Abstract
A collocation procedure with polynomial and piecewise polynomial approximation is considered for second order functional differential equations with two side-conditions. The piecewise polynomials are taken in the classC 1 and reduce to polynomials of increasing degree on each interval of a suitable assigned partition. Appropriate choices of the partition are made, according to the jump discontinuities in the derivatives caused by the functional argument, in order to optimize the rate of convergence.
Zusammenfassung
Es wird eine Kollokationsmethode mit polynomialer und stückweise polynomialer Approximation für Funktional-Differentialgleichungen zweiter Ordnung mit zweiseitigen Bedingungen betrachtet. Die stückweisen Polynome werden in der KlasseC 1 genommen und bestehen aus Polynomen mit ansteigendem Grad in jedem Intervall einer günstig gegebenen Partition. Günstige Wahlen der Partition werden gemacht, gemäß der Diskontinuitäten der Ableitungen, die durch die Funktional-Argumente verursacht werden, dermaßen, daß die Konvergenzgeschwindigkeit optimal werde.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ascher, U., Russel, R. D.: Reformulation of boundary value problems into “standard” form. SIAM. Rev.23, 238–254 (1981).
Bellen, A.: Cohen's iteration process for boundary value problems for functional differential equations. Rend. Ist. Mat. Univ. Trieste11, 32–46 (1979).
Bernsfeld, S. R., Lakshmikantham, V.: Introduction to nonlinear boundary value problems. New York: Academic Press 1974.
Chocholaty, P., Slahor, L.: A method to boundary value problems for delay equations. Numer. Math.33, 69–75 (1979).
De Boor, C., Swartz B.: Collocation at gaussian points. SIAM J. Numer. Anal.10, 582–606 (1973).
De Nevers, K., Schmitt, K.: An application of the shooting method to boundary value problems for second order delay equations. J. Math. Anal. Appl.36, 588–597 (1971).
Driver, R. D.: Ordinary and delay differential equations. (Applied Math. Science Series.) Berlin-Heidelberg-New York: Springer 1976.
Grimm, L. J., Schmitt, K.: Boundary value problems for delay-differential equations. Bull. Amer. Math. Soc.74, 997–1000 (1978).
Reddien, G. W., Travis, C. C.: Approximation methods for boundary value problems of differential equations with functional arguments. J. Math. Anal. Appl.46, 62–74 (1974).
Szegö, G.: Orthogonal polynomials. A.M.S. Coll. Publ. 23. Providence, R.I.: 1939.
Vainikko, G. M.: On the convergence of the collocation method for nonlinear differential equations. USSR Comp. Math. and Math. Phys.6, 35–42 (1966).
Author information
Authors and Affiliations
Additional information
This work was supported by the C.N.R. (Italian National Council of Researches) within the project “Informatica” subproject “Sofmat”.
Rights and permissions
About this article
Cite this article
Bellen, A., Zennaro, M. A collocation method for boundary value problems of differential equations with functional arguments. Computing 32, 307–318 (1984). https://doi.org/10.1007/BF02243775
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02243775