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Discontinuous solutions of neutral delay differential equations. (English) Zbl 1105.34055
It is well known that the solutions of delay differential and implicit and explicit neutral delay differential equations (NDDEs) may have discontinuous derivatives, but it has not been sufficiently appreciated that the solutions of NDDEs – and, therefore, solutions of delay differential algebraic equations – need not to be continuous. Numerical codes for solving differential equations, with or without retarded arguments, are generally based on the assumption that a solution is continuous.
In this paper, the authors illustrate and explain how the discontinuities arise, and present methods to deal with these problems computationally. These methods include discontinuity tracking; perturbing the initial function; direct treatment of NDDEs in Hale’s form; singular perturbation methods for NDDEs in Hale’s form. The investigation of a simple example is followed by a discussion of more general NDDEs and further mathematical details.

MSC:
34K40 Neutral functional-differential equations
34K05 General theory of functional-differential equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
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[1] 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
[2] Baker, C.T.H., Retarded differential equations, J. comput. appl. math., 125, 309-335, (2000) · Zbl 0970.65079
[3] C.T.H. Baker, C.A.H. Paul, Piecewise continuous solutions of neutral delay differential equations, MCCM Tech. Rep. 417, 2004, Manchester, ISSN: 1360-1725
[4] Baker, C.T.H.; Paul, C.A.H.; Tian, H.J., Differential algebraic equations with after-effect, J. comput. appl. math., 140, 63-80, (2002) · Zbl 0996.65077
[5] Bellen, A.; Zennaro, M., Numerical methods for delay differential equations, ISBN: 0-19-850654-6, (2003), Oxford University Press Oxford · Zbl 0749.65042
[6] El’sgol’ts, L.E.; Norkin, S.B., Introduction to the theory and application of differential equations with deviating arguments, (1973), Academic Press New York · Zbl 0287.34073
[7] 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
[8] Feldstein, M.A.; Neves, K.W., High order methods for state-dependent delay differential equations with nonsmooth solutions, SIAM J. numer. anal., 21, 844-863, (1984) · Zbl 0572.65062
[9] Guglielmi, N.; Hairer, E., On the error control in the numerical integration of implicit delay differential equations, in: Proceedings NUMDIFF10, Halle, 2003, Abstracts, pp. 23-24 at:
[10] Hartung, F.; Herdman, T.L.; Turi, J., On existence and numerical approximation for neutral equations with state-dependent delays, Appl. numer. math., 24, 393-409, (1997) · Zbl 0939.65101
[11] Hauber, R., Numerical treatment of retarded differential-algebraic equations by collocation methods, J. adv. comput. math., 7, 573-592, (1997) · Zbl 0891.65080
[12] Jackiewicz, Z.; Lo, E., The numerical integration of neutral functional-differential equations by fully implicit one-step methods, Z. angew. math. mech., 75, 207-221, (1995) · Zbl 0830.65079
[13] Jankowski, T.; Kwapisz, M., Convergence of numerical methods for systems of neutral functional-differential-algebraic equations, Appl. math., 40, 457-472, (1995) · Zbl 0853.65077
[14] Kolmanovskiĭ, V.B.; Myshkis, A., Introduction to the theory and applications of functional-differential equations, ISBN: 0-7923-5504-0, (1999), Kluwer Academic Dordrecht · Zbl 0917.34001
[15] Liu, Y., Numerical solution of implicit neutral functional-differential equations, SIAM J. numer. anal., 36, 516-528, (1999) · Zbl 0920.65045
[16] Neves, K.W., Automatic integration of functional differential equations: an approach, ACM trans. math. software, 1, 357-368, (1975) · Zbl 0315.65045
[17] 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
[18] C.A.H. Paul, A User-Guide to Archi—An explicit Runge-Kutta code for solving delay and neutral differential equations and parameter estimation problems, MCCM Tech. Rep. 283, 1997, Manchester, ISSN 1360-1725
[19] Willé, D.R.; Baker, C.T.H., The tracking of derivative discontinuities in systems of delay differential equations, Appl. numer. math., 9, 209-222, (1992) · Zbl 0747.65054
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