The stability of modified Runge-Kutta methods for the pantograph equation
HTML articles powered by AMS MathViewer
- by M. Z. Liu, Z. W. Yang and Y. Xu;
- Math. Comp. 75 (2006), 1201-1215
- DOI: https://doi.org/10.1090/S0025-5718-06-01844-8
- Published electronically: May 3, 2006
- PDF | Request permission
Abstract:
In the present paper, the modified Runge-Kutta method is constructed, and it is proved that the modified Runge-Kutta method preserves the order of accuracy of the original one. The necessary and sufficient conditions under which the modified Runge-Kutta methods with the variable mesh are asymptotically stable are given. As a result, the $\theta$-methods with $\tfrac 12\leq \theta \leq 1$, the odd stage Gauss-Legendre methods and the even stage Lobatto IIIA and IIIB methods are asymptotically stable. Some experiments are given.References
- Alfredo Bellen, Preservation of superconvergence in the numerical integration of delay differential equations with proportional delay, IMA J. Numer. Anal. 22 (2002), no. 4, 529–536. MR 1936518, DOI 10.1093/imanum/22.4.529
- A. Bellen, N. Guglielmi, and L. Torelli, Asymptotic stability properties of $\Theta$-methods for the pantograph equation, Appl. Numer. Math. 24 (1997), no. 2-3, 279–293. Volterra centennial (Tempe, AZ, 1996). MR 1464729, DOI 10.1016/S0168-9274(97)00026-3
- Alfredo Bellen and Marino Zennaro, Numerical methods for delay differential equations, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 2003. MR 1997488, DOI 10.1093/acprof:oso/9780198506546.001.0001
- Jack Carr and Janet Dyson, The functional differential equation $y’(x)=ay(\lambda x)+by(x)$, Proc. Roy. Soc. Edinburgh Sect. A 74 (1974/75), 165–174 (1976). MR 442421, DOI 10.1017/s0308210500016632
- G. A. Derfel, Kato problem for functional-differential equations and difference Schrödinger operators, Order, disorder and chaos in quantum systems (Dubna, 1989) Oper. Theory Adv. Appl., vol. 46, Birkhäuser, Basel, 1990, pp. 319–321. MR 1124676
- Saber N. Elaydi, An introduction to difference equations, 2nd ed., Undergraduate Texts in Mathematics, Springer-Verlag, New York, 1999. MR 1711587, DOI 10.1007/978-1-4757-3110-1
- L. Fox, D. F. Mayers, J. R. Ockendon, and A. B. Tayler, On a functional differential equation, J. Inst. Math. Appl. 8 (1971), 271–307. MR 301330
- Roger A. Horn and Charles R. Johnson, Matrix analysis, Cambridge University Press, Cambridge, 1985. MR 832183, DOI 10.1017/CBO9780511810817
- A. Iserles, On the generalized pantograph functional-differential equation, European J. Appl. Math. 4 (1993), no. 1, 1–38. MR 1208418, DOI 10.1017/S0956792500000966
- Arieh Iserles, Numerical analysis of delay differential equations with variable delays, Ann. Numer. Math. 1 (1994), no. 1-4, 133–152. Scientific computation and differential equations (Auckland, 1993). MR 1340650
- Arieh Iserles and József Terjéki, Stability and asymptotic stability of functional-differential equations, J. London Math. Soc. (2) 51 (1995), no. 3, 559–572. MR 1332892, DOI 10.1112/jlms/51.3.559
- Arieh Iserles, Exact and discretized stability of the pantograph equation, Appl. Numer. Math. 24 (1997), no. 2-3, 295–308. Volterra centennial (Tempe, AZ, 1996). MR 1464730, DOI 10.1016/S0168-9274(97)00027-5
- Tosio Kato and J. B. McLeod, The functional-differential equation $y^{\prime } \,(x)=ay(\lambda x)+by(x)$, Bull. Amer. Math. Soc. 77 (1971), 891–937. MR 283338, DOI 10.1090/S0002-9904-1971-12805-7
- Toshiyuki Koto, Stability of Runge-Kutta methods for the generalized pantograph equation, Numer. Math. 84 (1999), no. 2, 233–247. MR 1730016, DOI 10.1007/s002110050470
- Yun Kang Liu, Stability analysis of $\theta$-methods for neutral functional-differential equations, Numer. Math. 70 (1995), no. 4, 473–485. MR 1337227, DOI 10.1007/s002110050129
- Yunkang Liu, On the $\theta$-method for delay differential equations with infinite lag, J. Comput. Appl. Math. 71 (1996), no. 2, 177–190. MR 1399890, DOI 10.1016/0377-0427(95)00222-7
- Yunkang Liu, Numerical investigation of the pantograph equation, Appl. Numer. Math. 24 (1997), no. 2-3, 309–317. Volterra centennial (Tempe, AZ, 1996). MR 1464731, DOI 10.1016/S0168-9274(97)00028-7
- J.R. Ockendon and A.B. Tayler, The dynamics of a current collection system for an electric locomotive, Proc. Roy. Soc. Edinburg Sect. A, 322 (1971), pp. 447–468.
- Yang Xu and MingZhu Liu, $\scr H$-stability of Runge-Kutta methods with general variable stepsize for pantograph equation, Appl. Math. Comput. 148 (2004), no. 3, 881–892. MR 2024551, DOI 10.1016/S0096-3003(02)00947-5
Bibliographic Information
- M. Z. Liu
- Affiliation: Department of Mathematics, Harbin Institute of Technology, Harbin 150001, People’s Republic of China
- Email: mzliu@hope.hit.edu.cn
- Z. W. Yang
- Affiliation: Department of Mathematics, Harbin Institute of Technology, Harbin 150001, People’s Republic of China
- Y. Xu
- Affiliation: Department of Mathematics, Harbin Institute of Technology, Harbin 150001, People’s Republic of China
- Received by editor(s): September 13, 2004
- Published electronically: May 3, 2006
- Additional Notes: This paper was supported by the National Natural Science Foundation of China (10271036).
- © Copyright 2006 American Mathematical Society
- Journal: Math. Comp. 75 (2006), 1201-1215
- MSC (2000): Primary 65L02, 65L05; Secondary 65L20
- DOI: https://doi.org/10.1090/S0025-5718-06-01844-8
- MathSciNet review: 2219025