## Approximation properties of quadrature methods for Volterra integral equations of the first kind

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- by P. P. B. Eggermont PDF
- Math. Comp.
**43**(1984), 455-471 Request permission

## Abstract:

We present a unifying analysis of quadrature methods for Volterra integral equations of the first kind that are zero-stable and have an asymptotic repetition factor. We show that such methods are essentially collocation-projection methods with underlying subspaces that have nice approximation properties, and which are stable as projection methods. This is used to derive asymptotically optimal error estimates under minimal smoothness conditions. The class of quadrature methods covered includes the cyclic linear multistep and the reducible quadrature methods, but not (really) Runge-Kutta methods.## References

- Celia Andrade and S. McKee,
*On optimal high accuracy linear multistep methods for first kind Volterra integral equations*, BIT**19**(1979), no. 1, 1–11. MR**530109**, DOI 10.1007/BF01931215 - K. Atkinson and A. Sharma,
*A partial characterization of poised Hermite-Birkhoff interpolation problems*, SIAM J. Numer. Anal.**6**(1969), 230–235. MR**264828**, DOI 10.1137/0706021 - Melvin S. Berger,
*Nonlinearity and functional analysis*, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1977. Lectures on nonlinear problems in mathematical analysis. MR**0488101** - Hermann Brunner,
*A survey of recent advances in the numerical treatment of Volterra integral and integro-differential equations*, J. Comput. Appl. Math.**8**(1982), no. 3, 213–229. MR**682889**, DOI 10.1016/0771-050X(82)90044-4 - 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 - P. P. B. Eggermont,
*Collocation as a projection method and superconvergence for Volterra integral equations of the first kind*, Treatment of integral equations by numerical methods (Durham, 1982) Academic Press, London, 1982, pp. 131–138. MR**755348** - Charles J. Gladwin,
*Quadrature rule methods for Volterra integral equations of the first kind*, Math. Comp.**33**(1979), no. 146, 705–716. MR**521284**, DOI 10.1090/S0025-5718-1979-0521284-2 - C. J. Gladwin,
*On optimal integration methods for Volterra integral equations of the first kind*, Math. Comp.**39**(1982), no. 160, 511–518. MR**669643**, DOI 10.1090/S0025-5718-1982-0669643-1 - P. A. W. Holyhead, S. McKee, and P. J. Taylor,
*Multistep methods for solving linear Volterra integral equations of the first kind*, SIAM J. Numer. Anal.**12**(1975), no. 5, 698–711. MR**413564**, DOI 10.1137/0712052 - Malcolm S. Keech,
*A third order, semi-explicit method in the numerical solution of first kind Volterra integral equations*, Nordisk Tidskr. Informationsbehandling (BIT)**17**(1977), no. 3, 312–320. MR**474918**, DOI 10.1007/bf01932151 - M. A. Krasnosel′skiĭ, G. M. Vaĭnikko, P. P. Zabreĭko, Ya. B. Rutitskii, and V. Ya. Stetsenko,
*Approximate solution of operator equations*, Wolters-Noordhoff Publishing, Groningen, 1972. Translated from the Russian by D. Louvish. MR**0385655** - George G. Lorentz, Kurt Jetter, and Sherman D. Riemenschneider,
*Birkhoff interpolation*, Encyclopedia of Mathematics and its Applications, vol. 19, Addison-Wesley Publishing Co., Reading, Mass., 1983. MR**680938** - S. McKee,
*Best convergence rates of linear multistep methods for Volterra first kind equations*, Computing**21**(1978/79), no. 4, 343–358 (English, with German summary). MR**620379**, DOI 10.1007/BF02248734 - Theodor Meis,
*Eine spezielle Integralgleichung erster Art*, Numerical treatment of differential equations (Proc. Conf., Math. Forschungsinst., Oberwolfach, 1976) Lecture Notes in Math., Vol. 631, Springer, Berlin, 1978, pp. 106–120 (German). MR**0487337** - James L. Phillips,
*The use of collocation as a projection method for solving linear operator equations*, SIAM J. Numer. Anal.**9**(1972), 14–28. MR**307516**, DOI 10.1137/0709003 - Friedrich Stummel,
*Diskrete Konvergenz linearer Operatoren. I*, Math. Ann.**190**(1970/71), 45–92 (German). MR**291870**, DOI 10.1007/BF01349967 - Friedrich Stummel,
*Diskrete Konvergenz linearer Operatoren. II*, Math. Z.**120**(1971), 231–264 (German). MR**291871**, DOI 10.1007/BF01117498 - Friedrich Stummel,
*Diskrete Konvergenz linearer Operatoren. III*, Linear operators and approximation (Proc. Conf., Math. Res. Inst., Oberwolfach, 1971) Internat. Ser. Numer. Math., Vol. 20, Birkhäuser, Basel, 1972, pp. 196–216 (German). MR**0410431** - P. J. Taylor,
*The solution of Volterra integral equations of the first kind using inverted differentiation formulae*, Nordisk Tidskr. Informationsbehandling (BIT)**16**(1976), no. 4, 416–425. MR**433930**, DOI 10.1007/bf01932725 - P. J. Taylor,
*Applications of results of Vainikko to Volterra integral equations*, Treatment of integral equations by numerical methods (Durham, 1982) Academic Press, London, 1982, pp. 185–195. MR**755354** - G. Vainikko,
*Approximative methods for nonlinear equations (two approaches to the convergence problem)*, Nonlinear Anal.**2**(1978), no. 6, 647–687. MR**512161**, DOI 10.1016/0362-546X(78)90013-5 - Gennadi Vainikko,
*Funktionalanalysis der Diskretisierungsmethoden*, B. G. Teubner Verlag, Leipzig, 1976 (German). Mit Englischen und Russischen Zusammenfassungen; Teubner-Texte zur Mathematik. MR**0468159** - P. H. M. Wolkenfelt,
*Reducible quadrature methods for Volterra integral equations of the first kind*, BIT**21**(1981), no. 2, 232–241. MR**627884**, DOI 10.1007/BF01933168 - P. H. M. Wolkenfelt,
*Modified multilag methods for Volterra functional equations*, Math. Comp.**40**(1983), no. 161, 301–316. MR**679447**, DOI 10.1090/S0025-5718-1983-0679447-2

## Additional Information

- © Copyright 1984 American Mathematical Society
- Journal: Math. Comp.
**43**(1984), 455-471 - MSC: Primary 65R20; Secondary 45L05
- DOI: https://doi.org/10.1090/S0025-5718-1984-0758194-3
- MathSciNet review: 758194