Uniform error estimates of Galerkin methods for monotone AbelVolterra integral equations on the halfline
Author:
P. P. B. Eggermont
Journal:
Math. Comp. 53 (1989), 157189
MSC:
Primary 65R20; Secondary 45D05, 47H17
MathSciNet review:
969485
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Abstract: We consider Galerkin methods for monotone AbelVolterra integral equations of the second kind on the halfline. The theory follows from Kolodner's theory of monotone Hammerstein, equations. We derive the theory from the theory by relating the  and spectra of operators of the form to one another. Here denotes convolution, and and . As an extra condition we need , with . We also prove the discrete analogue. In particular, we verify that the Galerkin matrix satisfies the "discrete" conditions.
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 [1]
 C. T. H. Baker & G. F. Miller (editors), Treatment of Integral Equations by Numerical Methods, Academic Press, London, 1982. MR 755336 (85e:65005)
 [2]
 V. L. Bakke & Z. Jackiewicz, "Stability analysis of product methods for Abel integral equations of the second kind," Numer. Math., v. 48, 1986, pp. 127136. MR 824164 (87f:65149)
 [3]
 M. S. Berger, Nonlinearity and Functional Analysis, Academic Press, New York, 1977. MR 0488101 (58:7671)
 [4]
 H. Brunner, "The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes," Math. Comp., v. 45, 1985, pp. 417437. MR 804933 (87b:65223)
 [5]
 H. Brunner & P. J. van der Houwen, The Numerical Solution of Volterra Equations, NorthHolland, Amsterdam, 1986. MR 871871 (88g:65136)
 [6]
 N. G. de Bruijn & P. Erdös, "On a recursion formula and some Tauberian theorems," J. Res. Nat. Bur. Standards, v. 50, 1953, pp. 161164. MR 0054745 (14:973e)
 [7]
 J. R. Cannon, The OneDimensional Heat Equation, AddisonWesley, Reading, Mass., 1984. MR 747979 (86b:35073)
 [8]
 P. P. B. Eggermont, "A generalization of W. A. J. Luxemburg's asymptotic problem concerning the Laplace transform," Indag. Math., v. 42, 1981, pp. 257265.
 [9]
 P. P. B. Eggermont, "On monotone AbelVolterra integral equations on the half line," Numer. Math., v. 52, 1988, pp. 6579. MR 918317 (88m:65204)
 [10]
 P. P. B. Eggermont, "Galerkin methods for Abeltype integral equations of the first kind," SIAM J. Numer. Anal., v. 25, 1988, pp. 10931117. MR 960868 (89k:65161)
 [11]
 P. P. B. Eggermont, "Approximation of nonsmooth solutions of AbelVolterra integral equations," manuscript, 1986.
 [12]
 A. Erdélyi et al., Higher Transcendental Functions, Vol. 3, McGrawHill, New York, 1955.
 [13]
 G. A. Hufford, "An integral equation approach to the problem of wave propagation over an irregular surface," J. Math. Mech., v. 9, 1952, pp. 391404. MR 0044350 (13:407d)
 [14]
 D. Kershaw, "Some results for AbelVolterra equations of the second kind," in [1], pp. 273282 and 459461. MR 755362 (85m:65132)
 [15]
 I. I. Kolodner, "Equations of Hammerstein in Hilbert space," J. Math. Mech., v. 13, 1964, pp. 701750. MR 0171184 (30:1415)
 [16]
 S.O. Londen, "On some nonintegrable Volterra kernels with integrable resolvents including some applications to Riesz potentials," J. Integral Equations, v. 10, 1985, pp. 241289. MR 831246 (87f:45017)
 [17]
 Ch. Lubich, "Fractional linear multistep methods for AbelVolterra integral equations of the second kind," Math. Comp., v. 45, 1985, pp. 463469. MR 804935 (86j:65181)
 [18]
 Ch. Lubich, "On the numerical solution of Volterra equations with unbounded nonlinearity," J. Integral Equations, v. 10, 1986, pp. 175183. MR 831243 (87k:65154)
 [19]
 W. A. J. Luxemburg, "An asymptotic problem concerning the Laplace transform," Applicable Anal., v. 9, 1979, pp. 6170. MR 516522 (81b:44003)
 [20]
 C. C. Mei & E. O. Tuck, "Forward scattering by long thin bodies," SIAM J. Appl. Math., v. 39, 1980, pp. 178191. MR 585838 (81m:76006)
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 O. Nevanlinna, "On the stability of discrete Volterra equations," in [1], pp. 139147. MR 755349 (85f:65126)
 [22]
 S. L. PaveriFontana & R. Rigacci, "A singularly perturbed singular integrodifferential problem from analytical chemistry," in Numerical Analysis of Singular Perturbation Problems (P. W. Hemker and J. J. H. Miller, eds.), Academic Press, New York, 1979, pp. 475484. MR 556538 (81a:65128)
 [23]
 E. M. Stein & G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J., 1971. MR 0304972 (46:4102)
 [24]
 H. J. J. te Riele, "Collocation methods for weakly singular second kind Volterra integral equations with nonsmooth solutions," IMA J. Numer. Anal., v. 2, 1982, pp. 437449. MR 692290 (84g:65167)
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DOI:
http://dx.doi.org/10.1090/S0025571819890969485X
PII:
S 00255718(1989)0969485X
Article copyright:
© Copyright 1989 American Mathematical Society
