A new symmetric two-step P-stable Obrechkoff method with 12 algebraic order for the numerical solution of second-order IVPs
Subject Areas : Statisticsali shokri 1 , abbas-ali shokri 2 , Mohammad Mehdizadeh Khalsarei 3 , firoz pashaie 4
1 - Department of Mathematics, Faculty of Basic Science, University of Maragheh, Maragheh, Iran
2 - Department of Mathematics, Ahar Branch, Islamic Azad University-Ahar-Iran
3 - Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh,Iran
4 - Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh,Iran
Keywords: روش های چندمشتقی, مسائل مقدار اولیه, ناحیه پایداری, فاز تاخیری,
Abstract :
A new two-step implicit P-stable Obrechkoff of twelfth algebraic order with vanished phase-lag and its first, second and third derivatives is constructed in this paper. The purpose of this paper is to develop an efficient algorithm for the approximate solution of the second order iniitial value problems that have oscillatory or periodic solutions. This algorithm belongs in the category of the multistep and multiderivative methods. The advantage of the new methods in comparison with similar methods, in terms of efficiency, accuracy and stability, have been showed by the implementation of them in some important problems, including the undamped Duffing equation, etc. -------------- A new two-step implicit P-stable Obrechkoff of twelfth algebraic order with vanished phase-lag and its first, second and third derivatives is constructed in this paper. The purpose of this paper is to develop an efficient algorithm for the approximate solution of the second order iniitial value problems that have oscillatory or periodic solutions. This algorithm belongs in the category of the multistep and multiderivative methods. The advantage of the new methods in comparison with similar methods, in terms of efficiency, accuracy and stability, have been showed by the implementation of them in some important problems, including the undamped Duffing equation, etc.
[1] Achar, S. D., Symmetric multistep Obrechkoff methods with zero phase-lag for periodic initial value problems of second order differential equations, Applied Mathematics and Computation 218 (2011) 2237-2248.
[2] Alolyan, I., Simos, T. E., Mulitstep methods with vanished phase-lag and its first and second derivatives for the numerical integration of the Schrӧdinger equation, Journal of Mathematical Chemistry 48(4) (2010) 1092-1143.
[3] Anantha Krishnaiah, U. A., P-stable Obrechkoff methods with minimal phase-lag for periodic initial value problems, Mathematics of Computation 49(180) (1987) 553-559.
[4] Chawla, M. M., Rao, P. S., A Numerov-type method with minimal phase-lag for the integration of second order periodic initial value problems. ii: Explicit method, Journal of Computational and Applied Mathematics 15 (1986) 329-337.
[5] Dahlquist, G., On accuracy and unconditional stability of linear multistep methods for second order differential equations, BIT Numerical Mathematics 18(2) (1978) 133-136.
[6] Ramos, H., Vigo-Aguiar, J., On the frequency choice in trigonometrically fitted methods, Applied Mathematics Letters 23(11) (2010) 1378-1381.
[7] Raptis, A. D., Allison, A. C., Exponential-fitting methods for the numerical solution of the Schrӧdinger equation., Computer Physics Communications 14 (1978) 1-5.
[8] Raptis, A. D., Exponentially-fitted solutions of the eigenvalue Schrӧdinger equation with automatic error control, Computer Physics Communications 28 (1983) 427-431.
[9] Shokri, A., A new eight-order symmetric two-step multiderivative method for the numerical solution of second-order IVPs with oscillating solutions, Numerical Algorithms 77(1) (2018) 95-109.
[10] Shokri, A., The symmetric two-step P-stable nonlinear predictor-corrector methods for the numerical solution of second order initial value problems, Bulletin of the Iranian Mathematical Society 41(1) (2015) 191-205.
[11] Shokri, A., Saadat, H., Trigonometrically fitted high-order predictor corrector method with phase-lag of order infinity for the numerical solution of radial Schrӧdinger equation, Journal of Mathematical Chemistry 52(7) (2014) 1870-1894.
[12] Shokri, A., Shokri, A. A., Implicit one-step L-stable generalized hybrid methods for the numerical solution of first order initial value problems, Iranian Journal of Mathematical Chemistry 4(2) (2013) 201-212.
[13] Shokri, A., Shokri, A. A., Mostafavi, Sh., Saadat, H., Trigonometrically fitted two-step obrechkoff methods for the numerical solution of periodic initial value problems, Iranian Journal of Mathematical Chemistry 6(2) (2015) 145-161.
[14] Shokri, A., Tahmourasi, M., A new two-step Obrechkoff method with vanished phase-lag and some of its derivatives for the numerical solution of radial Schrӧdinger equation and related IVPs with oscillating solutions, Iranian Journal of Mathematical Chemistry 8(2) (2017) 137-159.
[15] Shokri, A., Mehdizadeh Khalsaraei,
M., Tahmourasi, M., Garcia-Rubio, R., 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, Numerical Algorithms 80(2) (2019) 557-593.
[16] Mehdizadeh Khalsaraei, M., Shokri, A., An explicit six-step singularly P-stable Obrechkoff method for the numerical solution of second-order oscillatory initial value problems, Numerical Algorithms (2019) https://doi.org/10.1007/s11075-019-00784-w.
[17] Mehdizadeh Khalsaraei, M., Shokri, A., A new explicit singularly P-stable four-step method for the numerical solution of second order IVPs, Iranian Journal of Mathematical Chemistry (2020) DOI:10.22052/ijmc.2020.207671.1472.
[18] Mehdizadeh Khalsaraei, M., Shokri, A., Molayi, M., The new high approximation of stiff systems of first order IVPs arising from chemical reactions by k-step L-stable hybrid methods, Iranian Journal of Mathematical Chemistry 10(2) (2019) 181-193.
[19] Shokri, A., Mehdizadeh Khalsaraei, M., A new family of explicit linear two-step singularly P-stable Obrechkoff methods for the numerical solution of second-order IVPs, Applied Mathematics and Computation (2020) DOI: 10.1016/ j.amc.2020.125116.
[20] Vanden Berghe, G., Van Daele, M., Trigonometric polynomial or exponential fitting approach, Journal of Computational and Applied Mathematics 233 (2009) 969-979.
[21] Lambert, J. D., Watson, I. A., Symmetric multistep methods for periodic initial value problems, IMA Journal of Applied Mathematics 18(2) (1976) 189-202.
[22] Simos, T. E., A P-stable complete in phase Obrechkoff trigonometric fitted method for periodic initial value problems, Proceedings: Mathematical and Physical Sciences 441(1912) (1993) 283-289.
[23] Van Daele, M., Vanden Berghe, G., P-stable exponentially fitted Obrechkoff methods of arbitrary order for second order differential equations, Numerical Algorithms 46 (2007) 333-350.
[24] Wang, Z., Zhao, D., Dai, Y. and Wu, D., An improved trigonometrically fitted P-stable Obrechkoff method for periodic initial value problems Proceedings of The Royal Society A 461 (2005) 1639-1658.
[25] Shokri, A., Saadat, H., High phase-lag order trigonometrically fitted two-step Obrechkoff methods for the numerical solution of periodic initial value problems, Numerical Algorithms 68 (2015) 337-354.
[26] عبدی، علی، حسینی، سید احمد، (1396)، روشهای عددی همتافته و متقارن برای حل عددی برخی مدلهای ریاضی اجرام سماوی. پژوهشهای نوین در ریاضی، دوره 3 شماره 11 صفحه 109-118.
