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A new family of three-stage two-step P-stable multiderivative methods with vanished phase-lag and some of its derivatives for the numerical solution of radial Schrödinger equation and IVPs with oscillating solutions

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Abstract

A new family of three-stage two-step methods are presented in this paper. These methods are of algebraic order 12 and have an important P-stability property. To make these methods, vanishing phase-lag and some of its derivatives have been used. The main structure of these methods are multiderivative, and the combined phases have been applied for expanding stability interval and for achieving P-stability. The advantage of the new methods in comparison with similar methods, in terms of efficiency, accuracy, and stability, has been showed by the implementation of them in some important problems, including the radial time-independent Schrödinger equation during the resonance problems with the use of the Woods-Saxon potential, undamped Duffing equation, etc.

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Acknowledgments

The authors wish to thank the Professor T. Mitsui for his careful reading the original draft of this article patiently and providing valuable feedback in order to correct it. The authors also express their gratitude to the anonymous referees who read the paper accurately and presented elaborate recommendations.

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Authors and Affiliations

  1. Faculty of Mathematical Science, University of Maragheh, Maragheh, Iran

    Ali Shokri, Mohammad Mehdizadeh Khalsaraei & Mortaza Tahmourasi

  2. IME Univeristy Salamanca, Salamanca, Spain

    Raquel Garcia-Rubio

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  1. Ali Shokri
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  2. Mohammad Mehdizadeh Khalsaraei
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  3. Mortaza Tahmourasi
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  4. Raquel Garcia-Rubio
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Correspondence to Ali Shokri.

Appendix: Coefficients of the first method (Method NMI)

Appendix: Coefficients of the first method (Method NMI)

$$\begin{array}{@{}rcl@{}} a_{1}&=&{\frac {14}{15}}+ 1/5\,(((276480\,{v}^{4}-19906560 ) \cos (v ) -24\,{v}^{12}-432\,{v}^{10}-1152\,{v}^{ 8}+ 209664\,{v}^{6}\\ &&-345600\,{v}^{4}-19491840\,{v}^{2}+ 59719680 ) (\sin (v ) )^{4}\\ &&+ 42\, (v ) ((-{\frac {12}{7}}\,{v}^{10}-{\frac {920}{7}}\,{v}^{8}+{ \frac {9984}{7}}\,{v}^{6}+{\frac {51840}{7}}\,{v}^{4}\\ &&-{\frac {933120}{ 7}}\,{v}^{2}+{\frac {1589760}{7}} ) \cos (v ) +{v}^{ 12}-{\frac {312}{7}}\,{v}^{10}\\ &&+{\frac {3688}{7}}\,{v}^{8}+ 4992\,{v}^{6 }-{\frac {708480}{7}}\,{v}^{4}+{\frac {2810880}{7}}\,{v}^{2}-{\frac { 3179520}{7}} ) (\sin (v ) )^{3}\\ &&+ ((6\,{v}^{14}-272\,{v}^{12}+ 4032\,{v}^{10}-2880\,{v}^{8} -574272\,{v}^{6}+ 8156160\,{v}^{4}\\ &&-40849920\,{v}^{2}+ 99532800 ) \cos (v ) -396\,{v}^{12}+ 19944\,{v}^{10}-269568\,{v}^{8}\\ &&+ 1744128\,{v}^{6}-9020160\,{v}^{4}+ 61171200\,{v}^{2}-139345920 ) (\sin (v ) )^{2}-36\, (v ) ((-{\frac {232}{3}}\,{v}^{10}\\ &&+{\frac {6400}{3}}\,{v}^{8} -25664\,{v}^{6}+ 147840\,{v}^{4}-472320\,{v}^{2}+ 529920\\ &&+{v}^{12} ) \cos (v ) -529920 + 7/6\,{v}^{12}-60\,{v}^{10}+{ \frac {1652}{3}}\,{v}^{8}+ 6944\,{v}^{6}\\ &&-121920\,{v}^{4}+ 472320\,{v}^{2 } ) \sin (v ) + (-6\,{v}^{14}\\ &&+ 272\,{v}^{12}- 4752\,{v}^{10}+ 27072\,{v}^{8}+ 352512\,{v}^{6}-7948800\,{v}^{4}+ 40642560\,{v}^{2}-79626240 ) \cos (v ) \\ \end{array} $$
$$\begin{array}{@{}rcl@{}} &&+ 168\,{v}^{12 }-6912\,{v}^{10}+ 119808\,{v}^{8}\\ &&-1251072\,{v}^{6}+ 7948800\,{v}^{4}- 40642560\,{v}^{2}+ 79626240 ) {v}^{-6} (((576 \,{v}^{4}-41472 ) \cos (v ) +{v}^{10}\\ &&-12\,{v}^{8}-24 \,{v}^{6}-288\,{v}^{4}-17280\,{v}^{2}+ 124416 ) (\sin (v ) )^{4}- (v ) (({v}^ {8}-60\,{v}^{6}\\ &&+ 360\,{v}^{4}+ 7776\,{v}^{2}-38016 ) \cos (v ) + 6\,{v}^{8}-24\,{v}^{6}+ 1512\,{v}^{4}-28224\,{v}^{2}\\ &&+ 76032 ) (\sin (v ) )^{3}+ ((18\,{v}^{8}-588\,{v}^{6}+ 4464\,{v}^{4}-46656\,{v}^{2}+ 207360 ) \cos (v ) + 600\,{v}^{6}\\&&-5616\,{v}^{4}+ 70848\,{v}^{2}\\ &&- 290304 ) (\sin (v ) )^{2}+ (v ) (({v}^{8}+ 12\,{v}^{6}-2664\,{v}^{4}+ 29376\,{v}^ {2}-76032 ) \cos (v ) -6\,{v}^{8}+ 144\,{v}^{6}\\ &&+ 1800 \,{v}^{4}-29376\,{v}^{2}+ 76032 ) \sin (v ) + (36\,{v}^{8}-504\,{v}^{6}-2304\,{v}^{4}+ 48384\,{v}^{2}-165888 ) \cos (v )\\ && -{v}^{10}+ 12\,{v}^{8}-72\,{v}^{6}+ 2304\,{v}^{4}- 48384\,{v}^{2}+ 165888 )^{-1}\\ &&+ 1/5\, (((192\, {v}^{6}+ 11520\,{v}^{4}-276480 ) \cos (v ) + 6\,{v}^{ 10}-160\,{v}^{8}+ 2208\,{v}^{6}-20160\,{v}^{4}-178560\,{v}^{2}\\ &&+ 552960 ) (\sin (v ) )^{2}-30\, (v ) (({v}^{8}+ 80\,{v}^{4}+ 1824\,{v}^{2}-{\frac {168 }{5}}\,{v}^{6}-2880 ) \cos (v ) + 440\,{v}^{4}\\ &&-2496\, {v}^{2}+ 2880 ) \sin (v ) + (-1392\,{v}^{6}+ 23040\,{v}^{4}-190080\,{v}^{2}+ 552960 ) \cos (v ) \\ &&+ 6 \,{v}^{10}-64\,{v}^{8}-3168\,{v}^{6}+ 14400\,{v}^{4}+ 190080\,{v}^{2}- 552960 ) {v}^{-6} (({v}^{6}-12\,{v}^{4}+ 48\,{v}^{2}\\ && + 576\,\cos (v ) -1152 ) (\sin (v ) )^{2}- (v ) (({v}^{4}-60\,{ v}^{2}+ 432 ) \cos (v ) + 84\,{v}^{2}-432 ) \sin (v )\\ && + (-12\,{v}^{4}+ 144\,{v}^{2}-1152 ) \cos (v ) +{v}^{6}-12\,{v}^{4}-144\,{v}^{2}+ 1152 )^{-1} \end{array} $$
$$\begin{array}{@{}rcl@{}} a_{2}&=&1/30\, ((v ) (-19906560\,{v}^{2} ({v}^ {4}-72 ) (\cos (v ) )^{2}\\ &&+ 576\,{v}^{2 } ({v}^{12}-90\,{v}^{10}+ 1248\,{v}^{8}-2880\,{v}^{6}+ 43200\,{v}^ {4}+ 1503360\,{v}^{2}-12441600 ) \cos (v ) \\ &&+{v}^{20}- 42\,{v}^{18}+ 936\,{v}^{16}-13248\,{v}^{14}+ 123840\,{v}^{12}-203904\,{v }^{10}-12897792\,{v}^{8}\\ &&+ 201553920\,{v}^{6}-2233516032\,{v}^{4}+ 2508226560\,{v}^{2}+ 19349176320 ) (\sin (v ) )^{6}+ (-576\,{v}^{4} ({v}^{10}\\ &&-210\,{v}^{8}+ 6768 \,{v}^{6}-34560\,{v}^{4}-699840\,{v}^{2}+ 3991680 ) (\cos (v ) )^{2}-2\,{v}^{4} ({v}^{16}\\ &&-126\,{v}^{ 14}+ 3888\,{v}^{12}-52416\,{v}^{10}+ 274752\,{v}^{8}-611712\,{v}^{6}- 81699840\,{v}^{4}+ 1420830720\,{v}^{2}\\ &&-4598415360 ) \cos (v ) -6\,{v}^{20}-6\,{v}^{18}+ 3168\,{v}^{16}+ 23472\,{v}^{14}- 2452032\,{v}^{12}+ 41423616\,{v}^{10}\\&&-458307072\,{v}^{8}\\ &&+ 2518677504\,{v }^{6}-507617280\,{v}^{4}-18632540160\,{v}^{2}+ 15049359360 ) (\sin (v ) )^{5}\\&&+ (v ) (({v}^{18}-210\,{v}^{16}\\ &&+ 12600\,{v}^{14}-281664\,{v}^{12}+ 1318464\,{v}^{10}+ 38683008\,{v}^{8}\\ &&-427368960\,{v}^{6}+ 2730682368\,{v}^{4}-16124313600\,{v}^{2}+ 19349176320 ) (\cos (v ) )^{2}\\ &&-6\,{v}^{2} ({v}^{16}+ 87\,{v}^{14}-2892\,{ v}^{12}+ 78360\,{v}^{10}-2782080\,{v}^{8}+ 54086400\,{v}^{6}\\ &&-419800320\, {v}^{4}+ 2118804480\,{v}^{2}-5971968000 ) \cos (v ) + {v}^{20}-42\,{v}^{18}+ 2748\,{v}^{16}-80784\,{v}^{14}\\ &&+ 1310112\,{v}^{12} -22400064\,{v}^{10}\\ &&+ 311071104\,{v}^{8}-2357061120\,{v}^{6}+ 12943494144 \,{v}^{4}-18632540160\,{v}^{2}-42998169600 ) (\sin (v ) )^{4}\\ &&+ ((12\,{v}^{18}+ 528\,{v}^ {16}-78696\,{v}^{14}+ 1943136\,{v}^{12}-28206144\,{v}^{10}+ 481966848\,{ v}^{8}-4885318656\,{v}^{6}\\ &&+ 17886044160\,{v}^{4}-18632540160\,{v}^{2}+ 15049359360 ) (\cos (v ) )^{2}\\ &&+ 252\,{ v}^{4} ({v}^{14}-{\frac {1297}{21}}\,{v}^{12}+{\frac {16066}{7}} \,{v}^{10}-49272\,{v}^{8}+{\frac {4620624}{7}}\,{v}^{6}\\ &&-{\frac { 52156800}{7}}\,{v}^{4}+{\frac {342627840}{7}}\,{v}^{2}-109486080 ) \cos (v ) -12\,{v}^{20}\\ &&+ 528\,{v}^{18}-9072\,{v}^{ 16}-72432\,{v}^{14}+ 5008608\,{v}^{12}\\ &&-90298368\,{v}^{10}+ 1276985088\,{ v}^{8}-7268133888\,{v}^{6}+ 8390615040\,{v}^{4}+ 20782448640\,{v}^{2}\\ &&- 15049359360 ) (\sin (v ) )^{3}- (v ) (({v}^{18}-30\,{v}^{16}-13104\,{v}^{14 }+ 1031472\,{v}^{12}\\ &&-27824256\,{v}^{10}+ 336866688\,{v}^{8}-2049753600\, {v}^{6}+ 7355225088\,{v}^{4}-24007311360\,{v}^{2}\\ &&+ 23648993280 ) (\cos (v ) )^{2}-60\,{v}^{2} ({v}^{16 }-{\frac {661}{10}}\,{v}^{14}\\ \end{array} $$
$$\begin{array}{@{}rcl@{}} &&+ 1398\,{v}^{12}+{\frac {50652}{5}}\,{v}^{ 10}-795312\,{v}^{8}+ 11524032\,{v}^{6}-83441664\,{v}^{4}+ 386684928\,{v}^{2}\\&&-859963392 ) \cos (v ) \\ &&+{v}^{20}-42\,{v}^{18}- 828\,{v}^{16}+ 77616\,{v}^{14}-954144\,{v}^{12}-14482368\,{v}^{10}+ 342921600\,{v}^{8}\\&&-3297894912\,{v}^{6}\\ &&+ 19190172672\,{v}^{4}- 34040217600\,{v}^{2}-23648993280 ) (\sin (v ) )^{2}\\ &&+ 2\,{v}^{2} ((-33\,{v}^{16}+ 3252\,{ v}^{14}-58104\,{v}^{12}-982368\,{v}^{10}+ 40290048\,{v}^{8}-470956032\, {v}^{6}\\ &&+ 2494291968\,{v}^{4}-5255331840\,{v}^{2}+ 1074954240 ) (\cos (v ) )^{2}+{v}^{2} ({v}^{16}- 7716\,{v}^{12}\\ &&+ 266976\,{v}^{10}-1651104\,{v}^{8}-46738944\,{v}^{6}+ 823219200\,{v}^{4}-4807434240\,{v}^{2}\\ &&+ 9196830720 ) \cos (v ) -3\,{v}^{18}+ 267\,{v}^{16}-6120\,{v}^{14}-3672\,{v}^{12}+ 1233792\,{v}^{10}+ 6448896\,{v}^{8}\\&&-352263168\,{v}^{6}\\ &&+ 2313142272\,{v}^{4}-3941498880\,{v}^{2}-1074954240 ) \sin (v ) -{v}^{3} ((648\,{v}^{14}\\ &&-23616\,{v}^{12}+ 245376\,{v}^{10}+ 957312\,{v}^{8}-41928192\,{v}^{6}+ 410323968\,{v}^{4}\\ &&-1791590400\,{v}^{2}+ 5016453120 ) (\cos (v ) )^{2}+ (-66\,{v}^{16}+ 2904\,{v}^{14}\\ &&-41616\,{v}^{12}-108864\,{v}^{10}+ 4555008\,{v}^{8}+ 60549120\,{v}^{6}-1502945280\,{v}^{4}\\ &&+ 10271784960\,{v }^{2}-22932357120 ) \cos (v ) +{v}^{18}-42\,{v}^{16} + 984\,{v}^{14}-21168\,{v}^{12}\\ &&+ 485568\,{v}^{10}-5512320\,{v}^{8}- 18620928\,{v}^{6}+ 1092621312\,{v}^{4}\\ &&-8480194560\,{v}^{2}+ 17915904000 ) ) {v}^{-5} (((576\,{v}^{4}-41472 ) \cos (v ) +{v}^{10}-12\,{v}^{8}\\&&-24\,{v}^{6}-288\, {v}^{4}-17280\,{v}^{2}\\ &&+ 124416 ) (\sin (v ) )^{4}- (v ) (({v}^{8}-60\,{v}^{6}+ 360\,{v}^{4}+ 7776\,{v}^{2}-38016 ) \cos (v ) \\&&+ 6\,{v}^{8}-24\,{v}^{6}+ 1512\,{v}^{4}-28224\,{v}^{2}\\ &&+ 76032 ) (\sin (v ) )^{3}+ ((18\,{v}^{8}-588\,{ v}^{6}+ 4464\,{v}^{4}\\ &&-46656\,{v}^{2}+ 207360 ) \cos (v ) + 600\,{v}^{6}-5616\,{v}^{4}+ 70848\,{v}^{2}\\ &&-290304 ) (\sin (v ) )^{2}+ (v ) (({v}^{8}+ 12\,{v}^{6}-2664\,{v}^{4}+ 29376\,{v}^{2}-76032 ) \cos (v ) \\&&-6\,{v}^{8}+ 144\,{v}^{6}\\ &&+ 1800\,{v}^{4}- 29376\,{v}^{2}+ 76032 ) \sin (v ) + (36\,{v}^{8 }-504\,{v}^{6}-2304\,{v}^{4}\\ &&+ 48384\,{v}^{2}-165888 ) \cos (v ) -{v}^{10}+ 12\,{v}^{8}-72\,{v}^{6}\\ &&+ 2304\,{v}^{4}- 48384\,{v}^{2}+ 165888 )^{-1} (({v}^{6}-12\,{v}^{4} + 48\,{v}^{2}+ 576\,\cos (v ) \\&&-1152 ) (\sin (v ) )^{2}\\ &&- (v ) (({v}^{4}-60\,{v}^{2}+ 432 ) \cos (v ) + 84\,{v}^{2}-432 ) \sin (v ) \\ &&+ (-12\,{v}^{4}+ 144\,{v}^{2}-1152 ) \cos (v ) +{v}^{6}-12\,{v}^{4}-144\,{v}^{2}+ 1152 )^{-1} , \end{array} $$
$$\begin{array}{@{}rcl@{}} a_{3}&=&1/20\, ((v ) (552960\,{v}^{2} ({v}^{2} + 24 ) ({v}^{4}-72 ) (\cos (v ) )^{2}+ 1536\,{v}^{2} ({v}^{12}\\ &&+ 15/2\,{v}^{10}-342\,{v}^{8 }+ 360\,{v}^{6}-23760\,{v}^{4}-246240\,{v}^{2}+ 3110400 ) \cos (v ) +{v}^{20}\\&&-12\,{v}^{18}-344\,{v}^{16}\\ &&+ 5952\,{v}^{14}- 84480\,{v}^{12}+ 711936\,{v}^{10}+ 5280768\,{v}^{8}-91238400\,{v}^{6}+ 1250131968\,{v}^{4}\\&&-1672151040\,{v}^{2}\\ &&-12899450880 ) (\sin (v ) )^{6}+ (-1536\,{v}^{4} ({v}^ {10}-{\frac {45}{2}}\,{v}^{8}-1242\,{v}^{6}+ 16200\,{v}^{4}\\ &&+ 131760\,{v}^{2}-997920 ) (\cos (v ) )^{2}-2\,{v}^{4} ({v}^{16}-36\,{v}^{14}-712\,{v}^{12}+ 33984\,{v}^{10}-200448 \,{v}^{8}\\ &&+ 2135808\,{v}^{6}+ 16588800\,{v}^{4}\\ &&-814510080\,{v}^{2}+ 3065610240 ) \cos (v ) -6\,{v}^{20}+ 24\,{v}^{18}- 3192\,{v}^{16}+ 69792\,{v}^{14}+ 799488\,{v}^{12}\\ &&-18722304\,{v}^{10}+ 215405568\,{v}^{8}-1443557376\,{v}^{6}+ 338411520\,{v}^{4}\\ &&+ 12421693440 \,{v}^{2}-10032906240 ) (\sin (v ) )^ {5}\\ \end{array} $$
$$\begin{array}{@{}rcl@{}} &&+ (v ) (({v}^{18}-60\,{v}^{16}-1080\,{v}^ {14}+ 127296\,{v}^{12}\\ &&-1886976\,{v}^{10}-12379392\,{v}^{8}+ 202798080\,{ v}^{6}-1541763072\,{v}^{4}+ 10749542400\,{v}^{2}\\ &&-12899450880 ) (\cos (v ) )^{2}+ 24\,{v}^{2} ({v}^{16 }-{\frac {253}{6}}\,{v}^{14}+ 243\,{v}^{12}-1900\,{v}^{10}\\ &&-116880\,{v}^ {8}+ 6312960\,{v}^{6}-56246400\,{v}^{4}+ 311662080\,{v}^{2}-995328000 ) \cos (v ) +{v}^{20}\\ &&-12\,{v}^{18}-32\,{v}^{16}+ 29616\,{v}^{14}-406368\,{v}^{12}+ 6967296\,{v}^{10}\\ &&-153052416\,{v}^{8}+ 1294341120\,{v}^{6}-7832733696\,{v}^{4}+ 12421693440\,{v}^{2}+ 28665446400 ) (\sin (v ) )^{4}\\ &&+ ((-18\,{v}^{18}+ 988\,{v}^{16}-2256\,{v}^{14}-382464\,{v}^{12}+ 6086016\,{v}^{10}-214728192\,{v}^{8}+ 2961598464\,{v}^{6}\\ &&- 11924029440\,{v}^{4}+ 12421693440\,{v}^{2}-10032906240 ) (\cos (v ) )^{2}\\ &&+ 72\,{v}^{4} ({v}^{14}-17\,{v }^{12}-{\frac {4166}{3}}\,{v}^{10}+ 66008\,{v}^{8}-933488\,{v}^{6}\\ &&+ 12804480\,{v}^{4}-103150080\,{v}^{2}+ 255467520 ) \cos (v ) -12\,{v}^{20}\\ &&+ 168\,{v}^{18}+ 4128\,{v}^{16}-37632\,{v}^{14}- 1821312\,{v}^{12}+ 30753792\,{v}^{10}\\ &&-624748032\,{v}^{8}+ 4344440832\,{v }^{6}-5593743360\,{v}^{4}-13854965760\,{v}^{2}+ 10032906240 ) (\sin (v ) )^{3}\\ &&- (v ) (({v}^{18}+ 300\,{v}^{16}-12264\,{v}^{14}-133248\,{v}^{12}+ 10917504\,{v}^{10}-169696512\,{v}^{8}+ 1125964800\,{v}^{6}\\ &&-4346099712\, {v}^{4}+ 16004874240\,{v}^{2}-15765995520 ) (\cos (v ) )^{2}-60\,{v}^{2} ({v}^{16}\\ &&-{\frac {103}{5}}\,{ v}^{14}-502\,{v}^{12}+{\frac {32952}{5}}\,{v}^{10}+ 267552\,{v}^{8}- 5622912\,{v}^{6}+ 45508608\,{v}^{4}-233902080\,{v}^{2}\\ &&+ 573308928 ) \cos (v ) +{v}^{20}-12\,{v}^{18}-848\,{v}^{16}- 14064\,{v}^{14}+ 853536\,{v}^{12}+ 2080512\,{v}^{10}\\ &&-163226880\,{v}^{8}+ 1831984128\,{v}^{6}-11917559808\,{v}^{4}+ 22693478400\,{v}^{2}+ 15765995520 ) (\sin (v ) )^{2}\\ &&+ 2\,{v}^{2} ((-18\,{v}^{16}+ 52\,{v}^{14}+ 29616\,{v}^{12}+ 55872\, {v}^{10}-16630272\,{v}^{8}+ 250933248\,{v}^{6}-1530150912\,{v}^{4}\\ &&+ 3503554560\,{v}^{2}-716636160 ) (\cos (v ) )^{2}+{v}^{2} ({v}^{16}-976\,{v}^{12}-83904\,{v}^{10}\\ &&+ 1607616\,{v}^{8}+ 12220416\,{v}^{6}-419420160\,{v}^{4}\\ &&+ 2939535360\,{v}^{2}-6131220480 ) \cos (v ) -3\,{v}^{18}+ 72\,{v}^{16} + 3660\,{v}^{14}-53712\,{v}^{12}\\ &&-730368\,{v}^{10}+ 4409856\,{v}^{8}+ 168486912\,{v}^{6}-1409384448\,{v}^{4}+ 2627665920\,{v}^{2}\\ &&+ 716636160 ) \sin (v ) -{v}^{3} ((288\,{v}^{14}- 2976\,{v}^{12}-19584\,{v}^{10}-85248\,{v}^{8}\\ &&+ 8598528\,{v}^{6}- 140838912\,{v}^{4}+ 875888640\,{v}^{2}\\ &&-3344302080 ) (\cos (v ) )^{2}+ (-36\,{v}^{16}+ 104\,{v}^{14}+ 23904\,{v}^{12}-157824\,{v}^{10}-3036672\,{v}^{8}\\ &&-8294400\,{v}^{6}+ 736542720\,{v}^{4}-6210846720\,{v}^{2}+ 15288238080 ) \cos (v ) +{v}^{18}\\ &&-12\,{v}^{16}-536\,{v}^{14}+ 13632\,{v}^{12} -237312\,{v}^{10}+ 3121920\,{v}^{8}\\ &&-304128\,{v}^{6}-595703808\,{v}^{4}+ 5334958080\,{v}^{2}-11943936000 ) ) {v}^{-5} (((576\,{v}^{4}-41472 ) \cos (v )\\ && +{v}^{10}-12\,{v}^{8}-24\,{v}^{6}-288\,{v}^{4}-17280\,{v}^{2}+ 124416 ) (\sin (v ) )^{4}- (v ) (({v}^{8}\\ &&-60\,{v}^{6}+ 360\,{v}^{4}+ 7776\,{v}^{2 }-38016 ) \cos (v ) + 6\,{v}^{8}-24\,{v}^{6}+ 1512\,{v }^{4}-28224\,{v}^{2}\\ &&+ 76032 ) (\sin (v ) )^{3}+ ((18\,{v}^{8}-588\,{v}^{6}+ 4464\,{v}^{4}- 46656\,{v}^{2}+ 207360 ) \cos (v ) \\ &&+ 600\,{v}^{6}-5616 \,{v}^{4}+ 70848\,{v}^{2}-290304 ) (\sin (v ) )^{2}+ (v ) (({v}^{8}\\ &&+ 12\,{v}^{6}- 2664\,{v}^{4}+ 29376\,{v}^{2}-76032 ) \cos (v ) -6\,{ v}^{8}+ 144\,{v}^{6}+ 1800\,{v}^{4}-29376\,{v}^{2}\\&&+ 76032 ) \sin (v ) + (36\,{v}^{8}\\ &&-504\,{v}^{6}-2304\,{v}^{4}+ 48384\,{v}^{2}-165888 ) \cos (v ) -{v}^{10}+ 12\,{v}^ {8}-72\,{v}^{6}+ 2304\,{v}^{4}-48384\,{v}^{2}\\ &&+ 165888 )^{-1} (({v}^{6}-12\,{v}^{4}+ 48\,{v}^{2}+ 576\,\cos (v ) -1152 ) (\sin (v ) )^{2} - (v ) (({v}^{4}-60\,{v}^{2}+ 432 ) \cos (v ) + 84\,{v}^{2}\\ &&-432 ) \sin (v ) + (-12\,{v}^{4}+ 144\,{v}^{2}-1152 ) \cos (v ) + {v}^{6}-12\,{v}^{4}-144\,{v}^{2}+ 1152 )^{-1} , \end{array} $$
$$\begin{array}{@{}rcl@{}} a_{4}&=&{\frac {1}{240}}\, (((138240\,{v}^{4}-9953280 ) \cos (v ) -12\,{v}^{12}-216\,{v}^{10}\\ &&-576\,{v}^{8 }+ 104832\,{v}^{6}-172800\,{v}^{4}-9745920\,{v}^{2}+ 29859840 ) (\sin (v ) )^{4}+ 21\, (v ) ((-{\frac {12}{7}}\,{v}^{10}\\ \end{array} $$
$$\begin{array}{@{}rcl@{}} &&-{\frac {920}{7}}\,{v}^{8}+{ \frac {9984}{7}}\,{v}^{6}+{\frac {51840}{7}}\,{v}^{4}-{\frac {933120}{ 7}}\,{v}^{2}+{\frac {1589760}{7}} ) \cos (v )\\ && +{v}^{12}-{\frac {312}{7}}\,{v}^{10}+{\frac {3688}{7}}\,{v}^{8}+ 4992\,{v}^{6 }-{\frac {708480}{7}}\,{v}^{4}+{\frac {2810880}{7}}\,{v}^{2}\\ &&-{\frac { 3179520}{7}} ) (\sin (v ) )^{3}+ ((3\,{v}^{14}-136\,{v}^{12}+ 2016\,{v}^{10}-1440\,{v}^{8} -287136\,{v}^{6}+ 4078080\,{v}^{4}\\ &&-20424960\,{v}^{2}+ 49766400 ) \cos (v ) -198\,{v}^{12}+ 9972\,{v}^{10}-134784\,{v}^{8}+ 872064\,{v}^{6}\\ &&-4510080\,{v}^{4}+ 30585600\,{v}^{2}-69672960 ) (\sin (v ) )^{2}-18\, (v ) ((-{\frac {232}{3}}\,{v}^{10}\\ &&+{\frac {6400}{3}}\,{v}^{8} -25664\,{v}^{6}+ 147840\,{v}^{4}-472320\,{v}^{2}+ 529920+{v}^{12} ) \cos (v ) -529920 + 7/6\,{v}^{12}\\ &&-60\,{v}^{10}+{ \frac {1652}{3}}\,{v}^{8}+ 6944\,{v}^{6}-121920\,{v}^{4}+ 472320\,{v}^{2 } ) \sin (v ) + (-3\,{v}^{14}\\ &&+ 136\,{v}^{12}- 2376\,{v}^{10}+ 13536\,{v}^{8}+ 176256\,{v}^{6}-3974400\,{v}^{4}\\ &&+ 20321280\,{v}^{2}-39813120 ) \cos (v )+ 84\,{v}^{12} -3456\,{v}^{10}+ 59904\,{v}^{8}-625536\,{v}^{6}\\ &&+ 3974400\,{v}^{4}- 20321280\,{v}^{2}+ 39813120 ) {v}^{-6} (((576 \,{v}^{4}-41472 ) \cos (v ) \\ &&+{v}^{10}-12\,{v}^{8}-24 \,{v}^{6}-288\,{v}^{4}-17280\,{v}^{2}+ 124416 ) (\sin (v ) )^{4}- (v ) (({v}^ {8}-60\,{v}^{6}+ 360\,{v}^{4}\\ &&+ 7776\,{v}^{2}-38016 ) \cos (v ) + 6\,{v}^{8}-24\,{v}^{6}+ 1512\,{v}^{4}\\ &&-28224\,{v}^{2}+ 76032 ) (\sin (v ) )^{3}+ ((18\,{v}^{8}-588\,{v}^{6}+ 4464\,{v}^{4}-46656\,{v}^{2}\\ &&+ 207360 ) \cos (v ) + 600\,{v}^{6}-5616\,{v}^{4}+ 70848\,{v}^{2}- 290304 ) (\sin (v ) )^{2}\\ &&+ (v ) (({v}^{8}+ 12\,{v}^{6}-2664\,{v}^{4}+ 29376\,{v}^ {2}-76032 ) \cos (v ) -6\,{v}^{8}\\ &&+ 144\,{v}^{6}+ 1800 \,{v}^{4}-29376\,{v}^{2}+ 76032 ) \sin (v ) + (36\,{v}^{8}-504\,{v}^{6}-2304\,{v}^{4}+ 48384\,{v}^{2}\\ &&-165888 ) \cos (v ) -{v}^{10}+ 12\,{v}^{8}-72\,{v}^{6}+ 2304\,{v}^{4}- 48384\,{v}^{2}+ 165888 )^{-1} , \end{array} $$
$$\begin{array}{@{}rcl@{}} a_{5}&=&{\frac {1}{60}}\, (((96\,{v}^{6}+ 5760\,{v}^{4}- 138240 ) \cos (v ) + 3\,{v}^{10}-80\,{v}^{8}+ 1104\,{v }^{6}-10080\,{v}^{4}-89280\,{v}^{2}\\ &&+ 276480 ) (\sin (v ) )^{2}-15\, (v ) (({v}^{8}+ 80\,{v}^{4}+ 1824\,{v}^{2}\\ &&-{\frac {168}{5}}\,{v}^{6}-2880 ) \cos (v ) + 440\,{v}^{4}-2496\,{v}^{2}+ 2880 ) \sin (v ) + (-696\,{v}^{6}\\ &&+ 11520\,{v}^{4}- 95040\,{v}^{2}+ 276480 ) \cos (v ) + 3\,{v}^{10}-32\,{ v}^{8}-1584\,{v}^{6}+ 7200\,{v}^{4}+ 95040\,{v}^{2}\\ &&-276480 ) (({v}^{6}-12\,{v}^{4}+ 48\,{v}^{2}+ 576\,\cos (v ) -1152 ) (\sin (v ) )^{2}\\ &&- (v ) (({v}^{4}-60\,{v}^{2}+ 432 ) \cos (v ) + 84\,{v}^{2}-432 ) \sin (v )\\ && + (-12\,{v}^{4}+ 144\,{v}^{2}-1152 ) \cos (v ) + {v}^{6}-12\,{v}^{4}-144\,{v}^{2}+ 1152 )^{-1}{v}^{-6} , \end{array} $$
$$\begin{array}{@{}rcl@{}} a_{6}&=&{\frac {1}{240}}\, (((-39813120\,{v}^{4}+ 2866544640 ) \cos (v ) + 4608\,{v}^{12}-884736\,{v}^{ 8}\\ &&-23224320\,{v}^{6}+ 238878720\,{v}^{4}+ 2627665920\,{v}^{2}- 17199267840 ) (\sin (v ) )^{8}+ 32\, (v ) ((720\,{v}^{10}\\ &&+ 43200\,{v}^{8}-570240 \,{v}^{6}-2177280\,{v}^{4}+ 44789760\,{v}^{2}-82114560 ) \cos (v )\\ && +{v}^{16}+{\frac {135}{2}}\,{v}^{14}-1512\,{v}^{12}+ 9396\,{v}^{10}-196992\,{v}^{8}+ 233280\,{v}^{6}\\ &&+ 36495360\,{v}^{4}- 264384000\,{v}^{2}+ 410572800 ) (\sin (v ) )^{7}+ ((-8\,{v}^{18}-64\,{v}^{16}+ 2424\,{v}^{14} \\ &&+ 219456\,{v}^{12}-3294144\,{v}^{10}-9234432\,{v}^{8}+ 228925440\,{v}^{6 }-1388482560\,{v}^{4}+ 13974405120\,{v}^{2}\\ &&-54464348160 ) \cos (v ) -3\,{v}^{20}+ 157\,{v}^{18}-3308\,{v}^{16}+ 42432\,{v}^{14}-639360\,{v}^{12}-3669120\,{v}^{10}\\&&+ 196584192\,{v}^{8}\\ \end{array} $$
$$\begin{array}{@{}rcl@{}} &&-1534464000 \,{v}^{6}+ 6484561920\,{v}^{4}-40609382400\,{v}^{2}+ 126127964160 ) (\sin (v ) )^{6}\\ &&+ 12\, (({v}^{18}-76\,{v}^{16}+ 1976\,{v}^{14}-13244\,{v}^{12}-164616\,{ v}^{10}+ 2408256\,{v}^{8}\\ &&-31576320\,{v}^{6}+ 396956160\,{v}^{4}- 1990656000\,{v}^{2}+ 2846638080 ) \cos (v ) + 95915520 \,{v}^{6}\\&&+ 3/2\,{v}^{18}-900357120\,{v}^{4}\\ &&-{\frac {469}{6}}\,{v}^{16}+ 3982970880\,{v}^{2}+ 2171\,{v}^{14}-5474304000-62280\,{v}^{12}+ 1084728 \,{v}^{10}\\&&-9859104\,{v}^{8} ) (v ) (\sin (v ) )^{5}\\ &&+ ((-116\,{v}^{18}+ 7412\,{ v}^{16}-158856\,{v}^{14}+ 2171232\,{v}^{12}-41693760\,{v}^{10}+ 542688768\,{v}^{8}\\&&-3327713280\,{v}^{6}\\ &&+ 14875176960\,{v}^{4}- 74769039360\,{v}^{2}+ 189191946240 ) \cos (v ) + 3\,{v }^{20}-166\,{v}^{18}\\&&+ 2152\,{v}^{16}+ 54816\,{v}^{14}\\ &&-2298528\,{v}^{12}+ 61729920\,{v}^{10}-815885568\,{v}^{8}+ 5315051520\,{v}^{6}-25455513600 \,{v}^{4}\\&&+ 123978055680\,{v}^{2}\\ &&-292387553280 ) (\sin (v ) )^{4}-6\, (v ) ((- 2401296\,{v}^{10}+ 27104832\,{v}^{8}-301870080\,{v}^{6}\\&&+{v}^{18}+ 2421135360\,{v}^{4}\\ &&-{\frac {232}{3}}\,{v}^{16}-9382625280\,{v}^{2}- 1012\,{v}^{14}+ 12262440960 + 121552\,{v}^{12} ) \cos (v ) \\&&-1778\,{v}^{14}-3236198400\,{v}^{4}\\ &&+{\frac {277}{3}}\,{v}^{16 }+ 12209356800\,{v}^{2}+ 2473584\,{v}^{10}-15765995520-71456\,{v}^{12}+ 410664960\,{v}^{6}\\&&-35445312\,{v}^{8} ) (\sin (v ) )^{3}\\ &&+ ((-100\,{v}^{18}+ 6404\,{v}^{16}- 104592\,{v}^{14}-1049184\,{v}^{12}+ 52842240\,{v}^{10}-645954048\,{v}^{ 8}\\ &&+ 4191989760\,{v}^{6}-23589273600\,{v}^{4}+ 110839726080\,{v}^{2}- 229323571200 ) \cos (v )\\ && + 3\,{v}^{20}-139\,{v}^{18}+ 3844\,{v}^{16}-127296\,{v}^{14}+ 3260448\,{v}^{12}\\ &&-62138880\,{v}^{10}+ 671721984\,{v}^{8}-4626616320\,{v}^{6}+ 28446474240\,{v}^{4}- 135683112960\,{v}^{2}\\ &&+ 275188285440 ) (\sin (v ) )^{2}-6\, ((841536\,{v}^{10}-14676480\,{ v}^{8}+ 217589760\,{v}^{6}+{v}^{18}\\&&-1630126080\,{v}^{4}\\ &&-{\frac {320}{3} }\,{v}^{16}+ 5653463040\,{v}^{2}+ 4176\,{v}^{14}-7007109120-72752\,{v}^{ 12} ) \cos (v ) \\&&-217589760\,{v}^{6}+ 3\,{v}^{18}\\ &&+ 1630126080\,{v}^{4}-{\frac {730}{3}}\,{v}^{16}-5653463040\,{v}^{2}+ 6480\,{v}^{14}+ 7007109120-61168\,{v}^{12}\\&&-254016\,{v}^{10}\\ &&+ 14676480\,{ v}^{8} ) (v ) \sin (v ) + (116\,{ v}^{18}-5520\,{v}^{16}+ 73632\,{v}^{14}-87552\,{v}^{12}-4078080\,{v}^{ 10}\\ &&+ 51535872\,{v}^{8}-869253120\,{v}^{6}+ 9714401280\,{v}^{4}\\ &&- 49686773760\,{v}^{2}+ 91729428480 ) \cos (v ) -3\,{v} ^{20}+ 148\,{v}^{18}-2688\,{v}^{16}+ 30048\,{v}^{14}\\ &&-327168\,{v}^{12}+ 4078080\,{v}^{10}-51535872\,{v}^{8}+ 869253120\,{v}^{6}-9714401280\,{v} ^{4}+ 49686773760\,{v}^{2}\\ &&-91729428480 ) {v}^{-8} (((-331776\,{v}^{4}+ 23887872 ) \cos (v ) -1152 \,{v}^{10}+ 13824\,{v}^{8}+ 27648\,{v}^{6}\\ &&+ 331776\,{v}^{4}+ 11943936\,{v}^{2}-119439360 ) (\sin (v ) )^{8}+ 2\, ((576\,{v}^{8}-34560\,{v}^{6}+ 207360\,{v}^{4}\\ &&+ 3483648\,{ v}^{2}-19906560 ) \cos (v ) +{v}^{14}-72\,{v}^{12}+ 1128\,{v}^{10}-576\,{v}^{8}-10368\,{v}^{6}+ 1036800\,{v}^{4}\\ &&-2092032\,{v}^{4}+ 20832768\,{v}^{2}-59719680 ) \cos (v ) -784\,{v}^{10 }+ 1/3\,{v}^{14}+ 28368\,{v}^{8}\\&&-355968\,{v}^{6}+ 4713984\,{v}^{4}\\ &&- 40559616\,{v}^{2}+ 106168320 ) (v ) (\sin (v ) )^{5}+ (({v}^{16}-22\,{v}^{14}+ 1128\,{v}^{12}-22752\,{v}^{10}+ 576576\,{v}^{8}\\ &&-9963648\,{v}^{6}+ 73571328\,{v}^{4}-359811072\,{v}^{2}+ 979402752 ) \cos (v ) -48\,{v}^{14}+ 1512\,{v}^{12}\\&&-5040\,{v}^{10}-695808\,{v}^{8}\\ \end{array} $$
$$\begin{array}{@{}rcl@{}} &&+ 14449536\,{v}^{6}-113052672\,{v}^{4}+ 561862656\,{v}^{2}-1457160192 ) (\sin (v ) )^{4}-12\, (v ) ((251\,{v}^{10}\\&&+ 6120\,{v}^{8}-181440\,{v}^{6}\\ &&+ 3131136\,{v}^{4}-22961664\,{v}^{2}+{v}^{14}+ 53084160\\ &&-{\frac {59}{2}}\, {v}^{12} ) \cos (v ) + 29431296\,{v}^{2}+ 1/6\,{v}^{14 }-66355200-12\,{v}^{12}-32\,{v}^{10}-4872\,{v}^{8}\\ &&+ 226944\,{v}^{6}- 4078080\,{v}^{4} ) (\sin (v ) )^{3}+ ((-{v}^{16}+ 23\,{v}^{14}+ 1104\,{v}^{12}-21240\,{v}^{10}\\ &&- 376704\,{v}^{8}+ 9455616\,{v}^{6}-83773440\,{v}^{4}+ 443916288\,{v}^{2}- 1051066368 ) \cos (v ) \\ &&+ 12\,{v}^{14}-1104\,{v}^{12}+ 13176\,{v}^{10}+ 342144\,{v}^{8}-9455616\,{v}^{6}+ 93394944\,{v}^{4}- 523542528\,{v}^{2}\\ &&+ 1242169344 ) (\sin (v ) )^{2}-6\, (({v}^{14}-42\,{v}^{12}-360\,{v}^{10}+ 13056\,{v}^{8}+ 101376\,{v}^{6}\\ &&-3787776\,{v}^{4}+ 25878528\,{v}^{2}- 53084160 ) \cos (v ) + 1224\,{v}^{10}-1/3\,{v}^{14}\\ &&- 18816\,{v}^{8}-101376\,{v}^{6}+ 3787776\,{v}^{4}-25878528\,{v}^{2}+ 53084160 ) (v ) \sin (v ) \\ &&+ (-{v} ^{16}+ 24\,{v}^{14}-504\,{v}^{12}+ 11520\,{v}^{10}-138240\,{v}^{8}\\ &&+ 19243008\,{v}^{4}-159252480\,{v}^{2}+ 382205952 ) \cos (v ) + 48\,{v}^{14}-1224\,{v}^{12}+ 2304\,{v}^{10}\\ &&+ 138240\,{v}^{8}- 19243008\,{v}^{4}+ 159252480\,{v}^{2}-382205952 )^{-1} . \end{array} $$

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Shokri, A., Khalsaraei, M.M., Tahmourasi, M. et al. A new family of three-stage two-step P-stable multiderivative methods with vanished phase-lag and some of its derivatives for the numerical solution of radial Schrödinger equation and IVPs with oscillating solutions. Numer Algor 80, 557–593 (2019). https://doi.org/10.1007/s11075-018-0497-z

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