Trigonometrically-fitted higher order two derivative Runge-Kutta method for solving orbital and related periodical IVPs

Year 2019, Volume: 48 Issue: 5, 1312 - 1323, 08.10.2019

N. A. Ahmad N. Senu F. Ismail

Abstract

In this paper, a trigonometrically-fitted two derivative Runge-Kutta method (TFTDRK) of high algebraic order for the numerical integration of first order Initial Value Problems (IVPs) which possesses oscillatory solutions is constructed. Using the trigonometrically-fitted property, a sixth order four stage Two Derivative Runge-Kutta (TDRK) method is designed. The numerical experiments are carried out with the comparison with other existing Runge-Kutta methods (RK) to show the accuracy and efficiency of the derived methods.

Keywords

two derivative Runge-Kutta method, trigonometrically-fitted, ordinary differential equations, initial value problems

References

• [1] F. Adel, N. Senu, F. Ismail, and Z.A. Majid, A New Efficient Phase-Fitted and Amplification-Fitted Runge-Kutta Method for Oscillatory Problems, Int. J. Pure Appl. Math. 107, 69-86, 2016.
• [2] Z.A. Anastassi and T.E. Simos, Trigonometrically fitted Runge-Kutta methods for the numerical solution of the Schrödinger equation, J. Math. Chem. 37 (3), 281-293, 2005.
• [3] J.C. Butcher, On Runge-Kutta processes of high order, J. Aust. Math. Soc. 4 (2), 179-194, 1964.
• [4] R.P. Chan and A.Y. Tsai, On explicit two-derivative Runge-Kutta methods, Numer. Algorithms 53, 171–194, 2010.
• [5] Z. Chen, J. Li, R. Zhang and X. You, Exponentially Fitted Two-Derivative Runge- Kutta Methods for Simulation of Oscillatory Genetic Regulatory Systems, Comput. Math. Methods Med. 2015, 689137, 2015.
• [6] Z. Chen, X. You, X. Shu and M. Zhang, A new family of phase-fitted and amplification- fitted Runge-Kutta type methods for oscillators, J. Appl. Math. 2012, 1-27, 2012.
• [7] M.A. Demba, N. Senu and F. Ismail, Trigonometrically-fitted explicit four-stage fourth-order Runge-Kutta-Nyström method for the solution of initial value problems with oscillatory behavior, Global Journal of Pure and Applied Mathematics, 12 (1), 67-80, 2016.
• [8] M.A. Demba, N. Senu and F. Ismail, Fifth-Order Four-Stage Explicit Trigonometrically-Fitted Runge-Kutta-Nyström Methods, Recent Advances in Math- ematical Sciences, 27-36, 2016.
• [9] M.A. Demba, N. Senu and F. Ismail, A 5(4) Embedded Pair of Explicit Trigonometrically-Fitted Runge-Kutta-Nyström Methods for the Numerical Solution of Oscillatory Initial Value Problems, Math. Comput. Appl. 21 (4), 46, 2016.
• [10] M.A. Demba, N. Senu and F. Ismail, A symplectic explicit trigonometrically-fitted Runge-Kutta-Nyströ method for the numerical solution of periodic problems, Int. J. Appl. Eng. Res. 11 (11), 7495-7500, 2016.
• [11] M.A. Demba, N. Senu and F. Ismail, An Embedded 4(3) Pair of Explicit Trigonometrically-Fitted Runge-Kutta-Nyström Method for Solving Periodic Initial Value Problems, Appl. Math. Sci. 17, 819-838, 2017.
• [12] Y. Fang, X. You and Q. Ming, Exponentially Fitted Two-Derivative Runge-Kutta Methods For The Schrödinger Equation, Int. J. Mod. Phys. C 24 (10), 1350073, 2013.
• [13] F.A. Fawzi, N. Senu, F. Ismail, and Z.A. Majid, A Phase-Fitted and Amplification- Fitted Modified Runge-Kutta Method of Fourth Order for Periodic Initial Value Prob- lems, Research and Education in Mathematics (ICREM7), 2015 International Con- ference on. IEEE, 25-28, 2015.
• [14] A.A. Kosti, Z.A. Anastassi and T.E. Simos, An optimized explicit Runge-Kutta- Nyström method for the numerical solution of orbital and related periodical initial value problems, Comput. Phys. Commun. 183, 470-479, 2012.
• [15] T. Monovasilis, Z. Kalogiratou and T.E. Simos, Construction of Exponentially Fit- ted Symplectic Runge-Kutta-Nyström Methods from Partitioned RungeKutta Methods, Mediterr. J. Math. 13 (4), 2271-2285, 2015.
• [16] P. Pokorny, Continuation of periodic solutions of dissipative and conservative systems: application to elastic pendulum, Math. Probl. Eng. 2009, Article ID 104547, 2009.
• [17] T.E. Simos, Family of fifth algebraic order trigonometrically fitted Runge-Kutta meth- ods for the numerical solution of the Schrödinger equation, Comp. Mater. Sci. 34 (4), 342-354, 2005.
• [18] E. Stiefel and D.G. Bettis, Stabilization of Cowell’s method, Numer. Math. 13, 154- 175, 1969.
• [19] H. Van de Vyver, An Explicit Numerov-Type Method for Second-Order Differential Equations with Oscillating Solutions, Comput. Math. Appl. 53, 1339-1348, 2007.
• [20] Y. Zhang, H. Che, Y. Fang and X. You, A new trigonometrically fitted two-derivative Runge-Kutta method for the numerical solution of the Schrödinger equation and re- lated problems, J. Appl. Math. 2013, Article ID 937858, 2013.