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An explicit six-step singularly P-stable Obrechkoff method for the numerical solution of second-order oscillatory initial value problems

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Abstract

In this paper, we present an explicit six-step singularly P-stable Obrechkoff method of tenth algebraic order for solving second-order linear periodic and oscillatory initial value problems of ordinary differential equations. The advantage of this new singularly P-stable Obrechkoff method is that it is a high-order explicit method, and thus does not require additional predictor stages. The numerical stability and phase properties of the new method is analyzed. Four numerical examples show that the new explicit method is more accurate than Obrechkoff schemes of the same order when applied to the numerical solution of second-order initial value problems with highly oscillatory solutions.

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Acknowledgments

The authors wish to thank the anonymous referees for their careful reading of the manuscript and their fruitful comments and suggestions which improved the presentation of this paper.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh, Iran

    Mohammad Mehdizadeh Khalsaraei & Ali Shokri

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  1. Mohammad Mehdizadeh Khalsaraei
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Correspondence to Mohammad Mehdizadeh Khalsaraei.

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Appendix

Appendix

$$ \begin{array}{@{}rcl@{}} A_{00}&=&-~7200\left( v^{8}+\left( -~2s^{2}-\frac{781}{36}\right)v^{6}+\left( s^{4}+\frac{161}{6}s^{2} +\frac{965}{24}\right)v^{4}\right.\\ &&+\left( -~\frac{35}{12}s^{4}-15-\frac{155}{4}s^{2}\right)v^{2} + \left.15s^{2}+\frac{15}{8}s^{4}\right)v\left( \cos(v)\right)^{7}\\ &&+\left( \left( 55500v^{8}+ \left( -~72000s^{2}-257100\right)v^{6}\right.\right.\\ &&+ \left( 16500s^{4}+294600s^{2}+225000\right)v^{4}\\ &&+\left.\left( -~13500s^{4}-225000s^{2}-27000 \right)v^{2}+27000s^{2}\right)\sin(v)\\ &&- 96(s+v)(s-v)v^{5}\left( -v^{6}+\left( -\frac{29}{2}+s^{2}\right)v^{4}\right. +\left( -~\frac{31}{2}-\frac{29}{6}s^{2}\right)v^{2}-\frac{1}{2}s^{2}\\ &&-\left.\left.12\right)\right)\left( \cos(v)\right)^{6}-464\left( (s+v)(s-v)\left( \frac{7}{29}v^{4}+ \left( \frac{63}{29}+s^{2}\right)v^{2}+\frac{108}{29}-\frac{3}{29}s^{2}\right) v^{5}\sin(v)\right.\\ &&+ \frac{28}{29}v^{10}+\left( -\frac{17565}{116}-\frac{28}{29}s^{2} \right)v^{8}+\left( \frac{402437}{232}+\frac{17445}{58}s^{2}\right)v^{6}\\ &&+ \left( -\frac{234531}{116}s^{2}-\frac{951375}{464}-\frac{17325}{116}s^{4} \right)v^{4}+\left( \frac{104625}{ 232}s^{4}+\frac{50625}{116}+\frac {469125}{232}s^{2}\right)v^{2}\\ &&- \left.\frac{124875}{464}s^{2}\left( s^{2}+\frac{60}{37}\right)\right)v \left( \cos(v)\right)^{5}+\left( \left( -64v^{12}+ \left( 128s^{2}+960\right)v^{10}\right.\right.\\ &&+ \left( -~64s^{4}-960s^{2}-335925\right)v^{8}+\left( 495900s^{2}+915825\right)v^{6}\\ \end{array} $$
$$ \begin{array}{@{}rcl@{}} &&+ \left( -159975s^{4}-985950s^{2}-452250\right)v^{4}+\left( 172125s^{4}+479250s^{2}+20250\right)v^{2}\\ &&- \left.27000s^{4}-20250s^{2}\right)\sin(v)-16(s+v)(s-v)\left( 55v^{6}+\left( -56s^{2}+382\right)v^{4}\right.\\ &&+ \left.\left.\left( s^{4}+\frac{1005}{2}+\frac{21}{2}s^{2}\right)v^{2}+72+\frac {171}{2}s^{2}+3/2s^{4}\right)v^{5}\right)\left( \cos(v)\right)^{4}\\ &&+ 24\left( (s+v)(s-v)\left( -\frac{112}{3}v^{4}+\left( 11+\frac{361}{3}s^{2} \right)v^{2}+s^{4}+137s^{2}-12\right)v^{5}\sin(v)\right.\\ &&+ \frac{484}{3}v^{10}+ \left( -\frac{700}{3}s^{2}-{\frac {20863}{8}} \right) {v}^{8}+ \left( 72 {s}^{4}+{\frac {611387}{24}}+{\frac {19823}{4}} {s}^{2} \right) {v}^{6}\\ &&+ \left( -{\frac {136377}{4}} {s}^{2}-{\frac {18783}{ 8}} {s}^{4}-{\frac {220625}{8}} \right) {v}^{4}\\ &&+ \left.\left( {\frac { 101625}{8}} {s}^{4}+{\frac {109125}{4}} {s}^{2}+3375 \right) {v}^{2} -3375 {s}^{2}-{\frac {59625}{8}} {s}^{4} \right) v \left( \cos(v)\right)^{3}\\ &&+ \left( \left( 832 {v}^{12}+ \left( -1664 {s}^{2}-6624 \right) {v}^{10}+ \left( 832 {s}^{4}+7776 {s}^{2 }+86301 \right) {v}^{8}\right.\right.\\ &&+ \left( -1152 {s}^{4}-234801 {s}^{2}-341775 \right) {v}^{6}+ \left( 148500 {s}^{4}+437400 {s}^{2}+118125 \right) {v}^{4}\\ &&+ \left.\left( -185625 {s}^{4}-151875 {s}^{2}+10125 \right) {v}^{2}+33750 {s}^{4}-10125 {s}^{2} \right) \sin \left( v \right) \\ &&- 88 \left( s+v \right) \left( s-v \right) {v}^{5} \left( - {\frac {34}{11}} {v}^{6}+ \left( {\frac {23}{11}} {s}^{2}-{\frac { 667}{11}} \right) {v}^{4}\right.\\ &&+ \left.\left.\left( {s}^{4}-{\frac {903}{11}}+{\frac { 474}{11}} {s}^{2} \right) {v}^{2}+{\frac {36}{11}}-{\frac {225}{11}} {s}^{2}-{\frac {15}{11}} {s}^{4} \right) \right) \left( \cos(v)\right)^{2}\\ &&+ 120 v \left( \left( s+v \right) \left( s-v \right) \left( {\frac {186}{5}} {v}^{4}+ \left( -{\frac {251}{5}}-7 {s}^{2} \right) {v}^{2}+{s}^{4}+{\frac {156}{5}}-41 {s}^{2} \right) {v}^{5}\sin \left( v \right)\right.\\ &&- {\frac {536}{15}} {v}^{10}+ \left( {\frac {824}{15}} {s}^{2}-{\frac {191}{2}} \right) {v}^{8}+ \left( -{\frac {96}{5}} {s}^{4}-{\frac {4299}{10}}+{\frac {1139}{5}} {s}^{2} \right) {v}^{6}\\ &&+ \left( {\frac {5587}{5}} {s}^{2}-{\frac { 1323}{10}} {s}^{4}+{\frac {2025}{4}} \right) {v}^{4}+ \left( -{\frac {1775}{2}} {s}^{4}-{\frac {1875}{4}} {s}^{2}+{\frac {225}{2}} \right) {v}^{2}\\ &&- \left.{\frac {225}{2}} {s}^{2}+{\frac {1125}{2}} {s}^{4} \right) \cos(v) +32(s+v)(s-v)\left( \left( 24 {v}^{10}+ \left( -24 {s}^{2}-60 \right) {v}^{8}\right.\right.\\ &&+ \left( 72 {s}^{2}-{\frac {597}{2}} \right) {v}^{6}+ \left( - {\frac {1275}{16}}+{\frac {1275}{8}} {s}^{2} \right) {v}^{4}\\ &&+ \left.\left( {\frac {3375}{4}} {s}^{2}-{\frac {1125}{2}} \right) {v}^{2}-{\frac { 3375}{16}} {s}^{2}+{\frac {3375}{32}} \right) \sin \left( v \right) \\ &&+ \left.\left( 34 {v}^{6}+ \left( -35 {s}^{2}-107 \right) {v}^{4}+ \left( { s}^{4}+134 {s}^{2}-48 \right) {v}^{2}-3 {s}^{4}-15 {s}^{2}+9 \right) {v}^{5} \right), \end{array} $$
$$ \begin{array}{@{}rcl@{}} A_{10}&=&\left( s+v \right) \left( -{\frac {2925}{4}} \left( -{v}^{6} + \left( -{\frac {17}{39}}+{s}^{2} \right) {v}^{4} + \left( {\frac {265}{26}}-{\frac {175}{39}} {s}^{2} \right) {v}^{2} - { \frac {90}{13}}+{\frac {75}{26}} {s}^{2} \right) \!v \left( \cos \left( v \right) \right)^{6}\right.\\ &&+ \left( \left( -{\frac {12825}{8}} { v}^{6}+ \left( -{\frac {38175}{8}}+{\frac {20025}{8}} {s}^{2} \right) {v}^{4}\right.\right.\\ &&+ \left.\left( {\frac {32625}{4}}-{\frac {23625}{8}} {s}^{2 } \right) {v}^{2}-{\frac {3375}{2}}+{\frac {3375}{4}} {s}^{2} \right) \sin \left( v \right)\\ &&+ \left.12 {v}^{11}+ \left( 9-12 {s}^{2} \right) {v}^{9}+ \left( 51-41 {s}^{2} \right) {v}^{7}+ \left( -72 + 33 {s}^{2} \right) {v}^{5} \right) \left( \cos \left( v \right) \right)^{5}\\ \end{array} $$
$$ \begin{array}{@{}rcl@{}} &&+ \left( \left( 32 {v}^{9}+ \left( 90-32 {s}^{2} \right) {v}^{7}+ \left( 180-90 {s}^{2} \right) {v}^{5} \right) \sin \left( v \right)\right.\\ &&+ {v}^{10}+ \left( -2 {s}^{2}+7/2 \right) {v}^{8}+ \left( {s}^{4}-{\frac {1353}{16}}-21 {s}^{2} \right) {v}^{6}+ \left( -{\frac {242649}{16}}+3/2 {s}^{4}+{\frac {1737}{16}} {s}^{2} \right) {v}^{4}\\ &&+ \left.\left( -{\frac {57375}{16}} {s}^{2}+{\frac {941625} {32}} \right) {v}^{2}-{\frac {84375}{8}}+{\frac {97875}{32}} {s}^{2} \right) v \left( \cos \left( v \right) \right)^{4}\\ &&+ \left( \left( 5/2 {v}^{10}+ \left( 6-{s}^{2} \right) {v}^{8}+ \left( -{\frac { 153477}{32}}+18 {s}^{2}-3/2 {s}^{4} \right) {v}^{6}+ \left( {\frac { 796125}{32}}-{\frac {34875}{32}} {s}^{2} \right) {v}^{4}\right.\right.\\ &&+ \left.\left( -{ \frac {298125}{16}}+{\frac {70875}{32}} {s}^{2} \right) {v}^{2}-{ \frac {16875}{16}} {s}^{2}+{\frac {16875}{8}} \right) \sin \left( v \right)\\ && + \left.12 {v}^{11}+ \left( 151-12 {s}^{2} \right) {v}^{9}+ \left( 27 + 121 {s}^{2} \right) {v}^{7}+ \left( -39 {s}^{2}+108 \right) {v}^{5} \right) \left( \cos \left( v \right) \right)^{3}\\ &&+ 11/2 v \left( -{\frac {128}{11}} \left( 5/4 {v}^{4}+ \left( {s}^{2 }+{\frac {51}{32}} \right) {v}^{2}+{\frac {27}{8}}-{\frac {69}{32}} { s}^{2} \right) {v}^{5}\sin \left( v \right)\right.\\ &&+ {v}^{10}+ \left( -{\frac {7}{11}}-2 {s}^{2} \right) {v}^{8}+ \left( -{\frac {5739}{16}}+{s}^{4 }+{\frac {102}{11}} {s}^{2} \right) {v}^{6}\\ &&+ \left( -{\frac {15}{11}} {s}^{4}+{\frac {368523}{176}}+{\frac {56601}{176}} {s}^{2} \right) {v}^{4}+ \left( -{\frac {34875}{176}} {s}^{2}-{\frac {664875}{176}} \right) {v}^{2}\\ &&- \left.{\frac {23625}{176}} {s}^{2}+{\frac {23625}{22}} \right) \left( \cos \left( v \right) \right)^{2}+ \left( \left( - {\frac {59}{2}} {v}^{10}+ \left( -138 + 37 {s}^{2} \right) {v}^{8}\right.\right.\\ &&+ \left( {\frac {2691}{2}}-54 {s}^{2}-15/2 {s}^{4} \right) {v}^{6}+ \left( -{\frac {4725}{2}} {s}^{2}-{\frac {224925}{32}} \right) {v}^{ 4}\\ &&+ \left.\left( {\frac {106875}{16}}+{\frac {23625}{32}} {s}^{2} \right) { v}^{2}-{\frac {3375}{8}}+{\frac {3375}{16}} {s}^{2} \right) \sin \left( v \right)\\ && - \left.51 {v}^{11}+ \left( 51 {s}^{2}-37 \right) {v}^{9} + \left( -42-107 {s}^{2} \right) {v}^{7}+ \left( -36+6 {s}^{2} \right) {v}^{5} \right) \cos \left( v \right) \\ &&- 2 \left( -3 {v}^{5} \left( -13 {v}^{4}+ \left( 45+{s}^{2} \right) {v}^{2}-6+{s}^{2} \right) \sin \left( v \right) +{v}^{10}+ \left( 21-2 {s}^{2} \right) {v}^{8}\right.\\ &&+ \left( -{\frac {2727}{16}}+6 {s}^{2}+{s}^{4} \right) {v}^{6}+ \left( -3 {s}^{4}+{\frac {1575}{16}} {s}^{2}+{ \frac {4575}{16}} \right) {v}^{4}\\ &&+ \left.\left.\left( -{\frac {8625}{16}} {s}^{2} -{\frac {21375}{32}} \right) {v}^{2}+{\frac {3375}{32}} {s}^{2}+{ \frac {3375}{16}} \right) v \right) \left( s-v \right), \end{array} $$
$$ \begin{array}{@{}rcl@{}} A_{20}&=&\left( \left( -~{\frac {3375}{2}} {v}^{3}+{\frac {3375}{2}} v-2250 {v}^{5} \right) \left( \cos \left( v \right) \right)^{5}\right.\\ &&+ \left( \left( {\frac {7875}{4}} {v}^{2}-{\frac {4275} {4}} {v}^{6}-{\frac {975}{2}} {v}^{4}-{\frac {3375}{4}} \right) \sin \left( v \right)\right.\\ &&+ \left.{v}^{5} \!\left( - ~2 {v}^{6} + \left( {s}^{2} + 24 \right) {v}^{4} + \left( {\frac {267}{2}}+{s}^{4}+1/2 {s}^{2} \right) {v}^{2}+3/2 {s}^{4}-36-3/2 {s}^{2} \right) \right) \left( \cos \left( v \right) \right)^{4}\\ &&- 3/2 \left( {v}^{5} \left( 10 {v}^{4}+ \left( 21+{s}^{2} \right) {v}^{2}+{s}^{4}-84-{s}^ {2} \right) \sin \left( v \right)\right.\\ && + \left.4/3 {v}^{8}+2250+4 {v}^{6}-6750 {v}^{2}+5238 {v}^{4} \right) v \left( \cos \left( v \right) \right)^{3}\\ &&+ \left( \left( -{\frac {86625}{16}} {v}^{2}-6 {v}^{8}+{\frac { 16875}{16}}+{\frac {101625}{8}} {v}^{4}-{\frac {77751}{16}} {v}^{6} \right) \sin \left( v \right)\right.\\ \end{array} $$
$$ \begin{array}{@{}rcl@{}} &&+\!\left.11/2 \left( \!-2 {v}^{6} + \left( {s}^{2} + {\frac {217}{11}} \right) {v}^{4} + \left( -{\frac {26}{11}} {s}^{2} + {s}^{4}-45 \right) {v}^{2} - {\frac {15}{11}} {s}^{4} + {\frac {108}{ 11}}+{\frac {15}{11}} {s}^{2} \right) {v}^{5} \right) \left( \cos \left( v \right) \right)^{2}\\ &&- 15/2 \left( {v}^{5} \left( 10 {v}^{4}+ \left( {s}^{2}-{\frac {83}{5}} \right) {v}^{2}-{s}^{2}+{\frac {108 }{5}}+{s}^{4} \right) \sin \left( v \right)\right.\\ &&+ \left.{\frac {3825}{4}} {v}^{2 }+8/5 {v}^{6}+{\frac {56}{15}} {v}^{8}-{\frac {4488}{5}} {v}^{4}- 225 \right) v\cos \left( v \right) \\ &&+ \left( {\frac {4113}{4}} {v}^{6} +{\frac {12375}{8}} {v}^{2}-{\frac {30225}{8}} {v}^{4}-{\frac {3375} {16}}-24 {v}^{8} \right) \sin \left( v \right)\\ && - \left.2 \left( -2 {v}^{6 }+ \left( {s}^{2}+31 \right) {v}^{4}+ \left( {s}^{4}-4 {s}^{2}-66 \right) {v}^{2}-3 {s}^{4}+3 {s}^{2}+9 \right) {v}^{5} \right) \left( s+v \right) \left( s-v \right), \end{array} $$

and

$$ \begin{array}{@{}rcl@{}} A&=&v^{5}(2\cos(v)^{4}v^{2}-3\cos(v)^{3}\sin(v)v+3\cos(v)^{4}+11\cos(v)^{2}v^{2}\\ &&-15\cos(v)\sin(v)v-15\cos(v)^{2}-4v^{2}+12). \end{array} $$

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Khalsaraei, M.M., Shokri, A. An explicit six-step singularly P-stable Obrechkoff method for the numerical solution of second-order oscillatory initial value problems. Numer Algor 84, 871–886 (2020). https://doi.org/10.1007/s11075-019-00784-w

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