The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes
Author:
Hermann Brunner
Journal:
Math. Comp. 45 (1985), 417437
MSC:
Primary 65R20; Secondary 45D05
MathSciNet review:
804933
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Abstract: Since the solution of a secondkind Volterra integral equation with weakly singular kernel has, in general, unbounded derivatives at the left endpoint of the interval of integration, its numerical solution by polynomial spline collocation on uniform meshes will lead to poor convergence rates. In this paper we investigate the convergence rates with respect to graded meshes, and we discuss the problem of how to select the quadrature formulas to obtain the fully discretized collocation equation.
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 P. R. Beesack, "More generalised discrete Gronwall inequalities," Preprint, 1983. MR 832170 (87e:26023)
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 C. de Boor, "Good approximation by splines with variable knots," in Spline Functions and Approximation Theory (A. Meir and A. Sharma, eds.), BirkhäuserVerlag, Basel, 1973, pp. 5772. MR 0403169 (53:6982)
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 H. Brunner, "The numerical solution of integral equations with weakly singular kernels," in Numerical Analysis, Dundee 1983 (D. F. Griffiths, ed.), Lecture Notes in Math., Vol. 1066, SpringerVerlag, Berlin, 1984, pp. 5071. MR 760457 (85j:65043)
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 H. Brunner & I. G. Graham, "Product integration for weakly singular Volterra integral equations." (To appear.)
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 H. G. Burchard, "On the degree of convergence of piecewise polynomial approximation on optimal meshes. II," Trans. Amer. Math. Soc., v. 234, 1977, pp. 531559. MR 0481758 (58:1857)
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 G. A. Chandler, Superconvergence of Numerical Methods to Second Kind Integral Equations, Ph. D. Thesis, Australian National University, Canberra, 1979.
 [8]
 J. Dixon & S. McKee, Singular Gronwall Inequalities, Numerical Analysis Report NA/83/44, Hertford College, University of Oxford, 1983.
 [9]
 I. G. Graham, The Numerical Solution of Fredholm Integral Equations of the Second Kind, Ph. D. Thesis, University of New South Wales, Kensington, 1980.
 [10]
 I. G. Graham, "Galerkin methods for second kind integral equations with singularities", Math. Comp., v. 39, 1982, pp. 519533. MR 669644 (84d:65090)
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 I. G. Graham, Estimates for the Modulus of Smoothness, Research Report No. 22, Dept. of Mathematics, University of Melbourne, 1982.
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 F. de Hoog & R. Weiss, "On the solution of a Volterra integral equation with a weakly singular kernel," SIAM J. Math. Anal., v. 4, 1973, pp. 561573. MR 0337045 (49:1818)
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 R. B. Kearfott, "A sine approximation for the indefinite integral," Math. Comp., v. 41, 1983, pp. 559572. MR 717703 (85g:65029)
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 S. McKee, "Generalised discrete Gronwall lemmas," Z. Angew. Math. Mech., v. 62, 1982, pp. 429434. MR 682559 (84d:26016)
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 J. R. Rice, "On the degree of convergence of nonlinear spline approximation," in Approximation with Special Emphasis on Spline Functions (I. J. Schoenberg. ed.), Academic Press, New York, 1969, pp. 349365. MR 0267324 (42:2226)
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 H. J. J. te Riele, "Collocation methods for weakly singular secondkind Volterra integral equations with nonsmooth solution," IMA J. Numer. Anal., v. 2, 1982, pp. 437449. MR 692290 (84g:65167)
 [19]
 C. Schneider, "Produktintegration mit nichtäquidistanten Stützstellen," Numer. Math., v. 35, 1980, pp. 3543. MR 583654 (81j:65049)
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 C. Schneider, "Product integration for weakly singular integral equations," Math. Comp., v. 36, 1981, pp. 207213. MR 595053 (82c:65090)
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 L. L. Schumaker, Spline Functions: Basic Theory, Wiley, New York, 1981. MR 606200 (82j:41001)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198508049333
PII:
S 00255718(1985)08049333
Keywords:
Volterra integral equations,
weakly singular kernels,
polynomial spline collocation,
graded meshes
Article copyright:
© Copyright 1985
American Mathematical Society
