An application of the hybrid methods to the numerical solution of ordinary differential equations of second order
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Keywords:
задача Коши, обыкновенные дифференциальные уравненияAbstract
Here for solving ODE of the second order we construct a new class of hybrid methods of multistep type. Taking into account that implicit methods are more precise we consider a question on definition of implicit character of hybrid methods and construct methods with the order of accuracy p = 6 for k = 2.References
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[2] Gear C.S. Hybrid methods for initial value problems in ordinary differential equations // SIAM, J. Numer. Anal. -v. 2.- 1965.- P. 69-86.
[3] Butcher J.C. A modified multistep method for the numerical integration of ordinary differential equations // J. Assoc. Comput. Math.- v.12.- 1965.- P.124-135.
[4] Мехтиева Г.Ю., Ибрагимов В.Р. Построение гибридных методов с помощью методов Рунге-Кутты // Вестник БГУ. -2006.- № 3.- P.17-22.
[5] Ибрагимов В.Р. Один нелинейный метод численного решения задачи Коши для обыкновенных дифференциальных уравнений // Диф. урав. и применения Труды докл. Второй международ. конф. Руссе. Болгария. -1982. -P.310-319.
[6] Bokhoven W.M.G. Efficient higher order imlicit one-step methods for integration of stiff differ. eq-s // BIT.20.-1980.- P.34-43.
[7] Ehigie J.O., Okunuga S.A., Sofoluwe A.B., Akanbi M.A. On generalized 2-step continuous linear multistep method of hybrid type for the integration of second order ordinary differential equations // Archives of Applied Research. -2(6),-2010.-P.362-372.
[8] Mehdiyeva G.Yu., Imanova M.N., Ibrahimov V.R. On an application of hybrid method to solving second ordinary differential equations The international conference on applied mathematics, modeling And computational science // Waterloo, Сanada. -July 25 - 29.-2011. - P. 363.
[9] 15. Dahlquist, G. Stability and Error bounds in the numerical integration of ordinary differential equations // Uppsala, Almqvist and Wiksells boktr, No.13. -1959. P.5-92.
[10] Enrite W.H. Second derivative multistep methods for stiff ordinary differential equations // SIAM, J.Numer.Anal. -1974. -№2. -P.321-332.
[11] Ибрагимов В.Р. Об одной связи между порядком и степенью для устойчивой формулы с забеганием вперед // Ж.Вычис. мат. и мат. физ. -№7.- 1990.- C.1045-1056.
[12] Ibrahimov V.R. On the maximum degree of the k-step Obrechkoff’s method // Bulletin of Iranian Mathematical Society. -Vol. 20. -2002. -No.1, P.1-28.
[13] Hairier E., Norsett S.P., Wanner G.Solving ordinary differential equations //М., Mir. -1990. - P. 512.
[14] Mehdiyeva G.Yu., Imanova M.N., Ibrahimov V.R. On one generalization of hybrid methods. Proceedings of the 4th international conference on approximation methodsn and numerical modeling in environment and natural resources // Saidia, Morocco.- may 23-26. -2011. -P.543-547.
[2] Gear C.S. Hybrid methods for initial value problems in ordinary differential equations // SIAM, J. Numer. Anal. -v. 2.- 1965.- P. 69-86.
[3] Butcher J.C. A modified multistep method for the numerical integration of ordinary differential equations // J. Assoc. Comput. Math.- v.12.- 1965.- P.124-135.
[4] Мехтиева Г.Ю., Ибрагимов В.Р. Построение гибридных методов с помощью методов Рунге-Кутты // Вестник БГУ. -2006.- № 3.- P.17-22.
[5] Ибрагимов В.Р. Один нелинейный метод численного решения задачи Коши для обыкновенных дифференциальных уравнений // Диф. урав. и применения Труды докл. Второй международ. конф. Руссе. Болгария. -1982. -P.310-319.
[6] Bokhoven W.M.G. Efficient higher order imlicit one-step methods for integration of stiff differ. eq-s // BIT.20.-1980.- P.34-43.
[7] Ehigie J.O., Okunuga S.A., Sofoluwe A.B., Akanbi M.A. On generalized 2-step continuous linear multistep method of hybrid type for the integration of second order ordinary differential equations // Archives of Applied Research. -2(6),-2010.-P.362-372.
[8] Mehdiyeva G.Yu., Imanova M.N., Ibrahimov V.R. On an application of hybrid method to solving second ordinary differential equations The international conference on applied mathematics, modeling And computational science // Waterloo, Сanada. -July 25 - 29.-2011. - P. 363.
[9] 15. Dahlquist, G. Stability and Error bounds in the numerical integration of ordinary differential equations // Uppsala, Almqvist and Wiksells boktr, No.13. -1959. P.5-92.
[10] Enrite W.H. Second derivative multistep methods for stiff ordinary differential equations // SIAM, J.Numer.Anal. -1974. -№2. -P.321-332.
[11] Ибрагимов В.Р. Об одной связи между порядком и степенью для устойчивой формулы с забеганием вперед // Ж.Вычис. мат. и мат. физ. -№7.- 1990.- C.1045-1056.
[12] Ibrahimov V.R. On the maximum degree of the k-step Obrechkoff’s method // Bulletin of Iranian Mathematical Society. -Vol. 20. -2002. -No.1, P.1-28.
[13] Hairier E., Norsett S.P., Wanner G.Solving ordinary differential equations //М., Mir. -1990. - P. 512.
[14] Mehdiyeva G.Yu., Imanova M.N., Ibrahimov V.R. On one generalization of hybrid methods. Proceedings of the 4th international conference on approximation methodsn and numerical modeling in environment and natural resources // Saidia, Morocco.- may 23-26. -2011. -P.543-547.
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Mehdieva, G. Y., Imanova, M. N., & Ibrahimov, V. R. (2012). An application of the hybrid methods to the numerical solution of ordinary differential equations of second order. Journal of Mathematics, Mechanics and Computer Science, 75(4), 46–54. Retrieved from https://bm.kaznu.kz/index.php/kaznu/article/view/158
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Computational Mathematics and mathematical modeling
