A spline product quasi-interpolation method for weakly singular Fredholm integral equations

SIAM Journal on Numerical Analysis

Volumn 46, Issue 4, 2008, Pages 1799-1820

  Vainikko, Eero ^a   Vainikko, Gennadi ^a  

^a UNIVERSITY OF TARTU   (Estonia)

Author keywords

Boundary singularities; Nystr m type methods; Product integration methods; Spline interpolation; Spline quasi interpolation; Weakly singular integral equations

Indexed keywords

INTEGRAL EQUATIONS; INTERPOLATION;

BOUNDARY SINGULARITIES; CHANGE OF VARIABLES; FREDHOLM INTEGRAL EQUATIONS; PRODUCT INTEGRATION METHODS; PRODUCT INTEGRATIONS; QUASI INTERPOLATIONS; SPLINE INTERPOLATION; WEAKLY SINGULAR INTEGRAL EQUATIONS; WEAKLY SINGULARS;

SPLINES;

EID: 55349112566     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/070693308     Document Type: Article

References (22)

1. P. M. ANSELONE, Collectively Compact Operator Approximation Theory, Prentice-Hall, Englewood Cliffs, NJ, 1971.
2. K. E. ATKINSON, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, Cambridge, UK, 1997.
3. H. BRUNNER, Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge University Press, Cambridge, UK, 2004.
4. C. DE BOOR, A Practical Guide to Splines, Springer, New York, 2001.
5. W. HACKBUSCH, Integral Equations, Birkhäuser, Basel, 1995.
6. N. P. KORNEYCHUK, Splines in the Approximation Theory, Nauka, Moscow, 1984 (in Russian).
7. R. KRESS, Linear Integral Equations, Springer, Berlin, 1989.
8. E. LEETMA AND G. VAINIKKO, Quasi-interpolation by splines on the uniform knot sets, Math. Model. Anal., 12 (2007), pp. 107-120.
9. G. MONEGATO AND L. SCUDERI, High order methods for weakly singular integral equations with nonsrnooth input functions, Math. Comp., 67 (1998), pp. 1493-1615.
10. A. PEDAS AND G. VAINIKKO, Smoothing transformation and piecewise polynomial collocation for weakly singular integral equations, Computing, 73 (2004), pp. 271-293.
11. A. PEDAS AND G. VAINIKKO, Integral equations with diagonal and boundary singularities of the kernel, Z. Anal. Anwend., 25 (2006), pp. 487-516.
12. A. PEDAS AND G. VAINIKKO, On the regularity of solutions to integral equations with nonsmooth kernels, J. Comput. Appl. Math., to appear.
13. C. SCHNEIDER, Product integration for weakly singular integral equations, Math. Comp., 36 (1981), pp. 207-213.
14. I. J. SCHOENBERG, Cardinal Spline Interpolation, SIAM, Philadelphia, 1973.
15. L. L. SCHUMAKER, Spline Functions: Basic Theory, Krieger, Malabar, FL, 1993.
16. S. B. STECHKIN AND YU. N. SUBBOTIN, Splines in Numerical Mathematics, Nauka, Moscow, 1976 (in Russian).
17. E. VAINIKKO AND G. VAINIKKO, A product quasi-interpolation method for weakly singular Volterra integral equations, in Numerical Analysis and Applied Mathematics, Vol. 936, T. E. Simos, ed., AIP Conference Proceedings, American Institute of Physics, Melville, NY, 2007, pp. 570-573.
18. G. VAINIKKO, Funktionalanalysis der Diskretisierungsmethoden, Teubner, Leipzig, 1976.
19. G. VAINIKKO, GMRES and discrete approximation of operators, Proc. Estonian Acad Sci. Phys. Math., 53 (2004), pp. 124-131.
20. G. VAINIKKO, Fast solvers of integral equations of the second kind: Quadrature methods, J. Integral Equations Appl., 17 (2005), pp. 91-120.
21. G. VAINIKKO, On the best approximation of function classes from values on a uniform grid in the real line, WSEAS Trans. Math., 4/6 (2006), pp. 523-528.
22. G. VAINIKKO, Error estimates for the cardinal spline interpolation, Z. Anal. Anwend., to appear.