Product integration for weakly singular integral equations
Author:
Claus Schneider
Journal:
Math. Comp. 36 (1981), 207213
MSC:
Primary 65R20; Secondary 45L05
MathSciNet review:
595053
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Abstract: The product integration method is used for the numerical solution of weakly singular integral equations of the second kind. These equations often have solutions which have derivative singularities at the endpoints of the range of integration. Therefore, the order of convergence results of de Hoog and Weiss for smooth solutions do not hold in general. In this paper it is shown that their results may be regained for the general case by using an appropriate nonuniform mesh. The spacing of the knot points is defined by the behavior of the solution at the endpoints. If the solution is smooth enough the mesh becomes uniform. Numerical examples are given.
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 P. M. Anselone, Collectively Compact Operator Approximation Theory and Applications to Integral Equations, PrenticeHall, Englewood Cliffs, N. J., 1971. MR 0443383 (56:1753)
 [2]
 K. Atkinson, "The numerical solution of Fredholm integral equations of the second kind," SIAM J. Numer. Anal., v. 4, 1967, pp. 337348. MR 0224314 (36:7358)
 [3]
 K. Atkinson, "The numerical solution of Fredholm integral equations of the second kind with singular kernels," Numer. Math., v. 19, 1972, pp. 248259. MR 0307512 (46:6632)
 [4]
 P. L. Auer & C. S. Gardner, "Note on singular integral equations of the KirkwoodRiseman type," J. Chem. Phys., v. 23, 1955, pp. 15451546.
 [5]
 P. L. Auer & C. S. Gardner, "Solution of the KirkwoodRiseman integral equation in the asymptotic limit," J. Chem. Phys., v. 23, 1955, pp. 15461547.
 [6]
 G. A. Chandler, Superconvergence of Numerical Solutions to Second Kind Integral Equations, Thesis, Australian National University, Canberra, 1979.
 [7]
 F. de Hoog & R. Weiss, "Asymptotic expansions for product integration," Math. Comp., v. 27, 1973, pp. 295306. MR 0329207 (48:7549)
 [8]
 L. M. Delves & J. Walsh (Editors), Numerical Solution of Integral Equations, Clarendon Press, Oxford, 1974. MR 0464624 (57:4551)
 [9]
 E. Hopf, Mathematical Problems of Radiative Equilibrium, StechertHafner Service Agency, New York, 1964. MR 0183517 (32:997)
 [10]
 H. G. Kaper & R. B. Kellogg, "Asymptotic behavior of the solution of the integral transport equation in slab geometry," SIAM J. Appl. Math., v. 32, 1977, pp. 191200. MR 0449378 (56:7682)
 [11]
 J. PitkÄranta, "On the differential properties of solutions to Fredholm equations with weakly singular kernels," J. Inst. Math. Appl., v. 24, 1979, pp. 109119. MR 544428 (80i:65157)
 [12]
 J. R. Rice, "On the degree of convergence of nonlinear spline approximation," in Approximations with Special Emphasis on Spline Functions (I. J. Schoenberg, Ed.), Academic Press, New York, 1969, pp. 349365. MR 0267324 (42:2226)
 [13]
 G. R. Richter, "On weakly singular Fredholm integral equations with displacement kernels," J. Math. Anal. Appl., v. 55, 1976, pp. 3242. MR 0407549 (53:11322)
 [14]
 C. Schneider, Beiträge zur numerischen Behandlung schwachsingulärer Fredholmscher Integralgleichungen zweiter Art, Thesis, Johannes GutenbergUniversität, Mainz, 1977.
 [15]
 C. Schneider, "Regularity of the solution to a class of weakly singular Fredholm integral equations of the second kind," Integral Equations Operator Theory, v. 2, 1979, pp. 6268. MR 532739 (80f:45002)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198105950530
PII:
S 00255718(1981)05950530
Article copyright:
© Copyright 1981
American Mathematical Society
