Product integration for weakly singular integral equations

Author:
Claus Schneider

Journal:
Math. Comp. **36** (1981), 207-213

MSC:
Primary 65R20; Secondary 45L05

DOI:
https://doi.org/10.1090/S0025-5718-1981-0595053-0

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.

**[1]**Philip M. Anselone,*Collectively compact operator approximation theory and applications to integral equations*, Prentice-Hall, Inc., Englewood Cliffs, N. J., 1971. With an appendix by Joel Davis; Prentice-Hall Series in Automatic Computation. MR**0443383****[2]**Kendall E. Atkinson,*The numerical solution of Fredholm integral equations of the second kind*, SIAM J. Numer. Anal.**4**(1967), 337–348. MR**0224314**, https://doi.org/10.1137/0704029**[3]**Kendall Atkinson,*The numerical solution of Fredholm integral equations of the second kind with singular kernels*, Numer. Math.**19**(1972), 248–259. MR**0307512**, https://doi.org/10.1007/BF01404695**[4]**P. L. Auer & C. S. Gardner, "Note on singular integral equations of the Kirkwood-Riseman type,"*J. Chem. Phys.*, v. 23, 1955, pp. 1545-1546.**[5]**P. L. Auer & C. S. Gardner, "Solution of the Kirkwood-Riseman integral equation in the asymptotic limit,"*J. Chem. Phys.*, v. 23, 1955, pp. 1546-1547.**[6]**G. A. Chandler,*Superconvergence of Numerical Solutions to Second Kind Integral Equations*, Thesis, Australian National University, Canberra, 1979.**[7]**Frank de Hoog and Richard Weiss,*Asymptotic expansions for product integration*, Math. Comp.**27**(1973), 295–306. MR**0329207**, https://doi.org/10.1090/S0025-5718-1973-0329207-0**[8]**L. M. Delves (ed.),*Numerical solution of integral equations*, Clarendon Press, Oxford, 1974. A collection of papers based on the material presented at a joint Summer School in July 1973, organized by the Department of Mathematics, University of Manchester, and the Department of Computational and Statistical Science, University of Liverpool. MR**0464624****[9]**E. Hopf,*Mathematical problems of radiative equilibrium*, Cambridge Tracts in Mathematics and Mathematical Physics, No. 31, Stechert-Hafner, Inc., New York, 1964. MR**0183517****[10]**Hans G. Kaper and R. Bruce Kellogg,*Asymptotic behavior of the solution of the integral transport equation in slab geometry*, SIAM J. Appl. Math.**32**(1977), no. 1, 191–200. MR**0449378**, https://doi.org/10.1137/0132016**[11]**J. Pitkäranta,*On the differential properties of solutions to Fredholm equations with weakly singular kernels*, J. Inst. Math. Appl.**24**(1979), no. 2, 109–119. MR**544428****[12]**John R. Rice,*On the degree of convergence of nonlinear spline approximation*, Approximations with Special Emphasis on Spline Functions (Proc. Sympos. Univ. of Wisconsin, Madison, Wis., 1969) Academic Press, New York, 1969, pp. 349–365. MR**0267324****[13]**G. R. Richter,*On weakly singular Fredholm integral equations with displacement kernels*, J. Math. Anal. Appl.**55**(1976), no. 1, 32–42. MR**0407549**, https://doi.org/10.1016/0022-247X(76)90275-4**[14]**C. Schneider,*Beiträge zur numerischen Behandlung schwachsingulärer Fredholmscher Integralgleichungen zweiter Art*, Thesis, Johannes Gutenberg-Universität, Mainz, 1977.**[15]**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**, https://doi.org/10.1007/BF01729361

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DOI:
https://doi.org/10.1090/S0025-5718-1981-0595053-0

Article copyright:
© Copyright 1981
American Mathematical Society