Continuous collocation approximations to solutions of first kind Volterra equations
HTML articles powered by AMS MathViewer
- by J.-P. Kauthen and H. Brunner PDF
- Math. Comp. 66 (1997), 1441-1459 Request permission
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.References
- Hermann Brunner, Discretization of Volterra integral equations of the first kind, Math. Comp. 31 (1977), no. 139, 708–716. MR 451794, DOI 10.1090/S0025-5718-1977-0451794-6
- Hermann Brunner, Discretization of Volterra integral equations of the first kind. II, Numer. Math. 30 (1978), no. 2, 117–136. MR 483586, DOI 10.1007/BF02042940
- 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, DOI 10.1007/BF02253135
- 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
- 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, DOI 10.1137/0720073
- Werner Greub, Linear algebra, 4th ed., Graduate Texts in Mathematics, No. 23, Springer-Verlag, New York-Berlin, 1975. MR 0369382, DOI 10.1007/978-1-4684-9446-4
- 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, DOI 10.1137/0720037
- 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
- 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, DOI 10.1007/978-3-662-09947-6
- Frank de Hoog and Richard Weiss, On the solution of Volterra integral equations of the first kind, Numer. Math. 21 (1973), 22–32. MR 371114, DOI 10.1007/BF01436183
- 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 373354, DOI 10.1137/0710057
- 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.
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
- Received by editor(s): March 16, 1995
- © Copyright 1997 American Mathematical Society
- 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