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

MathSciNet review:
1434941

Full-text PDF Free Access

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.**Hermann Brunner,*Discretization of Volterra integral equations of the first kind*, Math. Comp.**31**(1977), no. 139, 708–716. MR**0451794**, 10.1090/S0025-5718-1977-0451794-6**2.**Hermann Brunner,*Discretization of Volterra integral equations of the first kind. II*, Numer. Math.**30**(1978), no. 2, 117–136. MR**0483586****3.**H. Brunner,*Superconvergence of collocation methods for Volterra integral equations of the first kind*, Computing**21**(1978/79), no. 2, 151–157 (English, with German summary). MR**619921**, 10.1007/BF02253135**4.**H. Brunner and P. J. van der Houwen,*The numerical solution of Volterra equations*, CWI Monographs, vol. 3, North-Holland Publishing Co., Amsterdam, 1986. MR**871871****5.**P. P. B. Eggermont,*Collocation for Volterra integral equations of the first kind with iterated kernel*, SIAM J. Numer. Anal.**20**(1983), no. 5, 1032–1048. MR**714698**, 10.1137/0720073**6.**Werner Greub,*Linear algebra*, 4th ed., Springer-Verlag, New York-Berlin, 1975. Graduate Texts in Mathematics, No. 23. MR**0369382****7.**E. Hairer, Ch. Lubich, and S. P. Nørsett,*Order of convergence of one-step methods for Volterra integral equations of the second kind*, SIAM J. Numer. Anal.**20**(1983), no. 3, 569–579. MR**701097**, 10.1137/0720037**8.**E. Hairer, S. P. Nørsett, and G. Wanner,*Solving ordinary differential equations. I*, 2nd ed., Springer Series in Computational Mathematics, vol. 8, Springer-Verlag, Berlin, 1993. Nonstiff problems. MR**1227985****9.**E. Hairer and G. Wanner,*Solving ordinary differential equations. II*, Springer Series in Computational Mathematics, vol. 14, Springer-Verlag, Berlin, 1991. Stiff and differential-algebraic problems. MR**1111480****10.**Frank de Hoog and Richard Weiss,*On the solution of Volterra integral equations of the first kind*, Numer. Math.**21**(1973), 22–32. MR**0371114****11.**Frank de Hoog and Richard Weiss,*High order methods for Volterra integral equations of the first kind*, SIAM J. Numer. Anal.**10**(1973), 647–664. MR**0373354****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.

Retrieve articles in *Mathematics of Computation of the American Mathematical Society*
with MSC (1991):
65R20,
45L10

Retrieve articles in all journals with MSC (1991): 65R20, 45L10

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:
http://dx.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