos \left( v \right) \right) ^{2}{s}^{6}{v}^{4}\\ \end{aligned}$$
$$\begin{aligned}&+ 92160\, \left( \cos \left( v \right) \right) ^{2}{s}^{4}{v}^{6}-2088\,{v}^{12}\\&-123\, \left( \cos \left( v \right) \right) ^{2}{s}^{10}{v}^{4}+600\, \left( \cos \left( v \right) \right) ^{2}{s}^{8}{v}^{6}\\&- 1290\, \left( \cos \left( v \right) \right) ^{2}{s}^{6}{v}^{8}+1260\, \left( \cos \left( v \right) \right) ^{2}{s}^{4}{v}^{10}\\&-435\, \left( \cos \left( v \right) \right) ^{2}{s}^{2}{v}^{12}\\&+ 150\,\sin \left( v \right) {s}^{10}{v}^{5}-800\,\sin \left( v \right) {s}^{8}{v}^{7}\\&+1500\,\sin \left( v \right) {s}^{6}{v}^{9}-1200\,\sin \left( v \right) {s}^{4}{v}^{11}\\&+ 350\,\sin \left( v \right) {s}^{2}{v}^{13}-168\,\cos \left( v \right) \sin \left( v \right) {v}^{13}\\&-699\,\cos \left( v \right) {s}^{10}{v}^{4}+2760\,\cos \left( v \right) {s}^{8}{v}^{6}\\&- 4170\,\cos \left( v \right) {s}^{6}{v}^{8}+2700\,\cos \left( v \right) {s}^{4}{v}^{10}\\&-435\,\cos \left( v \right) {s}^{2}{v}^{12}-144\, \left( \cos \left( v \right) \right) ^{2}{s}^{10}{v}^{2}\\&+ 2448\,\sin \left( v \right) {s}^{10}v-11700\,\sin \left( v \right) {s}^{8}{v}^{3}\\&+20880\,\sin \left( v \right) {s}^{6}{v}^{5}+1500\,\cos \left( v \right) \sin \left( v \right) {s}^{8}{v}^{5}\\&- 2970\,\cos \left( v \right) \sin \left( v \right) {s}^{6}{v}^{7}+3300\,\cos \left( v \right) \sin \left( v \right) {s}^{4}{v}^{9}\\&-1395\,\cos \left( v \right) \sin \left( v \right) {s}^{2}{v}^{11}\\ \end{aligned}$$
$$\begin{aligned}&- 2448\,\cos \left( v \right) \sin \left( v \right) {s}^{10}v+11700\,\cos \left( v \right) \sin \left( v \right) {s}^{8}{v}^{3}\\&-20880\,\cos \left( v \right) \sin \left( v \right) {s}^{6}{v}^{5}\\&+ 8280\,\cos \left( v \right) \sin \left( v \right) {s}^{4}{v}^{7}+5\, \left( \cos \left( v \right) \right) ^{2}{s}^{10}{v}^{6}\\&-20\, \left( \cos \left( v \right) \right) ^{2}{s}^{8}{v}^{8}\\&+ 30\, \left( \cos \left( v \right) \right) ^{2}{s}^{6}{v}^{10}-20\, \left( \cos \left( v \right) \right) ^{2}{s}^{4}{v}^{12}\\&+5\, \left( \cos \left( v \right) \right) ^{2}{s}^{2}{v}^{14}\\&+ 65\,\cos \left( v \right) {s}^{10}{v}^{6}-260\,\cos \left( v \right) {s}^{8}{v}^{8}\\&+390\,\cos \left( v \right) {s}^{6}{v}^{10}\\&- 260\,\cos \left( v \right) {s}^{4}{v}^{12}+65\,\cos \left( v \right) {s}^{2}{v}^{14}\\&- 4320\,{s}^{6}{v}^{6}+6480\,{s}^{4}{v}^{8}-12\, \left( \cos \left( v \right) \right) ^{2}{v}^{14}\\&-156\,\cos \left( v \right) {v}^{14}+1044\, \left( \cos \left( v \right) \right) ^{2}{v}^{12}\\&- 840\,\sin \left( v \right) {v}^{13}\\&+4608\, \left( \cos \left( v \right) \right) ^{3}{s}^{10}-4608\, \left( \cos \left( v \right) \right) ^{3}{v}^{10}+1044\,\cos \left( v \right) {v}^{12}\\&- 9216\, \left( \cos \left( v \right) \right) ^{2}{s}^{10}+9216\, \left( \cos \left( v \right) \right) ^{2}{v}^{10}\\&-3348\,\sin \left( v \right) {v}^{11}+4608\,\cos \left( v \right) {s}^{10}\\ \end{aligned}$$
$$\begin{aligned}&- 4608\,\cos \left( v \right) {v}^{10}+30\,\cos \left( v \right) \sin \left( v \right) {s}^{10}{v}^{5}\\&- 160\,\cos \left( v \right) \sin \left( v \right) {s}^{8}{v}^{7}+300\,\cos \left( v \right) \sin \left( v \right) {s}^{6}{v}^{9}\\&- 240\,\cos \left( v \right) \sin \left( v \right) {s}^{4}{v}^{11}+70\,\cos \left( v \right) \sin \left( v \right) {s}^{2}{v}^{13}\\&- 267\,\cos \left( v \right) \sin \left( v \right) {s}^{10}{v}^{3}+180\, \left( \cos \left( v \right) \right) ^{2}{s}^{8}{v}^{4}\\&+ 2160\, \left( \cos \left( v \right) \right) ^{2}{s}^{6}{v}^{6}-3240\, \left( \cos \left( v \right) \right) ^{2}{s}^{4}{v}^{8}\\&-3840\, \left( \cos \left( v \right) \right) ^{2}{s}^{2}{v}^{10}\\&- 885\,\sin \left( v \right) {s}^{10}{v}^{3}+4980\,\sin \left( v \right) {s}^{8}{v}^{5}\\&-8550\,\sin \left( v \right) {s}^{6}{v}^{7}+3900\,\sin \left( v \right) {s}^{4}{v}^{9}\\&+ 1395\,\sin \left( v \right) {s}^{2}{v}^{11}-23040\, \left( \cos \left( v \right) \right) ^{3}{s}^{8}{v}^{2}\\&+46080\, \left( \cos \left( v \right) \right) ^{3}{s}^{6}{v}^{4}\\&- 46080\, \left( \cos \left( v \right) \right) ^{3}{s}^{4}{v}^{6}+3348\,\cos \left( v \right) \sin \left( v \right) {v}^{11}\\&-144\,\cos \left( v \right) {s}^{10}{v}^{2}+180\,\cos \left( v \right) {s}^{8}{v}^{4}\\&+ 2160\,\cos \left( v \right) {s}^{6}{v}^{6}-3240\,\cos \left( v \right) {s}^{4}{v}^{8}\\&+1920\,\cos \left( v \right) {s}^{2}{v}^{10}+80\,{s}^{10}{v}^{6}-320\,{s}^{8}{v}^{8}\\ \end{aligned}$$
$$\begin{aligned}&+ 480\,{s}^{6}{v}^{10}-320\,{s}^{4}{v}^{12}\\&+80\,{s}^{2}{v}^{14}-618\,{s}^{10}{v}^{4}+2040\,{s}^{8}{v}^{6}-1740\,{s}^{6}{v}^{8}-360\,{s}^{4}{v}^{10}\\&+ 870\,{s}^{2}{v}^{12}+288\,{s}^{10}{v}^{2}-360\,{s}^{8}{v}^{4}+1920\, \left( \cos \left( v \right) \right) ^{3}{s}^{2}{v}^{10}\\ T_{denom4} \left( v \right)= & {} \left( \cos \left( v \right) \right) ^{2}{v}^{4}+13\,\cos \left( v \right) {v}^{4}\\&+ 14\,\cos \left( v \right) {v}^{3}\sin \left( v \right) +16\,{v}^{4}+70\,{v}^{3}\sin \left( v \right) \\&- 87\, \left( \cos \left( v \right) \right) ^{2}{v}^{2} - 87\,{v}^{2}\cos \left( v \right) \\&- 279\,\cos \left( v \right) v\sin \left( v \right) +384\, \left( \cos \left( v \right) \right) ^{3}\\&+ 174\,{v}^{2}+279\,v\sin \left( v \right) -768\, \left( \cos \left( v \right) \right) ^{2}+384\,\cos \left( v \right) \end{aligned}$$

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Kovalnogov, V.N., Fedorov, R.V., Bondarenko, A.A. et al. New hybrid two-step method with optimized phase and stability characteristics. J Math Chem 56, 2302–2340 (2018). https://doi.org/10.1007/s10910-018-0894-5

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