Short proofs and a counterexample for analytical and numerical stability of delay equations with infinite memory

IMA Journal of Numerical Analysis

Volumn 26, Issue 1 SPEC. ISS., 2006, Pages 60-77

  Guglielmi, Nicola ^a  

^a UNIVERSITY OF L'AQUILA   (Italy)

Author keywords

Asymptotic stability; Infinite memory; Joint spectral radius; Linear systems of DDEs; Variable coefficient difference equations; methods

Indexed keywords

ASYMPTOTIC STABILITY; DIFFERENCE EQUATIONS; DIFFERENTIAL EQUATIONS; LINEAR SYSTEMS; MATRIX ALGEBRA; PANTOGRAPHS;

ASYMPTOTIC BEHAVIOUR; CONSTANT COEFFICIENTS; DELAY DIFFERENTIAL EQUATIONS; INFINITE MEMORY; JOINT SPECTRAL RADIUS; NUMERICAL SOLUTION; PANTOGRAPH EQUATION; VARIABLE COEFFICIENTS;

NUMERICAL METHODS;

EID: 30344459133     PISSN: 02724979     EISSN: 14643642     Source Type: Journal    
DOI: 10.1093/imanum/dri021     Document Type: Article

References (24)

1. BAKER, C. T. H. & BUCKWAR, E. (2000) Continuous Θ-methods for the stochastic pantograph equation. Electron. Trans. Numer. Anal., 11, 131-151.
2. BAKER, C. T. H., PAUL, C. A. H. & WILLE, D. R. (1995) Issues in the numerical solution of evolutionary delay differential equations. Adv. Comput. Math., 3, 171-196.
3. BELLEN, A. (2002) Preservation of superconvergence in the numerical integration of delay differential equations with proportional delay. IMA J. Numer. Anal., 22, 529-536.
4. BELLEN, A. GUGLIELMI, N. & TORELLI, L. (1997) Asymptotic stability properties of Theta-methods for the pantograph equation. Appl. Numer. Math., 24, 279-293.
5. BELLEN, A. & ZENNARO, M. (2003) Numerical Methods for Delay Differential Equations. Oxford: Oxford University Press.
6. BERGER, M. A. & WANG, Y. (1992) Bounded semigroups of matrices. Linear Algebra Appl., 166, 21-27.
7. BRUNNER, H., HU, Q. & LIN, Q. (2001) Geometric meshes in collocation methods for Volterra integral equations with proportional delays. IMA J. Numer. Anal., 21, 783-798.
8. BUHMANN, M. D. & ISERLES, A. (1993) Stability of the discretized pantograph differential equation. Math. Comput., 60, 575-589.
9. CARR, J. & DYSON, J. (1975a) The functional differential equation y′ (x) = ay (λx) + by (x). Proc. R. Soc. Edinb. A, 13, 165-174.
10. CARR, J. & DYSON, J. (1975b) The matrix functional differential equation y′ (x) = Ay (λx) + By (x). Proc. R. Soc. Edinb. 75A, 2, 5-22.
11. DAHLQUIST, G. (1959) Stability and error bounds in the numerical integration of ordinary differential equations. Trans. R. Inst. Technol. Stockh., 130, 87.
12. DEKKER, K. & VERWER, J. G. (1984) Stability of Runge-Kutta Methods for Stiff Nonlinear Differential Equations. CWI Monograph. Amsterdam: North Holland.
13. GILBERT, G. & DAVIES, H. A. H. (1966) Pantograph motion on a nearly uniform railway overhead line. Proc. IEE, 113, 485-492.
14. GUGLIELMI, N. & HAIRER, E. (2001) Implementing Radau IIA methods for stiff delay differential equations. Computing, 67, 1-12.
15. GUGLIELMI, N. & ZENNARO, M. (2003) Stability of one-leg Θ-methods for the variable coefficient pantograph equation on the quasi-geometric mesh. IMA J. Numer. Anal., 23, 421-438.
16. HUANG, C. & VANDEWALLE, S. (2005) Discretized stability and error growth of the non-autonomous pantograph equation. SIAM J. Numer. Anal. 42, 2020-2042.
17. ISERLES, A. (1993) On the generalized pantograph functional-differential equation. Eur. J. Appl. Math., 4, 1-38.
18. ISERLES, A. & TERJEKI, J. (1995) Stability and asymptotic stability of functional-differential equations. J. Lond. Math. Soc., 51, 559-572.
19. KATO, T. & MCLEOD, J. B. (1971) The functional-differential equation y′ (x) = ay (λx) + by (x). Bull. Am. Math. Soc., 77, 891-937.
20. KOTO, T. (1999) Stability of Runge-Kutta methods for the generalized pantograph equation. Numer. Math., 84, 233-247.
21. LIU, Y. (1996) Asymptotic behavior of functional differential equations with proportional time delays. Eur. J. Appl. Math., 7, 11-30.
22. OCKENDON, J. R. & TAYLER, A. B. (1971) The dynamics of a current collection system for an electric locomotive. Proc. R. Soc. Lond. A, 322, 447-468.
23. SHIH, M. H., WU, J. W. & PANG, C. T. (1997) Asymptotic stability and generalized Gelfand spectral radius formula. Linear Algebra Appl., 252, 61-70.
24. ZHANG, C. & SUN, G. (2002) The discrete dynamics of nonlinear infinite-delay differential equations. Appl. Math. Lett., 15, 521-526.