An Efficient Step Size Strategy for Numerical Approximation of Dynamical Systems with Hurwitz Stable

Year 2018, Volume: 6 Issue: 2, 290 - 298, 15.10.2018

Gülnur Çelik Kızılkan , Kemal Aydın

Abstract

In this study, we have given an algorithm and a step size strategy for numerical solution of Hurwitz stable differential equation systems. The algorithm is suited for implementation using computer algebra systems. So we also have given numerical examples from various field using this algorithm and a Maple procedure for the algorithm.

Keywords

Variable step size, Step size strategy, Hurwitz stable matrix, Differential equation system, Numerical integration

References

• [1] Bulgakov, H. Matrix Computations with Guaranteed Accuracy in Stabilty Theory, Selc¸uk University, Konya, 1995.
• [2] Bulgak, H. Pseudoeigenvalues, Spectral Portrait of the Matrices and Their Connections with Different Criteria of Stability, NATO ASI Series, Series C: Mathematical and Physical Sciences,536, 1999, 95 p.
• [3] Çelik Kızılkan, G. On the finding of step size in the numerical integtation of initial value peroblem, Master Thesis, Selc¸uk University Graduate Natural and Applied Sciences, Konya (in Turkish), 2004.
• [4] Çelik Kızılkan, G. Step size strategies on the numerical integration of the systems of differential equations, Ph.D. Thesis, Selc¸uk University Graduate Natural and Applied Sciences, Konya (in Turkish), 2009.
• [5] Çelik Kızılkan, G., Aydın, K. Step size strategy based on error analysis, SUFEFD, 25, 2005, pp. 79-86.
• [6] Çelik Kızılkan, G., Aydın, K. A new variable step size algorithm for Cauchy problem, Appl Math Comput, 183, 2006, pp. 878-884.
• [7] Çelik Kızılkan, G., Aydın, K. Step size strategies based on error analiysis for the linear systems, SDU Journal of Science (e- journal), 6(2), 2011, pp. 149-159.
• [8] Çelik Kızılkan, G., Aydın, K. Step size strategies for the numerical integration of systems of differential equations, J Comput Appl Math, 236(15), 2012, pp. 3805-3816.
• [9] Çıbıkdiken, A. O., Aydın, K. Computation of the monodromy matrix in floating point arithmetic with Wilkinson model, Comput Math Appl, 67(5), 2014, pp. 1186-1194.
• [10] Duman,A., Aydın, K. Sensitivity of Hurwitz stability of linear differential equation systems with constant coefficients, Int J Geom Methods Mod Phys, 14(6),2017, pp. 1750084.
• [11] El-Zahar, Essam R. An adaptative Step-Size Taylor Series Based Method and Application to Nonlinear Biochemical Reaction Model, Trends Appl Sci Res,7(11), 2012, pp.901-912.
• [12] Golub, G.H., Ortega, J.M. Scientific Computing and Differential Equations, Academic Press Limited, London, 1992.
• [13] Heath, M.T. Scientific Computing an Introductory Survey, 2nd Edition, McGraw-Hill, New York, 2002.
• [14] Loan, C.F.V. Introduction to Scientific Computing, Prentice Hill, United States of America, 2000.
• [15] Pastravanu, O., Voicu, M. Generalized matrix diagonal stability and linear dynamical systems, Linear Algebra Appl, 419, 2006, pp. 299-310.
• [16] Gustafson, G. Sytems of differential equations. Grant Gustafson’s home Page (Access date 10.10.2017).