Continuous collocation approximations

to solutions of first kind Volterra equations

Authors:
J.-P. Kauthen and H. Brunner

Journal:
Math. Comp. **66** (1997), 1441-1459

MSC (1991):
Primary 65R20, 45L10

DOI:
https://doi.org/10.1090/S0025-5718-97-00905-8

MathSciNet review:
1434941

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we give necessary and sufficient conditions for convergence of continuous collocation approximations of solutions of first kind Volterra integral equations. The results close some longstanding gaps in the theory of polynomial spline collocation methods for such equations. The convergence analysis is based on a Runge-Kutta or ODE approach.

**1.**H. Brunner,*Discretization of Volterra integral equations of the first kind*, Math. Comp., 31 (1977), 708-716. MR**56:10076****2.**H. Brunner,*Discretization of Volterra integral equations of the first kind (II)*, Numer. Math., 30 (1978), 117-136. MR**58:3578****3.**H. Brunner,*Superconvergence of collocation methods for Volterra integral equations of the first kind*, Computing, 21 (1979), 151-157. MR**83a:65125****4.**H. Brunner and P.J. van der Houwen,*The Numerical Solution of Volterra Equations*, North-Holland, Amsterdam, 1986. MR**88g:65136****5.**P.P.B. Eggermont,*Collocation for Volterra integral equations of the first kind with iterated kernel*, SIAM J. Numer. Anal., 20 (1983), 1032-1048. MR**85i:65170****6.**W. Greub,*Linear Algebra*, Fourth Edition, Springer-Verlag, New York Heidelberg Berlin, 1975. MR**51:5615****7.**E. Hairer, Ch. Lubich and S.P. Nø rsett,*Order of convergence of one-step methods for Volterra integral equations of the second kind*, SIAM J. Numer. Anal., 20 (1983), 569-579. MR**84g:65163****8.**E. Hairer, S.P. Nørsett and G. Wanner,*Solving Ordinary Differential Equations I. Nonstiff Problems*, Second Revised Edition, Springer-Verlag, Berlin Heidelberg, 1993. MR**94c:65005****9.**E. Hairer and G. Wanner,*Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems*, Springer-Verlag, Berlin Heidelberg, 1991. MR**92a:65016****10.**F. de Hoog and R. Weiss,*On the solution of Volterra integral equations of the first kind*, Numer. Math., 21 (1973), 22-32. MR**51:7335****11.**F. de Hoog and R. Weiss,*High order methods for Volterra integral equations of the first kind*, SIAM J. Numer. Anal., 10 (1973), 647-664. MR**51:9554****12.**H.S. Hung,*The numerical solution of differential and integral equations by spline functions*, MRC Tech. Summary Rep. 1053, Mathematics Research Center, University of Wisconsin, Madison, 1970.

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Additional Information

**J.-P. Kauthen**

Affiliation:
Institut de Mathématiques, Université de Fribourg, CH-1700 Fribourg, Switzerland

Email:
jean-paul.kauthen@unifr.ch, kauthen@bluewin.ch

**H. Brunner**

Affiliation:
Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Newfoundland, Canada A1C 5S7

Email:
hbrunner@morgan.ucs.mun.ca

DOI:
https://doi.org/10.1090/S0025-5718-97-00905-8

Keywords:
Integral equation,
collocation method,
Runge-Kutta method

Received by editor(s):
March 16, 1995

Article copyright:
© Copyright 1997
American Mathematical Society