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Baker, C. T. H.; Paul, C. A. H.; Tian, H.

Differential algebraic equations with after-effect. (English) Zbl 0996.65077

J. Comput. Appl. Math. 140, No. 1-2, 63-80 (2002).
Summary: We are concerned with the solution of delay differential algebraic equations. These are differential algebraic equations with after-effect, or constrained delay differential equations. The general semi-explicit form of the problem consists of a set of delay differential equations combined with a set of constraints that may involve retarded arguments. Even simply stated problems of this type can give rise to difficult analytical and numerical problems. The more tractable examples can be shown to be equivalent to systems of delay or neutral delay differential equations. Our purpose is to highlight some of the complexities and obstacles that can arise when solving these problems, and to indicate problems that require further research.

Cited in 16 Documents

MSC:

65L80 Numerical methods for differential-algebraic equations
34E15 Singular perturbations for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations
34A09 Implicit ordinary differential equations, differential-algebraic equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)

Keywords:

index reduction; singular perturbation; delay differential algebraic equations; neutral delay differential equations

Software:

RODAS
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Full Text: DOI

References:

[1] Arndt, H., Numerical solution of retarded initial value problems: local and global error and stepsize control, Numer. Math., 43, 343-360 (1984) · Zbl 0518.65053
[2] Ascher, U. M.; Petzold, L. R., The numerical solution of delay-differential-algebraic equations of retarded and neutral type, SIAM J. Numer. Anal., 32, 1635-1657 (1995) · Zbl 0837.65070
[3] Baker, C. T.H., Retarded differential equations, J. Comput. Appl. Math., 125, 309-335 (2000) · Zbl 0970.65079
[4] Baker, C. T.H.; Paul, C. A.H.; Willé, D. R., Issues in the numerical solution of evolutionary delay differential equations, Adv. Comput. Math., 3, 171-196 (1995) · Zbl 0832.65064
[5] A. Bellen, M. Zennaro, Numerical methods for delay differential equations, in preparation.; A. Bellen, M. Zennaro, Numerical methods for delay differential equations, in preparation. · Zbl 1038.65058
[6] Brenan, K. E.; Campbell, S. L.; Petzold, L. R., Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations (1996), SIAM: SIAM Philadelphia · Zbl 0844.65058
[7] Brown, P. N.; Hindmarsh, A. C.; Petzold, L. R., Consistent initial condition calculation for differential-algebraic systems, SIAM J. Sci. Comput., 19, 1495-1512 (1998) · Zbl 0915.65079
[8] Campbell, S. L., Singular linear systems of differential equations with delays, Appl. Anal., 11, 129-136 (1980) · Zbl 0444.34062
[9] S.L. Campbell, 2-D (differential-difference) implicit systems, Proceedings of the 13th IMACS World Congress on Scientific Computation, Dublin, 1991, pp. 1828-1829.; S.L. Campbell, 2-D (differential-difference) implicit systems, Proceedings of the 13th IMACS World Congress on Scientific Computation, Dublin, 1991, pp. 1828-1829.
[10] Campbell, S. L., Non-regular descriptor systems with delays, IMA J. Math. Control Inform., 12, 57-67 (1995) · Zbl 0835.34085
[11] Èl’sgol’c, L.È., Qualitative Methods in Mathematical Analysis. Qualitative Methods in Mathematical Analysis, Translations of Mathematical Monographs, Vol. 12 (1964), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0476.49001
[12] Enright, W. H.; Hayashi, H., Convergence analysis of the solution of retarded and neutral delay differential equations by continuous numerical methods, SIAM J. Numer. Anal., 35, 572-585 (1998) · Zbl 0914.65084
[13] Gear, C. W.; Petzold, L. R., ODE methods for the solution of differential-algebraic systems, SIAM J. Numer. Anal., 21, 367-384 (1984) · Zbl 0557.65053
[14] Hairer, E.; Nørsett, S. P.; Wanner, G., Solving Ordinary Differential Equations. I Non-stiff Problems (1980), Springer: Springer Berlin
[15] Hairer, E.; Jay, L., Implicit Runge-Kutta methods for higher index differential-algebraic systems, WSSIAA, 2, 213-224 (1993) · Zbl 0834.65062
[16] Hairer, E.; Lubich, C.; Roche, M., The Numerical Treatment of Differential Algebraic Systems by Runge-Kutta Methods. The Numerical Treatment of Differential Algebraic Systems by Runge-Kutta Methods, Lecture Notes in Mathematics, Vol. 1409 (1989), Springer: Springer Berlin · Zbl 0683.65050
[17] Hairer, E.; Wanner, G., Solving Ordinary Differential Equations. II Stiff and Differential-algebraic Problems (1996), Springer: Springer Berlin · Zbl 0859.65067
[18] Hauber, R., Numerical treatment of retarded differential-algebraic equations by collocation methods, Adv. Comput. Math., 7, 573-592 (1997) · Zbl 0891.65080
[19] In’t Hout, K. J., A new interpolation procedure for adapting Runge-Kutta methods to delay differential equations, BIT, 32, 634-649 (1992) · Zbl 0765.65069
[20] In’t Hout, K. J., The stability of a class of Runge-Kutta methods for delay differential equations, Appl. Numer. Math., 9, 347-355 (1992) · Zbl 0749.65048
[21] Jankowski, T.; Kwapisz, M., Convergence of numerical methods for systems of neutral functional differential-algebraic equations, Appl. Math., 6, 457-472 (1995) · Zbl 0853.65077
[22] Liu, Y., Runge-Kutta collocation methods for systems of functional differential and functional equations, Adv. Comput. Math., 11, 315-329 (1999) · Zbl 0948.65069
[23] Neves, K. W.; Feldstein, M. A., Characterization of jump discontinuities for state dependent delay differential equations, J. Math. Anal. Appl., 56, 689-707 (1976) · Zbl 0348.34054
[24] Oberle, H. J.; Pesch, H. J., Numerical treatment of delay differential equations by Hermite interpolation, Numer. Math., 37, 235-255 (1981) · Zbl 0469.65057
[25] Paul, C. A.H., Developing a delay differential equation solver, Appl. Num. Math., 9, 403-414 (1992) · Zbl 0779.65043
[26] C.A.H. Paul, The treatment of derivative discontinuities in differential equations, Proceedings HERMCA 98, LEA, Athens, 1999.; C.A.H. Paul, The treatment of derivative discontinuities in differential equations, Proceedings HERMCA 98, LEA, Athens, 1999.
[27] Petzold, L. R., Differential-algebraic equations are not ODEs, SIAM J. Sci. Statist. Comput., 3, 367-384 (1982) · Zbl 0482.65041
[28] L.R. Petzold, Numerical solution of differential-algebraic equations, in: M. Ainsworth, L. Levesley, W.A. Light, M. Marletta (Eds.), Theory and Numerics of Ordinary and Partial Differential Equations, Clarendon Press, Oxford, 1995.; L.R. Petzold, Numerical solution of differential-algebraic equations, in: M. Ainsworth, L. Levesley, W.A. Light, M. Marletta (Eds.), Theory and Numerics of Ordinary and Partial Differential Equations, Clarendon Press, Oxford, 1995.
[29] H. Tian, Ph.D. Thesis, University of Manchester, UK (due to be submitted 2000).; H. Tian, Ph.D. Thesis, University of Manchester, UK (due to be submitted 2000).
[30] Willé, D. R.; Baker, C. T.H., The tracking of derivative discontinuities in systems of delay differential equations, Appl. Num. Math., 9, 209-222 (1992) · Zbl 0747.65054
[31] Zennaro, M., Natural continuous extensions of Runge-Kutta methods, Math. Comput., 46, 119-133 (1986) · Zbl 0608.65043
[32] Zennaro, M., Delay differential equations: theory and numerics, (Ainsworth, M.; Levesley, L.; Light, W. A.; Marletta, M., Theory and Numerics of Ordinary and Partial Differential Equations (1995), Clarendon Press: Clarendon Press Oxford) · Zbl 0847.34072
[33] Zhu, W.; Petzold, L. R., Asymptotic stability of Hessenberg delay differential-algebraic equations of retarded or neutral type, Appl. Numer. Math, 27, 309-325 (1998) · Zbl 0937.34059
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