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Recent trends in the numerical solution of retarded functional differential equations*

Published online by Cambridge University Press:  08 May 2009

Alfredo Bellen ,
Stefano Maset ,
Marino Zennaro  and
Nicola Guglielmi
Alfredo Bellen
Affiliation:
Dipartimento di Matematica e Informatica, Università degli Studi di Trieste, I-34100 Trieste, ItalyE-mail:[email protected]
Stefano Maset
Affiliation:
Dipartimento di Matematica e Informatica, Università degli Studi di Trieste, I-34100 Trieste, ItalyE-mail:[email protected]
Marino Zennaro
Affiliation:
Dipartimento di Matematica e Informatica, Università degli Studi di Trieste, I-34100 Trieste, ItalyE-mail:[email protected]
Nicola Guglielmi
Affiliation:
Dipartimento di Matematica Pura e Applicata, Università degli Studi di L'Aquila, I-67100 L'Aquila, ItalyE-mail:[email protected]
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Abstract

Retarded functional differential equations (RFDEs) form a wide class of evolution equations which share the property that, at any point, the rate of the solution depends on a discrete or distributed set of values attained by the solution itself in the past. Thus the initial problem for RFDEs is an infinite-dimensional problem, taking its theoretical and numerical analysis beyond the classical schemes developed for differential equations with no functional elements. In particular, numerically solving initial problems for RFDEs is a diffcult task that cannot be founded on the mere adaptation of well-known methods for ordinary, partial or integro-differential equations to the presence of retarded arguments. Indeed, efficient codes for their numerical integration need speciffc approaches designed according to the nature of the equation and the behaviour of the solution.

By defining the numerical method as a suitable approximation of the solution map of the given equation, we present an original and unifying theory for the convergence and accuracy analysis of the approximate solution. Two particular approaches, both inspired by Runge–Kutta methods, are described. Despite being apparently similar, they are intrinsically different. Indeed, in the presence of speciffc types of functionals on the right-hand side, only one of them can have an explicit character, whereas the other gives rise to an overall procedure which is implicit in any case, even for non-stiff problems.

In the panorama of numerical RFDEs, some critical situations have been recently investigated in connection to speciffc classes of equations, such as the accurate location of discontinuity points, the termination and bifurcation of the solutions of neutral equations, with state-dependent delays, the regularization of the equation and the generalization of the solution behind possible termination points, and the treatment of equations stated in the implicit form, which include singularly perturbed problems and delay differential-algebraic equations as well. All these issues are tackled in the last three sections.

In this paper we have not considered the important issue of stability, for which we refer the interested reader to the comprehensive book by Bellen and Zennaro (2003).

Type
Research Article
Information
Acta Numerica , Volume 18 , May 2009 , pp. 1 - 110
Copyright
Copyright © Cambridge University Press 2009

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References

Baker, C. T. H. (1996), Numerical analysis of Volterra functional and integral equations, in The State of the Art in Numerical Analysis (Duff, I. S. and Watson, G. A., eds), Clarendon Press, Oxford.Google Scholar
Baker, C. T. H. (2000), ‘Retarded differential equations’, J. Comput. Appl. Math. 125, 309–335.CrossRefGoogle Scholar
Baker, C. T. H. and Paul, C. A. H. (2006), ‘Discontinuous solutions of neutral delay differential equations’, Appl. Numer. Math. 56, 284–304.CrossRefGoogle Scholar
Baker, C. T. H., Paul, C. A. H. and Willé, D. R. (1995 a), ‘Issues in the numerical solution of evolutionary delay differential equations’, Adv. Comput. Math. 3, 171–196.CrossRefGoogle Scholar
Baker, C. T. H., Paul, C. A. H. and Willé, D. R. (1995 b), A bibliography on the numerical solution of delay differential equations. NA Report 269, Department of Mathematics, University of Manchester.Google Scholar
Bellen, A. (1985), Constrained mesh methods for functional differential equations, in Delay Equations, Approximation and Application, Vol. 74 of Internat. Ser. Numer. Math., pp. 52–70.Google Scholar
Bellen, A. and Guglielmi, N. (2009), ‘Solving neutral delay differential equations with state dependent delays’, J. Comput. Appl. Math., in press.CrossRefGoogle Scholar
Bellen, A. and Zennaro, M. (2003), Numerical Methods for Delay Differential Equations, Numerical Mathematics and Scientific Computation, Oxford University Press, Oxford.CrossRefGoogle Scholar
Bellman, R. and Cooke, K. L. (1963), Differential-Difference Equations, Academic Press.Google Scholar
Brunner, H. (2004), Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge University Press, Cambridge.CrossRefGoogle Scholar
Butcher, J. C. (1976), ‘On the implementation of implicit Runge–Kutta methods’, BIT 6, 237–240.CrossRefGoogle Scholar
Castleton, R. N. and Grimm, L. J. (1973), ‘A first order method for differential equations of neutral type’, Math. Comput. 27, 571–577.CrossRefGoogle Scholar
Cooke, K. L. and Wiener, J. (1984), ‘Retarded differential equations with piecewise constant delays’, J. Math. Anal. Appl. 99, 265–297.CrossRefGoogle Scholar
Cryer, C. W. (1972), Numerical methods for functional differential equations, in Delay and Functional Differential Equations and their Applications (Schmitt, K., ed.), Academic Press, New York, pp. 17–101.CrossRefGoogle Scholar
Cryer, C. W. and Tavernini, L. (1972), ‘The numerical solution of Volterra functional differential equations by Euler’s method’, SIAM J. Numer. Anal. 9, 105–129.CrossRefGoogle Scholar
Diekmann, O., van Gils, S. A., Lunel, S. M. Verduyn and Walther, H. O. (1995), Delay Equations: Functional-, Complex-, and Nonlinear Analysis, AMS series, Springer, Berlin.CrossRefGoogle Scholar
Driver, R. D. (1977), Ordinary and Delay Differential Equations, Springer, Berlin.CrossRefGoogle Scholar
El'sgol'ts, L. E. and Norkin, S. B. (1973), Introduction to the Theory and Application of Differential Equations with Deviating Arguments, Academic Press, New York.Google Scholar
Enright, W. H. and Hayashi, H. (1997), ‘A delay differential equation solver based on a continuous Runge–Kutta method with defect control’, Numer. Algorithms 16, 349–364.CrossRefGoogle Scholar
Enright, W. H., Jackson, K. R., Nørsett, S. P. and Thomsen, P. G. (1988), ‘Effective solution of discontinuous IVPs using a Runge–Kutta formula pair with interpolants’, Appl. Math. Comput. 27, 313–335.Google Scholar
Feldstein, A. (1964), Discretization methods for retarded ordinary differential equation. PhD Thesis, Department of Mathematics, UCLA, Los Angeles.Google Scholar
Feldstein, A. and Neves, K. W. (1984), ‘High order methods for state-dependent delay differential equations with nonsmooth solutions’, SIAM J. Numer. Anal. 21, 844–863.CrossRefGoogle Scholar
Feldstein, A., Neves, K. W. and Thompson, S. (2006), ‘Sharpness results for state dependent delay differential equations: An overview’, Appl. Numer. Math. 56, 472–487.CrossRefGoogle Scholar
Filippov, A. F. (1964), ‘Differential equations with discontinuous right-hand sides’, Trans. Amer. Math. Soc. 42, 199–231.Google Scholar
Filippov, A. F. (1988), Differential Equations with Discontinuous Righthand Sides, Vol. 18 of Mathematics and its Applications (Soviet Series), Kluwer Academic, Dordrecht (translated from the Russian).CrossRefGoogle Scholar
Fusco, G. and Guglielmi, N. (2009), A regularization for discontinuous differential equations with application to state-dependent delay differential equations of neutral type. In preparation.Google Scholar
Guglielmi, N. (2005), ‘On the Newton iteration in the application of collocation methods to implicit delay equations’, Appl. Numer. Math. 53, 281–297.CrossRefGoogle Scholar
Guglielmi, N. and Hairer, E. (2001), ‘Implementing Radau IIA methods for stiff delay differential equations’, Computing 67, 1–12.CrossRefGoogle Scholar
Guglielmi, N. and Hairer, E. (2008), ‘Computing breaking points in implicit delay differential equations’, Adv. Comput. Math. 29, 229–247.CrossRefGoogle Scholar
Hairer, E. and Wanner, G. (1996), Solving Ordinary Differential Equations II: Stiff and Differential Algebraic Problems, Springer Series in Computational Mathematics, Springer, Berlin.CrossRefGoogle Scholar
Hale, J. K. (1977), Theory of Functional Differential Equations, Springer, New York.CrossRefGoogle Scholar
Hale, J. K. and Lunel, S. M. Verduyn (1993), Introduction to Functional Differential Equations, Applied Mathematical Sciences, Springer, New York.CrossRefGoogle Scholar
Hauber, R. (1997), ‘Numerical treatment of retarded differential-algebraic equations by collocation methods’, Adv. Comput. Math. 7, 573–592.CrossRefGoogle Scholar
Kolmanovskii, V. and Myshkis, A. (1992), Applied Theory of Functional Differential Equations, Kluwer, Dordrecht.CrossRefGoogle Scholar
Kolmanovskii, V. and Nosov, V. (1986), Stability of Functional Differential Equations, Academic Press, London.Google Scholar
Kuang, J. and Cong, Y. (2005), Stability of Numerical Methods for Delay Differential Equations, Science Press, Beijing.Google Scholar
Kuang, Y. (1993), Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston.Google Scholar
Maset, S. (2009), Theoretical and numerical analysis of retarded functional differential equations. In preparation.Google Scholar
Maset, S., Torelli, L. and Vermiglio, R. (2005), ‘Runge–Kutta methods for retarded functional differential equations’, Math. Models Methods Appl. Sci. 15, 1203–1251.CrossRefGoogle Scholar
Meinardus, G. and Nürnberger, G. (1985), Approximation theory and numerical methods for delay differential equations in Delay Equations, Approximation and Application, Vol. 74 of Internat. Ser. Numer. Math., pp. 13–40.CrossRefGoogle Scholar
Neves, K. W. (1975), ‘Automatic integration of functional differential equations: An approach’, ACM Trans. Math. Software 1, 357–368.CrossRefGoogle Scholar
Paul, C. A. H. (1994), A test set of functional differential equation. NA Report 243, Department of Mathematics, University of Manchester.Google Scholar
Shampine, L. F. and Gahinet, P. (2006), ‘Delay-differential-algebraic equations in control theory’, Appl. Numer. Math. 56, 574–588.CrossRefGoogle Scholar
Shampine, L. F. and Thompson, S. (2000), ‘Event location for ordinary differential equations’, Comput. Math. Appl. 39, 43–54.CrossRefGoogle Scholar
Tavernini, L. (1971), ‘One-step methods for the numerical solution of Volterra functional differential equations’, SIAM J. Numer. Anal. 4, 786–795.CrossRefGoogle Scholar
Waltman, P. (1978), A threshold model of antigen-stimulated antibody production, in Theoretical Immunology, Vol. 8 of Immunology series, Dekker, New York, pp. 437–453.Google Scholar
Wang, W. and Li, S. (2004), ‘Stability analysis of nonlinear delay differential equations of neutral type’, Math. Numer. Sin. 26, 303–314.Google Scholar
Willé, D. R. and Baker, C. T. H. (1992), ‘The tracking of derivative discontinuities in systems of delay differential equations’, Appl. Numer. Math. 9, 299–222.Google Scholar
Zennaro, M. (1995), Delay differential equations: Theory and numerics, in Theory and Numerics of Ordinary and Partial Differential Equations (Ainsworth, M., Levesley, J., Light, W. A. and Marletta, M., eds), Clarendon Press, Oxford, pp. 291–333.CrossRefGoogle Scholar