| | | |

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

#### N. A. AHMAD  , N. SENU  , F. ISMAİL 

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.
two derivative Runge-Kutta method, trigonometrically-fitted, ordinary differential equations, initial value problems
•  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.
•  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.
•  J.C. Butcher, On Runge-Kutta processes of high order, J. Aust. Math. Soc. 4 (2), 179-194, 1964.
•  R.P. Chan and A.Y. Tsai, On explicit two-derivative Runge-Kutta methods, Numer. Algorithms 53, 171–194, 2010.
•  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.
•  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.
•  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.
•  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.
•  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.
•  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.
•  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.
•  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.
•  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.
•  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.
•  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.
•  P. Pokorny, Continuation of periodic solutions of dissipative and conservative systems: application to elastic pendulum, Math. Probl. Eng. 2009, Article ID 104547, 2009.
•  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.
•  E. Stiefel and D.G. Bettis, Stabilization of Cowell’s method, Numer. Math. 13, 154- 175, 1969.
•  H. Van de Vyver, An Explicit Numerov-Type Method for Second-Order Differential Equations with Oscillating Solutions, Comput. Math. Appl. 53, 1339-1348, 2007.
•  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.
Birincil Dil en Matematik Matematik Orcid: 0000-0003-3007-0432Yazar: N. A. AHMAD (Sorumlu Yazar) Orcid: 0000-0001-8614-8281Yazar: N. SENU Orcid: 0000-0002-1548-8702Yazar: F. ISMAİL Yayımlanma Tarihi : 8 Ekim 2019
 Bibtex @araştırma makalesi { hujms629826, journal = {Hacettepe Journal of Mathematics and Statistics}, issn = {2651-477X}, eissn = {2651-477X}, address = {}, publisher = {Hacettepe Üniversitesi}, year = {2019}, volume = {48}, pages = {1312 - 1323}, doi = {}, title = {Trigonometrically-fitted higher order two derivative Runge-Kutta method for solving orbital and related periodical IVPs}, key = {cite}, author = {Ahmad, N. A. and Senu, N. and Ismai̇l, F.} } APA Ahmad, N , Senu, N , Ismai̇l, F . (2019). Trigonometrically-fitted higher order two derivative Runge-Kutta method for solving orbital and related periodical IVPs . Hacettepe Journal of Mathematics and Statistics , 48 (5) , 1312-1323 . Retrieved from https://dergipark.org.tr/tr/pub/hujms/issue/49321/629826 MLA Ahmad, N , Senu, N , Ismai̇l, F . "Trigonometrically-fitted higher order two derivative Runge-Kutta method for solving orbital and related periodical IVPs" . Hacettepe Journal of Mathematics and Statistics 48 (2019 ): 1312-1323 Chicago Ahmad, N , Senu, N , Ismai̇l, F . "Trigonometrically-fitted higher order two derivative Runge-Kutta method for solving orbital and related periodical IVPs". Hacettepe Journal of Mathematics and Statistics 48 (2019 ): 1312-1323 RIS TY - JOUR T1 - Trigonometrically-fitted higher order two derivative Runge-Kutta method for solving orbital and related periodical IVPs AU - N. A. Ahmad , N. Senu , F. Ismai̇l Y1 - 2019 PY - 2019 N1 - DO - T2 - Hacettepe Journal of Mathematics and Statistics JF - Journal JO - JOR SP - 1312 EP - 1323 VL - 48 IS - 5 SN - 2651-477X-2651-477X M3 - UR - Y2 - 2018 ER - EndNote %0 Hacettepe Journal of Mathematics and Statistics Trigonometrically-fitted higher order two derivative Runge-Kutta method for solving orbital and related periodical IVPs %A N. A. Ahmad , N. Senu , F. Ismai̇l %T Trigonometrically-fitted higher order two derivative Runge-Kutta method for solving orbital and related periodical IVPs %D 2019 %J Hacettepe Journal of Mathematics and Statistics %P 2651-477X-2651-477X %V 48 %N 5 %R %U ISNAD Ahmad, N. A. , Senu, N. , Ismai̇l, F. . "Trigonometrically-fitted higher order two derivative Runge-Kutta method for solving orbital and related periodical IVPs". Hacettepe Journal of Mathematics and Statistics 48 / 5 (Ekim 2019): 1312-1323 . AMA Ahmad N , Senu N , Ismai̇l F . Trigonometrically-fitted higher order two derivative Runge-Kutta method for solving orbital and related periodical IVPs. Hacettepe Journal of Mathematics and Statistics. 2019; 48(5): 1312-1323. Vancouver Ahmad N , Senu N , Ismai̇l F . Trigonometrically-fitted higher order two derivative Runge-Kutta method for solving orbital and related periodical IVPs. Hacettepe Journal of Mathematics and Statistics. 2019; 48(5): 1312-1323.

Makalenin Yazarları