High order methods for weakly singular integral equations with nonsmooth input functions

Monegato, G.; Scuderi, L.

doi:10.1090/S0025-5718-98-01005-9

High order methods for weakly singular integral equations with nonsmooth input functions
HTML articles powered by AMS MathViewer

by G. Monegato and L. Scuderi PDF

Math. Comp. 67 (1998), 1493-1515 Request permission

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.

References

Kendall E. Atkinson, A survey of numerical methods for the solution of Fredholm integral equations of the second kind, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1976. MR 0483585
Giuliana Criscuolo, Giuseppe Mastroianni, and Giovanni Monegato, Convergence properties of a class of product formulas for weakly singular integral equations, Math. Comp. 55 (1990), no. 191, 213–230. MR 1023045, DOI 10.1090/S0025-5718-1990-1023045-1
David Elliott and Siegfried Prössdorf, An algorithm for the approximate solution of integral equations of Mellin type, Numer. Math. 70 (1995), no. 4, 427–452. MR 1337225, DOI 10.1007/s002110050127
J. Elschner and I. G. Graham, An optimal order collocation method for first kind boundary integral equations on polygons, Numer. Math. 70 (1995), no. 1, 1–31. MR 1320699, DOI 10.1007/s002110050107
Ivan G. Graham, Galerkin methods for second kind integral equations with singularities, Math. Comp. 39 (1982), no. 160, 519–533. MR 669644, DOI 10.1090/S0025-5718-1982-0669644-3
Ivan G. Graham, Singularity expansions for the solutions of second kind Fredholm integral equations with weakly singular convolution kernels, J. Integral Equations 4 (1982), no. 1, 1–30. MR 640534
Konrad Jörgens, Linear integral operators, Surveys and Reference Works in Mathematics, vol. 7, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1982. Translated from the German by G. F. Roach. MR 647629
V. L. Makarov and V. V. Guminskiĭ, Solution of a singularly perturbed fourth-order nonstationary boundary value problem, Vychisl. Prikl. Mat. (Kiev) 70 (1990), 3–11 (Russian); English transl., J. Math. Sci. 70 (1994), no. 1, 1485–1491. MR 1086373, DOI 10.1007/BF02110998
G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York-Chicago, Ill.-Toronto, Ont., 1966. MR 0213785
Giuseppe Mastroianni and Siegfried Prössdorf, A quadrature method for Cauchy integral equations with weakly singular perturbation kernel, J. Integral Equations Appl. 4 (1992), no. 2, 205–228. MR 1172890, DOI 10.1216/jiea/1181075682
G. Mastroianni, M. G. Russo, Lagrange interpolation in weighted Besov spaces, to appear in Constr. Approx.
Solomon G. Mikhlin and Siegfried Prössdorf, Singular integral operators, Springer-Verlag, Berlin, 1986. Translated from the German by Albrecht Böttcher and Reinhard Lehmann. MR 881386, DOI 10.1007/978-3-642-61631-0
Giovanni Monegato, Product integration for one-dimensional integral equations of Fredholm type, Atti Sem. Mat. Fis. Univ. Modena 40 (1992), no. 2, 653–666. MR 1200314
G. Monegato and V. Colombo, Product integration for the linear transport equation in slab geometry, Numer. Math. 52 (1988), no. 2, 219–240. MR 923711, DOI 10.1007/BF01398690
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.
Paul Nevai, Mean convergence of Lagrange interpolation. III, Trans. Amer. Math. Soc. 282 (1984), no. 2, 669–698. MR 732113, DOI 10.1090/S0002-9947-1984-0732113-4
Lawrence F. Shampine, Stability properties of Adams codes, ACM Trans. Math. Software 4 (1978), no. 4, 323–329. MR 513704, DOI 10.1145/356502.356478
Siegfried Prössdorf and Bernd Silbermann, Numerical analysis for integral and related operator equations, Mathematische Lehrbücher und Monographien, II. Abteilung: Mathematische Monographien [Mathematical Textbooks and Monographs, Part II: Mathematical Monographs], vol. 84, Akademie-Verlag, Berlin, 1991 (English, with English and German summaries). MR 1206476
G. R. Richter, On weakly singular Fredholm integral equations with displacement kernels, J. Math. Anal. Appl. 55 (1976), no. 1, 32–42. MR 407549, DOI 10.1016/0022-247X(76)90275-4
Claus Schneider, Regularity of the solution to a class of weakly singular Fredholm integral equations of the second kind, Integral Equations Operator Theory 2 (1979), no. 1, 62–68. MR 532739, DOI 10.1007/BF01729361
Claus Schneider, Product integration for weakly singular integral equations, Math. Comp. 36 (1981), no. 153, 207–213. MR 595053, DOI 10.1090/S0025-5718-1981-0595053-0
Ian H. Sloan, Analysis of general quadrature methods for integral equations of the second kind, Numer. Math. 38 (1981/82), no. 2, 263–278. MR 638450, DOI 10.1007/BF01397095
Ian H. Sloan and William E. Smith, Properties of interpolatory product integration rules, SIAM J. Numer. Anal. 19 (1982), no. 2, 427–442. MR 650061, DOI 10.1137/0719027
R. V. Slonevskiĭ and B. Ĭ. Bandyrskiĭ, The nonlinear formula of Mikeladze-Aitken, Vestnik L′vov. Politekhn. Inst. 141 Differentsial′nye Uravneniya i ikh Prilozhen. (1980), 81–82, 123 (Russian). MR 587460
G. Vainikko and A. Pedas, The properties of solutions of weakly singular integral equations, J. Austral. Math. Soc. Ser. B 22 (1980/81), no. 4, 419–430. MR 626933, DOI 10.1017/S0334270000002769
G. Vainikko and P. Uba, A piecewise polynomial approximation to the solution of an integral equation with weakly singular kernel, J. Austral. Math. Soc. Ser. B 22 (1980/81), no. 4, 431–438. MR 626934, DOI 10.1017/S0334270000002770
C. Abaci, The Scientific Desk Library Documentation System, Raleigh, North Carolina, 1994.