Determine the Solution of Delay Differential Equations using Runge-Kutta Methods with Cubic-Spline Interpolation
Keywords:
Delay Differential Equations, Cubic-Spline In- terpolation, Runge-Kutta Methods, LipschitzAbstract
This paper describes some iterations for term delay in Delay Differential Equation (DDE), which is causing a huge number of iteration calculations. Time-delay was approximated using Cubic-Spline Interpolation, so DDE can rewrite as Differential Equations. Then, Runge-Kutta methods have been used to determine the solution of Differential equations from DDE.
References
J. J. Batzel and H. T. Tran, “Stability of the human respiratory control system i. analysis of a two-dimensional delay state-space model: I. analysis of a two-dimensional delay state-space model,” Journal of mathematical biology, vol. 41, pp. 45–79, 2000.
B. Balachandran, T. Kalm´ar-Nagy, and D. E. Gilsinn, Delay differential equations. Springer, 2009.
F. Karakoc¸ and H. Bereketo˘glu, “Solutions of delay differential equations by using differential transform method,” International Journal of Computer Mathematics, vol. 86, no. 5, pp. 914–923, 2009.
A. K. Alomari, M. S. M. Noorani, and R. Nazar, “Solution of delay differential equation by means of homotopy analysis method,” Acta Applicandae Mathematicae, vol. 108, pp. 395–412, 2009.
I. Ali, H. Brunner, and T. Tang, “A spectral method for pantograph-type delay differential equations and its convergence analysis,” Journal of Computational Mathematics, pp. 254–265, 2009.
H. Y. Seong and Z. A. Majid, “Solving neutral delay differential equations of pantograph type by using multistep block method,” in International Conference on Research and Education in Mathematics (ICREM7), 2015, pp. 56–60.
H. Y. Seong and Z. A. Majid, “Solving delay differential equations of pantograph type using predictor-corrector method,” in AIP Conference Proceedings, vol. 1660, no. 1, 2015, p. 050059.
F. Ismail, R. A. Al-Khasawneh, A. S. Lwin, and M. Suleiman, “Numerical treatment of delay differential equations by runge-kutta method using Hermite interpolation,” MATEMATIKA: Malaysian Journal of Industrial and Applied Mathematics, pp. 79–90, 2002.
H. J. Oberle and H. J. Pesch, “Numerical treatment of delay differential equations by hermite interpolation,” Numerische Mathematik, vol. 37, pp. 235–255, 1981.
A. Bellen and M. Zennaro, Numerical methods for delay differential equations. Oxford university press, 2013.
H. Su, S.-P. Yang, and L.-P. Wen, “Stability and convergence of the two parameter cubic spline collocation method for delay differential equations,” Computers & Mathematics with Applications, vol. 62, no. 6, pp. 2580–2590, 2011.
