zbMATH — the first resource for mathematics

Computing breaking points in implicit delay differential equations. (English) Zbl 1160.65041
The authors are concerned with systems of implicit delay differential equations. The key aim of the paper is the computation of breaking points and the authors’ approach is to give a detailed description of the differences between the approaches that can be used, and the results obtained, for delay differential equations, as compared to ordinary differential equations (ODEs).
The paper is organised as follows: After a brief introduction, the authors focus on the ways in which Runge-Kutta methods are extended from methods for ODEs to apply to delay equations. As is well known, exact solutions of delay equations may not be smooth everywhere, and these breaking points need to be detected in the numerical schemes. These ideas are reviewed and a new algorithm is presented and discussed, in particular in relation to the ways in which breaking points are detected. Results show how the breaking point detected by the numerical scheme approximates the true breaking point of the underlying equation and the authors are able to give a new convergence result for the scheme. The paper concludes with some numerical experiments.

65L20 Stability and convergence of numerical methods for ordinary differential equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)
65L05 Numerical methods for initial value problems
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65L70 Error bounds for numerical methods for ordinary differential equations
Full Text: DOI
[1] Bellman, R., Cooke, K.L.: Differential-Difference Equations. Academic, New York (1963) · Zbl 0105.06402
[2] 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
[3] Brunner, H.: Collocation Methods for Volterra Integral and Related Functional Differential Equations. Cambridge University Press, Cambridge (2005) · Zbl 1059.65122
[4] Butcher, J.C.: The adaptation of stride to delay differential equations. Appl. Numer. Math. 9, 415–425 (1992) · Zbl 0776.65049 · doi:10.1016/0168-9274(92)90031-8
[5] Bellen, A., Zennaro, M.: Numerical Methods for Delay Differential Equations. Oxford University Press, Oxford (2003) · Zbl 1038.65058
[6] Castleton, R., Grimm, L.: A first order method for differential equations of neutral type. Math. Comp. 27, 571–577 (1973) · Zbl 0294.65044 · doi:10.1090/S0025-5718-1973-0343621-9
[7] Enright, W.H., Hayashi, H.: A delay differential equation solver based on a continuous Runge-Kutta method with defect control. Numer. Algorithms 16, 349–364 (1997) · Zbl 1005.65071 · doi:10.1023/A:1019107718128
[8] Enright, W.H., Hayashi, H.: The evaluation of numerical software for delay differential equations. In: The Quality of Numerical Software: Assessment and Enhancements, pp. 179–192. Chapman and Hall, London (1997)
[9] El’sgol’ts, L.E., Norkin, S.B.: Introduction to the Theory and Application of Differential Equations with Deviating Arguments. Academic, New York(1973)
[10] Feldstein, 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 · doi:10.1137/0721055
[11] Guglielmi, N., Hairer, E.: Implementing Radau-IIA methods for stiff delay differential equations. Computing 67, 1–12 (2001) · Zbl 0986.65069 · doi:10.1007/s006070170013
[12] Hauber, R.: Numerical treatment of retarded differential-algebraic equations by collocation methods. Adv. Comput. Math. 7, 573–592 (1997) · Zbl 0891.65080 · doi:10.1023/A:1018919508111
[13] Hairer, E., Nørsett, S.P., Wanner, G.: Solving ordinary differential equations I. Nonstiff Problems. In: Springer Series in Computational Mathematics, 2nd edition, vol. 8. Springer-Verlag, Berlin (1993) · Zbl 0789.65048
[14] Hairer, E., Wanner, G.: Solving ordinary differential equations II. Stiff and Differential-Algebraic Problems. In: Springer Series in Computational Mathematics, 2nd edition, vol. 14. Springer-Verlag, Berlin (1996) · Zbl 0859.65067
[15] Jackiewicz, Z., Lo, E.: The numerical solution of neutral functional differential equations by adams predictor-corrector methods. Appl. Numer. Math. 8, 477–491 (1991) · Zbl 0748.65057 · doi:10.1016/0168-9274(91)90110-L
[16] Kuang, Y.: On neutral delay logistic Gause-type predator-prey systems. Dynam. Stability Systems 6(2), 173–189 (1991) · Zbl 0728.92016
[17] Paul, C.A.H.: A Test Set of Functional Differential Equations. Numerical Analysis Report 243, University Manchester (1994)
[18] Waltman, P.: A threshold model of antigen-stimulated antibody production. In: Theoretical immunology. Immunology Ser., vol. 8, pp. 437–453. Dekker, New York (1978)
[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 · doi:10.1016/0168-9274(92)90016-7
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.