Piecewise polynomial collocation for Fredholm integro-differential equations with weakly singular kernels

SIAM Journal on Numerical Analysis

Volumn 43, Issue 5, 2005, Pages 1897-1911

  Parts, Inga ^a   Pedas, Arvet ^a   Tamme, Enn ^a  

^a UNIVERSITY OF TARTU   (Estonia)

Author keywords

Attainable order of convergence; Fredholm integro differential equation; Graded grid; Piecewise polynomial collocation method; Weakly singular kernel

Indexed keywords

BOUNDARY VALUE PROBLEMS; CONVERGENCE OF NUMERICAL METHODS; INTEGRODIFFERENTIAL EQUATIONS; PARAMETER ESTIMATION; PROBLEM SOLVING;

ATTAINABLE ORDER OF CONVERGENCE; FREDHOLM INTEGRO-DIFFERENTIAL EQUATION; GRADED GRID; PIECEWISE POLYNOMIAL COLLOCATION METHOD; WEAKLY SINGULAR KERNEL;

POLYNOMIALS;

EID: 33750151083     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/040612452     Document Type: Article

References (26)

1. K. E. ATKINSON, The Numerical Solution of Integral Equations of the Second Kind, Cambridge Monogr. Appl. Comput. Math. 4, Cambridge University Press, Cambridge, UK, 1997.
2. C. T. H. BAKER, The Numerical Treatment of Integral Equations, Monogr. Numer. Anal., Clarendon Press, Oxford, UK, 1977.
3. H. BRUNNER, Polynomial spline collocation methods for Volterra integro-differential equations with weakly singular kernels, IMA J. Numer. Anal., 6 (1986), pp. 221-239.
4. H. BRUNNER, Collocation Methods for Volterra Integral and Related Functional Differential Equations, Cambridge Monogr. Appl. Comput. Math. 15, Cambridge University Press. Cambridge, UK, 2004.
5. H. BRUNNER AND P. J. VAN DER HOUWEN, The Numerical Solution of Volterra Equations, CWI Monogr. 3, North-Holland, Amsterdam, 1986.
6. H. BRUNNER, A. PEDAS, AND G. VAINIKKO, Piecewise polynomial collocation methods for linear Volterra integro-differential equations with weakly singular kernels, SIAM J. Numer. Anal., 39 (2001), pp. 957-982.
7. H. BRUNNER, A. PEDAS, AND G. VAINIKKO, A spline collocation method for linear Volterra integro-differential equations with weakly singular kernels, BIT, 41 (2001), pp. 891-900.
8. Z. CHEN, C. A. MICCHELLI, AND Y. XU, Fast collocation methods for second kind integral equations, SIAM J. Numer. Anal., 40 (2002), pp. 344-375.
9. M. GANESH AND I. H. SLOAN, Optimal order spline methods for nonlinear differential and integro-differential equations, Appl. Numer. Math., 29 (1999), pp. 445-478.
10. W. HACKBUSCH, Integral Equations. Theory and Numerical Treatment, Internat. Ser. Numer. Math 120, Birkhäuser-Verlag, Basel, Switzerland, 1996.
11. R. J. HANGELBROEK, H. G. KAPER, AND G. K. LEAF, Collocation methods for integro-differential equations, SIAM J. Numer. Anal., 14 (1977), pp. 377-390.
12. Q. HU, Geometric meshes and their application to Volterra Integro-differential equations with singularities, IMA J. Numer. Anal., 18 (1998), pp. 151-164.
13. Q. HU, Interpolation correction for collocation solution for Fredholm integro-differential equation, Math. Comp., 67 (1998), pp. 987-999.
14. H. KANEKO, R. NOREN, AND P. PADILLA, Superconvergence of the iterated collocation methods for Hammerstein equations, J. Comput. Appl. Math., 80 (1997), pp. 335-349.
15. R. KANGRO AND I. PARTS, Superconvergence in the maximum norm of a class of piecewise polynomial collocation methods for solving linear weakly singular Volterra integro-differential equations, J. Integral Equations Appl., 15 (2003), pp. 403-427.
16. A. KARAMETE AND M. SEZER, A Taylor collocation method for the solution of linear integro-differential equations, Int. J. Comput. Math., 79 (2002), pp. 987-1000.
17. R. KRESS, Linear Integral Equations, Springer-Verlag, Berlin, 1989.
18. I. PARTS AND A. PEDAS, Spline collocation methods for weakly singular Volterra integro-differential equations, in Numerical Mathematics and Advanced Applications, Enumath 2001, F. Brezzi, A. Buffa, S. Corsar, and A. Merli, eds., Springer-Verlag, Milano, 2003, pp. 919-928.
19. I. PARTS AND A. PEDAS, Collocation approximations for weakly singular Volterra integro-differential equations, Math. Model. Anal., 8 (2003), pp. 315-328.
20. T. TANG, Superconvergence of numerical solutions to weakly singular Volterra integro-differential equations, Numer. Math., 61 (1992), pp. 373-382.
21. T. TANG, A note on collocation methods for Volterra integro-differential equations with weakly singular kernels, IMA J. Numer. Anal., 13 (1993), pp. 93-99.
22. G. VAINIKKO, Multidimensional Weakly Singular Integral Equations, Lecture Notes in Math. 1549, Springer-Verlag, Berlin, 1993.
23. G. VAINIKKO AND A. PEDAS, The properties of solutions of weakly singular integral equations, J. Austral. Math. Soc. Ser. B, 22 (1981) pp. 419-430.
24. G. VAINIKKO, A. PEDAS, AND P. UBA, Methods for Solving Weakly Singular Integral Equations, University of Tartu, Tartu, Estonia, 1984 (in Russian).
25. W. VOLK, The numerical solution of linear integro-differential equations by projection methods, J. Integral Equations, 9 (1985), pp. 171-190.
26. W. VOLK, The iterated Galerkin methods for linear integro-differential equations, J. Comput. Appl. Math., 21 (1988), pp. 63-74.