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Keywords:
neutral differential systems with multiple delays; delay-dependent stability; Runge–Kutta method; Lagrange interpolation; argument principle
neutral differential systems with multiple delays; delay-dependent stability; Runge–Kutta method; Lagrange interpolation; argument principle
Summary:
In this paper, we are concerned with stability of numerical methods for linear neutral systems with multiple delays. Delay-dependent stability of Runge-Kutta methods is investigated, i. e., for delay-dependently stable systems, we ask what conditions must be imposed on the Runge-Kutta methods in order that the numerical solutions display stability property analogous to that displayed by the exact solutions. By means of Lagrange interpolation, Runge-Kutta methods can be applied to neutral differential systems with multiple delays. Based on the argument principle, sufficient conditions for delay-dependent stability of Runge-Kutta methods combined with Lagrange interpolation are presented. Numerical examples are given to illustrate the main results.
References:
[1] Bellen, A., Zennaro, M.: Numerical Methods for Delay Differential Equations. Oxford University Press, Oxford 2003. DOI 10.1093/acprof:oso/9780198506546.001.0001 | MR 1997488
[2] Brown, J. W., Churchill, R. V.: Complex Variables and Applications. McGraw-Hill Companies, Inc. and China Machine Press, Beijing 2004. MR 0112948
[3] Hale, J. K., Lunel, S. M. Verduyn: Strong stabilization of neutral functional differential equations. IMA J. Math. Control Info. 19 (2002), 5-23. DOI 10.1093/imamci/19.1_and_2.5 | MR 1899001
[4] Hu, G. D.: Stability criteria of linear neutral systems with distributed delays. Kybernetika 47 (2011), 273-284. MR 2828577
[5] Hu, G. D., Cahlon, B.: Estimations on numerically stable step-size for neutral delay differential systems with multiple delays. J. Comput. Appl. Math. 102 (1999), 221-234. DOI 10.1016/s0377-0427(98)00215-5 | MR 1674027
[6] Hu, G. D., Hu, G. D., Zou, X.: Stability of linear neutral systems with multiple delays: boundary criteria. Appl. Math. Comput. 148 (2004), 707-715. DOI 10.1016/s0096-3003(02)00929-3 | MR 2024535
[7] Huang, C., Vandewalle, S.: An analysis of delay-dependent stability for ordinary and partial differential equations with fixed and distributed delays. SIAM J. Scientific Computing 25 (2004), 1608-1632. DOI 10.1137/s1064827502409717 | MR 2087328
[8] Johnson, L. W., Riess, R. Dean, Arnold, J. T.: Introduction to Linear Algebra. Prentice-Hall, Englewood Cliffs 2000.
[9] Jury, E. I.: Theory and Application of $z$-Transform Method. John Wiley and Sons, New York 1964.
[10] Kim, A. V., Ivanov, A. V.: Systems with Delays. Scrivener Publishing LLC, Salem, Massachusetts 2015. DOI 10.1002/9781119117841 | MR 3496968
[11] Kolmanovskii, V. B., Myshkis, A.: Introduction to Theory and Applications of Functional Differential Equations. Kluwer Academic Publishers, Dordrecht 1999. DOI 10.1007/978-94-017-1965-0 | MR 1680144
[12] Lambert, J. D.: Numerical Methods for Ordinary Differential Systems. John Wiley and Sons, New York 1999. MR 1127425
[13] Lancaster, P., Tismenetsky, M.: The Theory of Matrices with Applications. Academic Press, Orlando 1985. MR 0792300
[14] Michiels, W., Niculescu, S.: Stability, Control and Computation for Time Delay Systems: An Eigenvalue Based Approach. SIAM, Philadelphia 2014. DOI 10.1137/1.9781611973631 | MR 3288751
[15] Tian, H., Kuang, J.: The stability of the $\theta$-methods in numerical solution of delay differential equations with several delay terms. J. Comput. Appl. Math. 58 (1995), 171-181. DOI 10.1016/0377-0427(93)e0269-r | MR 1343634
[16] Vyhlidal, T., Zitek, P.: Modification of Mikhaylov criterion for neutral time-delay systems. IEEE Trans. Automat. Control 54 (2009), 2430-2435. DOI 10.1109/tac.2009.2029301 | MR 2562848
[17] Wang, W.: Nonlinear stability of one-leg methods for neutral Volterra delay-integro-differential equations. Math. Comput. Simul. 97 (2014), 147-161. DOI 10.1016/j.matcom.2013.08.004 | MR 3137913
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