A New Two-step Obrechkoff Method with Vanished Phase-lag and Some of its Derivatives for the Numerical Solution of Radial Schrodinger Equation and Related IVPs with Oscillating Solutions

Document Type : Research Paper

Authors

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

Abstract

A new two-step implicit linear Obrechkoff twelfth algebraic order method with vanished phase-lag and its first, second, third and fourth derivatives is constructed in this paper. The purpose of this paper is to develop an efficient algorithm for the approximate solution of the one-dimensional radial Schrodinger equation and related problems. This algorithm belongs in the category of the multistep methods. In order to produce an efficient multistep method the phase-lag property and its derivatives are used. An error analysis and a stability analysis is also investigated and a comparison with other methods is also studied. The efficiency of the new methodology is proved via theoretical analysis and numerical applications.

Keywords

Main Subjects


[1] U. Ananthakrishnaiah, P–stable Obrechkoff methods with minimal phase–lag for periodic initial value problems, Math. Comput. 49 (1987) 553–559.
[2] M. Asadzadeh, D. Rostamy and F. Zabihi, Discontinuous Galerkin and multiscale variational schemes for a coupled damped nonlinear system of Schrödinger equations, J. Numer. Methods Partial Differential Equations 29 (6) (2013) 1912–1945.
[3] M. M. Chawla, P. S. Rao, A Numerov–type method with minimal phase–lag for the integration of second order periodic initial value problems. II: Explicit method, J. Comput. Appl. Math. 15 (1986) 329–337.
[4]  M. M. Chawla, P. S. Rao, An explicit sixth–order method with phase–lag of order eight for , J. Comput. Appl. Math. 17 (1987) 363–368.
[5] G. Dahlquist, On accuracy and unconditional stability of linear multistep methods for second order differential equations, BIT 18 (1978) 133–136.
[6] J. M. Franco, An explicit hybrid method of Numerov type for second–order periodic initial–value problems, J. Comput. Appl. Math. 59 (1995) 79–90.
[7] J. M. Franco, M. Palacios, High–order P–stable multistep methods, J. Comput. Appl. Math. 30 (1990) 1–10.
[8] W. Gautschi, Numerical integration of ordinary differential equations based on trigonometric polynomials, Numer. Math. 3 (1961) 381–397.
[9] A. Ibraheem, T. E. Simos, A family of high–order multistep methods with vanished phase–lag and its derivatives for the numerical solution of the Schrödinger equation, Comput. Math. Appl. 62 (2011) 3756–3774.
[10] A. Ibraheem, T. E. Simos, A family of ten–step methods with vanished phase–lag and its first derivative for the numerical solution of the Schrödinger equation, J. Math. Chem. 49 (2011) 1843–1888.
[11] A. Ibraheem, T. E. Simos, Mulitstep methods with vanished phase–lag and its first and second derivatives for the numerical integration of the Schrödinger equation, J. Math. Chem. 48 (2010) 1092–1143.
[12] L. Gr. Ixaru, M. Rizea, A Numerov–like scheme for the numerical solution of the Schrödinger equation in the deep continuum spectrum of energies, Comput. Phys. Commun. 19 (1) (1980) 23–27.
[13] M. K. Jain, R. K. Jain and U. Krishnaiah, Obrechkoff methods for periodic initial value problems of second order differential equations, J. Math. Phys. Sci. 15 (1981) 239–250.
[14] J. D. Lambert, I. A. Watson, Symmetric multistep methods for periodic initial value problems, IMA J. Appl. Math. 18 (1976) 189–202.
[15] G. D. Quinlan, S. Tremaine, Symmetric multistep methods for the numerical integration of planetary orbits, Astron.  J. 100 (1990) 1694–1700.
[16] D. P. Sakas, T. E. Simos, Multiderivative methods of eighth algebraic order with minimal phase–lag for the numerical solution of the radial Schrödinger equation, J. Comput. Appl. Math. 175 (2005) 161–172.
[17] A. Shokri, H. Saadat, Trigonometrically fitted high–order predictor–corrector method with phase–lag of order infinity for the numerical solution of radial Schrödinger equation, J. Math. Chem. 52 (2014) 1870–1894.
[18] A. Shokri, H. Saadat, High phase–lag order trigonometrically fitted two–step Obrechkoff methods for the numerical solution of periodic initial value problems, Numer.  Algor. 68 (2015) 337–354.
[19] A. Shokri, A. A. Shokri, Sh. Mostafavi, H. Saadat, Trigonometrically fitted two-step Obrechkoff methods for the numerical solution of periodic initial value problems, Iranian J. Math. Chem. 6 (2015) 145-161.
[20] T. E. Simos, A P-stable complete in phase Obrechkoff trigonometric fitted method for periodic initial value problems, Proc. Roy. Soc. London Ser. A. 441 (1993) 283–289.
[21] T. E. Simos, A two–step method with vanished phase–lag and its first two derivatives for the numerical solution of the Schrödinger equation, J. Math. Chem. 49 (2011) 2486–2518.
[22] T. E. Simos, Exponentially fitted multiderivative methods for the numerical solution of the Schrödinger equation, J. Math. Chem. 36 (2004) 13–27.
[23] T. E. Simos, Multiderivative methods for the numerical solution of the Schrödinger equation, MATCH Commun. Math. Comput. Chem. 50 (2004) 7–26.
[24] E. Steifel, D. G. Bettis, Stabilization of Cowells methods, Numer. Math. 13 (1969) 154–175.
[25] R. M. Thomas, Phase properties of high order, almost P–stable formulae, BIT 24 (1984) 225–238.
[26] M. Van Daele, G. Vanden Berghe, P-stable exponentially fitted Obrechkoff methods of arbitrary order for second order differential equations, Numer. Algor. 46 (2007) 333–350.
[27] Z. Wang, D. Zhao, Y. Dai and D. Wu, An improved trigonometrically fitted P–stable Obrechkoff method for periodic initial value problems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2005) 1639–1658.