AUTHORS: N. Senu, N. A. Ahmad, F. Ismail, N. Bachok

Download as PDF

ABSTRACT: Based on First Same As Last (FSAL) technique, an embedded trigonometrically-fitted Two Derivative Runge-Kutta method (TDRK) for the numerical solution of first order Initial Value Problems (IVPs) is developed. Using the trigonometrically-fitting technique, an embedded 5(4) pair explicit fifth-order TDRK method with a “small” principal local truncation error coefficient is derived. The numerical experiments are carried out and showed that our new method is more accurate and efficient when compared with other existing Runge-Kutta (RK) and TDRK methods of the same order.

KEYWORDS: Explicit methods, Embedded methods, First Same As Last technique, Initial Value Problems, TDRK methods, Variable step-size

REFERENCES:

[1] Chan, R. P., & Tsai, A. Y. 2010. On explicit two-derivative Runge-Kutta methods. Numerical Algorithms, 53(2-3), 171-194.

[2] Chan, R. P., Wang, S., & Tsai, A. Y. 2012, September. Two-derivative Runge-Kutta methods for differential equations. In Numerical Analysis and Applied Mathematics ICNAAM 2012: International Conference of Numerical Analysis and Applied Mathematics (Vol. 1479, No. 1, pp. 262-266). AIP Publishing.

[3] Chan, R. P., Wang, S., & Tsai, A. Y. 2012, Zhang, Y., Che, H., Fang, Y., & You, X. 2013. A new trigonometrically fitted two-derivative Runge-Kutta method for the numerical solution of the Schrödinger equation and related problems. Journal of Applied Mathematics, 2013.

[4] Fang, Y., You, X., & Ming, Q. 2013. Exponentially fitted two-derivative runge–kutta methods for the schrödinger equation. International Journal of Modern Physics C, 24(10), 1350073.

[5] Chen, Z., Li, J., Zhang, R., & You, X. 2015. Exponentially Fitted Two-Derivative RungeKutta Methods for Simulation of Oscillatory Genetic Regulatory Systems. Computational and mathematical methods in medicine, 2015.

[6] Bogacki, P., & Shampine, L. F. 1989. A 3 (2) pair of Runge-Kutta formulas. Applied Mathematics Letters, 2(4), 321-325.

[7] Dormand, J. R., & Prince, P. J. 1980. A family of embedded Runge-Kutta formulae. Journal of computational and applied mathematics, 6(1), 19-26.

[8] Tsitouras, C. 2011. Runge–Kutta pairs of order 5 (4) satisfying only the first column simplifying assumption. Computers & Mathematics with Applications, 62(2), 770- 775.

[9] Fang, Y., Song, Y., & Wu, X. 2008. New embedded pairs of explicit Runge–Kutta methods with FSAL properties adapted to the numerical integration of oscillatory problems. Physics Letters A, 372(44), 6551-6559.

[10] Chen, Z., You, X., Shu, X., & Zhang, M. 2012. A new family of phase-fitted and amplificationfitted Runge-Kutta type methods for oscillators. Journal of Applied Mathematics, 2012.

[11] Demba, M. A., Senu, N., & Ismail, F. 2016. 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.

[12] Demba, M. A., Senu, N., & Ismail, F. 2016. Fifth-Order Four-Stage Explicit Trigonometrically-Fitted Runge–Kutta– Nyström Methods. In Recent Advances in Mathematical Sciences (pp. 27-36). Springer Singapore.

[13] Fawzi, F. A., Senu, N., Ismail, F., & Majid, Z. A. (2015, August). A Phase-Fitted and Amplification-Fitted Modified Runge-Kutta Method of Fourth Order for Periodic Initial Value Problems. In Research and Education in Mathematics (ICREM7), 2015 International Conference on (pp. 25-28). IEEE.

[14] F. A. Fawzi, N. Senu, F. Ismail, and Z. A. Majid. 2016. A New Efficient Phase-Fitted and Amplification-Fitted Runge-Kutta Method for Oscillatory Problems. International Journal of Pure and Applied Mathematics, 107, 69-86.

[15] Senu, N. 2010. Runge-Kutta-Nystrom Methods for Solving Oscillatory Problems (Doctoral dissertation, Universiti Putra Malaysia).

[16] Rabiei, F., & Ismail, F. 2012. Fifth-order Improved Runge-Kutta method for solving ordinary differential equation. Australian Journal of Basic and Applied Sciences, 6(3), 97-105.

[17] Kosti, A. A., Anastassi, Z. A., & Simos, T. E. 2012. An optimized explicit Runge–Kutta– Nyström method for the numerical solution of orbital and related periodical initial value problems. Computer Physics Communications, 183(3), 470-479.

[18] Dormand, J. R., & Prince, P. 1978. New Runge-Kutta algorithms for numerical simulation in dynamical astronomy. Celestial Mechanics, 18(3), 223-232.

[19] Bu, S., Chung, W., & Kim, P. 2014. A Method for Improving the Embedded Runge-Kutta Fehlberg 4 (5).

[20] Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1996. Numerical recipes in C (Vol. 2). Cambridge: Cambridge university press.