Discretization of Volterra integral equations of the first kind
Author:
Hermann Brunner
Journal:
Math. Comp. 31 (1977), 708716
MSC:
Primary 65R05; Secondary 45L05
MathSciNet review:
0451794
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Abstract: We show that various (discrete) methods for the approximate solution of Volterra (and Abel) integral equations of the first kind correspond to some discrete version of the method of (recursive) collocation in the space of (continuous) piecewise polynomials. In a collocation method no distinction has to be made between equations with regular or weakly singular kernels; the regularity or nonregularity of the given integral operator becomes only relevant when selecting a discretization procedure for the moment integrals resulting from collocation. Similar results hold for equations of the second kind.
 [1]
C.
T. H. Baker, Methods for Volterra equations of first kind,
Numerical solution of integral equations (LiverpoolManchester Summer
School, 1973) Clarendon Press, Oxford, 1974, pp. 162–174. MR 0488900
(58 #8398)
 [2]
W.
Robert Boland and C.
S. Duris, Product type quadrature formulas, Nordisk Tidskr.
Informationsbehandling (BIT) 11 (1971), 139–158. MR 0292295
(45 #1382)
 [3]
H.
Brunner, The solution of nonlinear Volterra integral equations by
piecewise polynomials, Proceedings of the Manitoba Conference on
Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1971) Dept.
Comput. Sci., Univ. Manitoba, Winnipeg, Man., 1971, pp. 65–78.
MR
0337039 (49 #1812)
 [4]
H.
Brunner, Global solution of the generalized
Abel integral equation by implicit interpolation, Math. Comp. 28 (1974), 61–67. MR 0331830
(48 #10162), http://dx.doi.org/10.1090/S00255718197403318305
 [5]
H.
Brunner, On the approximate solution of firstkind integral
equations of Volterra type, Computing (Arch. Elektron. Rechnen)
13 (1974), no. 1, 67–79 (English, with German
summary). MR
0400754 (53 #4584)
 [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 0371114
(51 #7335)
 [7]
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
(51 #9554)
 [8]
F.
de Hoog and R.
Weiss, Implicit RungeKutta methods for second kind Volterra
integral equations, Numer. Math. 23 (1974/75),
199–213. MR 0373349
(51 #9549)
 [9]
H. S. HUNG, The Numerical Solution of Differential and Integral Equations by Spline Functions, MRC Technical Report #1053, Math. Res. Center, Univ. of Wisconsin, Madison, 1970.
 [10]
Peter
Linz, Numerical methods for Volterra integral equations of the
first kind., Comput. J. 12 (1969), 393–397. MR 0253577
(40 #6791)
 [11]
Peter
Linz, Numerial methods for Volterra integral equations with
singular kernels., SIAM J. Numer. Anal. 6 (1969),
365–374. MR 0260222
(41 #4850)
 [12]
Peter
Linz, Product integration methods for Volterra integral equations
of the first kind, Nordisk Tidskr. Informationsbehandling (BIT)
11 (1971), 413–421. MR 0314296
(47 #2848)
 [13]
J. N. LYNESS & J. J. KAGANOVE, EAR Report QR2 (May 1975 version), Argonne National Laboratory, Argonne, Ill., 1975.
 [14]
Richard
Weiss, Product integration for the
generalized Abel equation, Math. Comp. 26 (1972), 177–190.
MR
0299001 (45 #8050), http://dx.doi.org/10.1090/S00255718197202990017
 [15]
R.
Weiss and R.
S. Anderssen, A product integration method for a class of singular
first kind Volterra equations, Numer. Math. 18
(1971/72), 442–456. MR 0312759
(47 #1314)
 [16]
Andrew
Young, The application of approximate product integration to the
numerical solution of integral equations, Proc. Roy. Soc. London Ser.
A. 224 (1954), 561–573. MR 0063779
(16,179b)
 [1]
 C. T. H. BAKER, "Methods for Volterra equations of the first kind," Numerical Solution of Integral Equations (L. M. Delves & J. Walsh, Editors), Clarendon Press, Oxford, 1974, pp. 162174. MR 0488900 (58:8398)
 [2]
 W. R. BOLAND & C. S. DURIS, "Product type quadrature formulas," BIT, v. 11, 1971, pp. 139158. MR 45 #1382. MR 0292295 (45:1382)
 [3]
 H. BRUNNER, "The solution of nonlinear Volterra integral equations by piecewise polynomials," Proc. Manitoba Conf. on Numerical Mathematics (R. S. D. Thomas & H. C. Williams, Editors), Univ. of Manitoba, Winnipeg, Canada, 1971, pp. 6578. MR 49 #1812. MR 0337039 (49:1812)
 [4]
 H. BRUNNER, "Global solution of the generalized Abel integral equation by implicit interpolation," Math. Comp., v. 28, 1974, pp. 6167. MR 48 #10162. MR 0331830 (48:10162)
 [5]
 H. BRUNNER, "On the approximate solution of firstkind integral equations of Volterra type," Computing, v. 13, 1974, pp. 6779. MR 0400754 (53:4584)
 [6]
 F. de HOOG & R. WEISS, "On the solution of Volterra integral equations of the first kind," Numer. Math., v. 21, 1973, pp. 2232. MR 51 #7335. MR 0371114 (51:7335)
 [7]
 F. de HOOG & R. WEISS, "High order methods for Volterra integral equations of the first kind," SIAM J. Numer. Anal., v. 10, 1973, pp. 647664. MR 51 #9554. MR 0373354 (51:9554)
 [8]
 F. de HOOG & R. WEISS, "Implicit RungeKutta methods for second kind Volterra integral equations," Numer. Math., v. 23, 1974/75, pp. 199213. MR 51 #9549. MR 0373349 (51:9549)
 [9]
 H. S. HUNG, The Numerical Solution of Differential and Integral Equations by Spline Functions, MRC Technical Report #1053, Math. Res. Center, Univ. of Wisconsin, Madison, 1970.
 [10]
 P. LINZ, "Numerical methods for Volterra integral equations of the first kind," Comput. J., v. 12, 1969, pp. 393397. MR 40 #6791. MR 0253577 (40:6791)
 [11]
 P. LINZ, "Numerical methods for Volterra integral equations with singular kernels," SIAM J. Numer. Anal., v. 6, 1969, pp. 365374. MR 41 #4850. MR 0260222 (41:4850)
 [12]
 P. LINZ, "Product integration methods for Volterra integral equations," BIT, v. 11, 1971, pp. 413421. MR 47 #2848. MR 0314296 (47:2848)
 [13]
 J. N. LYNESS & J. J. KAGANOVE, EAR Report QR2 (May 1975 version), Argonne National Laboratory, Argonne, Ill., 1975.
 [14]
 R. WEISS, "Product integration for the generalized Abel equation," Math. Comp., v. 26, 1972, pp. 177190. MR 45 #8050. MR 0299001 (45:8050)
 [15]
 R. WEISS & R. S. ANDERSSEN, "A product integration method for a class of singular first kind Volterra equations," Numer. Math., v. 18, 1971/72, pp. 442456. MR 47 #1314. MR 0312759 (47:1314)
 [16]
 A. YOUNG, "The application of approximate productintegration to the numerical solution of integral equations," Proc. Roy. Soc. London Ser. A, v. 224, 1954, pp. 561573. MR 16, 179. MR 0063779 (16:179b)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197704517946
PII:
S 00255718(1977)04517946
Keywords:
Firstkind integral equations of Volterra and Abel type,
collocation by piecewise polynomials,
numerical quadrature,
discrete appoximations
Article copyright:
© Copyright 1977
American Mathematical Society
