High order methods

for weakly singular integral equations

with nonsmooth input functions

Authors:
G. Monegato and L. Scuderi

Journal:
Math. Comp. **67** (1998), 1493-1515

MSC (1991):
Primary 65R20

DOI:
https://doi.org/10.1090/S0025-5718-98-01005-9

MathSciNet review:
1604395

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Abstract | References | Similar Articles | Additional Information

Abstract: To solve one-dimensional linear weakly singular integral equations on bounded intervals, with input functions which may be smooth or not, we propose to introduce first a simple smoothing change of variable, and then to apply classical numerical methods such as product-integration and collocation based on global polynomial approximations. The advantage of this approach is that the order of the methods can be arbitrarily high and that the associated linear systems one has to solve are very well-conditioned.

**[1]**K. E. Atkinson,*A survey of numerical methods for the solution of Fredholm integral equations of the second kind,*SIAM, Philadelphia, 1976. MR**58:3577****[2]**G. Criscuolo, G. Mastroianni, G. Monegato,*Convergence properties of a class of product formulas for weakly singular integral equations,*Math. Comp. 55 (1990), 213-230. MR**90m:65230****[3]**D. Elliott, S. Prössdorf,*An algorithm for the approximate solution of integral equations of Mellin type,*Numer. Math. 70 (1995), 427-452. MR**96d:65206****[4]**J. Elschner, I. G. Graham,*An optimal order collocation method for first kind boundary integral equations on polygons,*Numer. Math. 70 (1995), 1-31. MR**95m:65215****[5]**I. G. Graham,*Galerkin methods for second kind integral equations with singularities,*Math. Comp. 39 (1982), 519-533. MR**84d:65090****[6]**I. G. Graham,*Singularity expansions for the solutions of second kind Fredholm integral equations with weakly singular convolution kernels,*J. Integral Equations 4 (1982), 1-30. MR**83e:45006****[7]**K. Jörgens,*Linear Integral Operators,*Pitman London, 1982. MR**83j:45001****[8]**R. Kress,*A Nyström method for boundary integral equations in domains with corners,*Numer. Math. 58 (1990), 145-161. MR**91m:65239****[9]**G. G. Lorentz,*Approximation of Functions,*Holt, Rinehart and Winston, New York, 1966. MR**35:4642****[10]**G. Mastroianni, S. Prössdorf,*A quadrature method for Cauchy integral equations with weakly singular perturbation kernel,*J. Integral Equations Appl. 4 (1992), 205-228. MR**93g:45013****[11]**G. Mastroianni, M. G. Russo,*Lagrange interpolation in weighted Besov spaces,*to appear in Constr. Approx.**[12]**S. G. Mikhlin, S. Prössdorf,*Singular Integral Operators,*Springer-Verlag, Berlin 1986. MR**88e:47097****[13]**G. Monegato,*Product integration for one-dimensional integral equations of Fredholm type,*Atti Sem. Mat. Fis. Univ. Modena**40**(1992), 653-666. MR**93j:45012****[14]**G. Monegato, V. Colombo,*Product integration for the linear transport equation in slab geometry,*Numer. Math. 52 (1988), 219-240; Errata, ibid.**53**(1988), 739. MR**88k:65133**; MR**89g:65139****[15]**G. Monegato, I. H. Sloan,*Numerical solution of the generalized airfoil equation for an airfoil with a flap,*SIAM J. Numer. Anal.**34**(1997), 2288-2305. CMP**98:04****[16]**P. Nevai,*Mean convergence of Lagrange interpolation.III,*Trans. Amer. Math. Soc. 282 (1984), 669-698. MR**85c:41009****[17]**J. Pitkäranta,*On the differential properties of solutions to Fredholm equations with weakly singular kernels,*J. Inst. Math. Appl. 24 (1979), 109-119. MR**80c:65157****[18]**S. Prössdorf, B. Silbermann,*Numerical Analysis for Integral and Related Operator Equations,*Akademie-Verlag, Berlin, and Birkhäuser-Verlag, Basel, 1991. MR**94f:65126a**; MR**94f:65126b****[19]**G. R. Richter,*On weakly singular Fredholm integral equations with displacement kernels,*J. Math. Anal. Appl. 55 (1976), 32-42. MR**53:11322****[20]**C. Schneider,*Regularity of the solution to a class of weakly singular Fredholm integral equations of the second kind,*Integral Equations Operator Theory 2 (1979), 62-68. MR**80f:45002****[21]**C. Schneider,*Product integration for weakly singular integral equations,*Math. Comp. 36 (1981), 207-213. MR**82c:65090****[22]**I. H. Sloan,*Analysis of general quadrature methods for integral equations of the second kind,*Numer. Math. 38 (1981), 263-278. MR**82m:65128****[23]**I. H. Sloan, W. E. Smith,*Properties of interpolatory product integration rules,*SIAM J. Numer. Anal. 19 (1982), 427-442. MR**83e:41032****[24]**W. E. Smith, I. H. Sloan,*Product-integration rules based on the zeros of orthogonal polynomials,*SIAM J. Numer. Anal. 17 (1980), 1-13. MR**81i:65018****[25]**G. Vainikko, A. Pedas,*The properties of solutions of weakly singular integral equations,*J. Austral. Math. Soc. Ser. B 22 (1981), 419-430. MR**82i:45014****[26]**G. Vainikko, P. Uba,*A piecewise polynomial approximation to the solution of an integral equation with weakly singular kernel,*J. Austral. Math. Soc. Ser. B 22 (1981), 431-438. MR**82h:65100****[27]**C. Abaci,*The Scientific Desk Library Documentation System,*Raleigh, North Carolina, 1994.

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Additional Information

**G. Monegato**

Affiliation:
Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Email:
monegato@polito.it

**L. Scuderi**

Affiliation:
Dipartimento di Matematica, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

Email:
scuderi@polito.it

DOI:
https://doi.org/10.1090/S0025-5718-98-01005-9

Received by editor(s):
July 22, 1996

Additional Notes:
This work was supported by the Ministero dell’Universitá e della Ricerca Scientifica e Tecnologica of Italy.

Dedicated:
Dedicated to Professor M. R. Occorsio on the occasion of his 65th birthday

Article copyright:
© Copyright 1998
American Mathematical Society