Peculiar spline collocation method for solving rough and stiff delay differential problems

F. Calio; E. Marchetti; R. Pavani

doi:10.33993/jnaat331-756

Authors

F. Calio Politecnico di Milano, Italy
E. Marchetti Politecnico di Milano, Italy
R. Pavani Politecnico di Milano, Italy

DOI:

https://doi.org/10.33993/jnaat331-756

Keywords:

NDDE, collocation spline, deficient spline, stiff and rough problems

Abstract views: 185

Abstract

As well known, solutions of delay differential equations (DDEs) are characterized by low regularity. In particular solutions of neutral delay differential equations (NDDEs) frequently exhibit discontinuities in the first derivative so that the differential problems become rough. The aim of this paper is to approximate the solutions of such rough delay differential problems by means of a peculiar deficient spline collocation method. Significant numerical examples are provided to enlighten the features of the proposed method.

Downloads

Download data is not yet available.

References

Asher, U. M. and Petzold, L. R., Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations, SIAM, Philadelphia, 1998, https://doi.org/10.1137/1.9781611971392 DOI: https://doi.org/10.1137/1.9781611971392

Bellen, A. and Micula, G., Spline approximations for neutral delay differential equations, Rev. Anal. Numér. Théor. Approx., 23, pp. 117-125, 1994.

Brugnano, L. and Trigiante, D., Solving Differential Problems by Multistep Initial and Boundary Value methods, Gordon and Breach, The Netherlands, 1998.

Caliò, F., Marchetti, E. Micula, G. and Pavani, R., A new deficient spline functions collocation method for the second order delay differential equations, PU.M.A., 13, N. 1-2, pp. 97-109, 2002.

Caliò, F., Marchetti, E., and Pavani, R. On existence and uniqueness of first order NDDE solution obtained by deficient spline collocation, Dip. di Matematica, Politecnico di Milano, Internal Report no. 482/P, 2001.

Caliò, F., Marchetti, E. and Pavani, R., An algorithm about the deficient spline collocation method for particular differential and integral equations with delay, Rend. Sem. Mat. Univ. Torino, 61 N. 3, pp. 287-300, 2004.

Driver, R. D., Ordinary and delay differential equations, Springer-Verlag, Berlin, 1977. DOI: https://doi.org/10.1007/978-1-4684-9467-9

Enright, W. H. and Hayashi, H., Some inherent difficulties and complications in the numerical solution of neutral delay differential equations, Proceedings of Fourth Hellenic-European Conference on Computer Mathematics and its Applications, E. A. Lipitakis Ed., Athens, pp. 139-149, 1998.

Guglielmi, N. and Hairer, E., Implementing Radau IIA methods for stiff delay differential equations, Computing, 65, 2001, https://doi.org/10.1007/s006070170013. DOI: https://doi.org/10.1007/s006070170013

Loscalzo, F. R. and Talbot, T. D., Spline function approximations for solutions of ordinary differential equations, SIAM J. Numer. Anal., 1, 433-445, 1967, https://doi.org/10.1137/0704038 DOI: https://doi.org/10.1137/0704038

Micula, G. and Micula, S., Handbook of Splines, Kluwer Acad. Publishers, Dordrecht-London-Boston, 1999. DOI: https://doi.org/10.1007/978-94-011-5338-6

Shampine, L. F. and Thompson, S., Solving Delay Differential Equations with dde23, tutorial available at http://www.runet.edu/~thompson/webddes/.

Wille', D. R. and Baker, C. T. H., DELSOL: a numerical code for the solution of systems of delay-differential equations, Appl. Numer. Math., 9, 223-234, 1992, https://doi.org/10.1016/0168-9274(92)90017-8 DOI: https://doi.org/10.1016/0168-9274(92)90017-8

Zennaro M., P-stability properties of Runge-Kutta methods for delay differential equations, Numer. Math., 49, pp. 305-318, 1986, https://doi.org/10.1007/bf01389632 DOI: https://doi.org/10.1007/BF01389632