Stability of one-leg Θ-methods for the variable coefficient pantograph equation on the quasi-geometric mesh

IMA Journal of Numerical Analysis

Volumn 23, Issue 3, 2003, Pages 421-438

  Guglielmi, N ^a   Zennaro, M ^b  

^a UNIVERSITY OF L'AQUILA   (Italy)

^b UNIVERSITY OF TRIESTE   (Italy)

Author keywords

Constrained integration; Delay differential equation; Joint spectral radius; Polytope extremal norm; Quasi geometric mesh; Variable coefficient pantograph equation

Indexed keywords

DIFFERENTIAL EQUATIONS; MATRIX ALGEBRA; MESH GENERATION; NUMERICAL METHODS; PANTOGRAPHS; STABILITY;

DELAY DIFFERENTIAL EQUATIONS; EXTREMAL; JOINT SPECTRAL RADIUS; PANTOGRAPH EQUATION; QUASI-GEOMETRIC MESH;

INTEGRAL EQUATIONS;

EID: 0038166019     PISSN: 02724979     EISSN: None     Source Type: Journal    
DOI: 10.1093/imanum/23.3.421     Document Type: Article

References (23)

1. BAKER, C. T. H. & TANG, A. (2000) Generalized Halanay inequalities for Volterra functional differential equations and applications. Volterra Equations and Applications. (C. Corduneanu & I. W. Sandberg, eds). London: Gordon and Breach, pp. 39-55.
2. BELLEN, A. (2002) Preservation of superconvergence in the numerical integration of delay differential equations with proportional delay. IMA J. Numer. Anal., 22, 529-536.
3. BELLEN, A. & ZENNARO, M. (2003) Numerical Methods for Delay Differential Equations, volume in press. Oxford: Oxford University Press.
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. BERGER, M. A. & WANG, Y. (1992) Bounded semigroups of matrices. Linear Algebra Appl., 166, 21-27.
6. BUHMANN, M. D. & ISERLES, A. (1992) On the dynamics of a discretized neutral equation. IMA J. Numer. Anal, 12, 339-363.
7. BUHMANN, M. D. & ISERLES, A. (1993) Stability of the discretized pantograph differential equation. Math. Comp., 60, 575-589.
8. BUHMANN, M. D., ISERLES, A. & NÖRSETT, S. P. (1993) Runge-Kutta methods for neutral differential equations. WSSIA A, 2, 85-98.
9. 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.
10. CARR, J. & DYSON, J. (1975) The functional differential equation y′(x) = a y (λ x) + b y(x). Proc. Roy. Soc. Edinburgh A, 13, 165-174.
11. DAUBECHIES, I. & LAGARIAS, J. C. (1992) Sets of matrices all infinite products of which converge. Linear Algebra Appl., 161, 227-263.
12. ELSNER, L. (1995) The generalized spectral-radius theorem: an analytic-geometric proof. Linear Algebra Appl., 220, 151-159.
13. GUGLIELMI, N. & ZENNARO, M. (2001a) On the asymptotic properties of a family of matrices. Linear Algebra Appl., 322, 169-192.
14. GUGLIELMI, N. & ZENNARO, M. (2001b) On the zero-stability of variable stepsize multistep methods: the spectral radius approach. Numer. Math., 88, 445-458.
15. ISERLES, A. (1993) On the generalized pantograph functional-differential equation. Europ. J. Appl. Math., 4, 1-38.
16. ISERLES, A. & TERJEKI, J. (1995) Stability and asymptotic stability of functional-differential equations. J. London Math. Soc., 51, 559-572.
17. JACKIEWICZ, Z. (1984) Asymptotic stability analysis of θ-methods for functional differential equations. Numer. Math., 43, 389-396.
18. KATO, T. & McLEOD, J. B. (1971) The functional-differential equation y′(x) = a y(λx) + b y(x). Bull. Amer. Math. Soc., 77, 891-937.
19. KOTO, T. (1999) Stability of Runge-Kutta methods for the generalized pantograph equation. Numer. Math., 84, 233-247.
20. LIU, Y. (1995) Stability analysis of θ-methods for neutral functional-differential equations. Numer. Math., 70, 473-485.
21. LIU, Y. (1996) On the θ-method for delay differential equations with infinite lag. J. Comput. Appl. Math., 71, 177-190.
22. MOHAMAD, S. & GOPALSAMY, K. (2000) Continuous and discrete Halanay-type inequalities. Bull. Austral. Math. Soc., 61, 371-385.
23. ROTA, G. C. & STRANG, G. (1960) A note on the joint spectral radius. Indag. Math., 22, 379-381.