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The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations

Authors: Hermann Brunner, Arvet Pedas and Gennadi Vainikko
Journal: Math. Comp. 68 (1999), 1079-1095
MSC (1991): Primary 65R20, 45E10, 45B05
Published electronically: February 8, 1999
MathSciNet review: 1642797
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Abstract: Second-kind Volterra integral equations with weakly singular kernels typically have solutions which are nonsmooth near the initial point of the interval of integration. Using an adaptation of the analysis originally developed for nonlinear weakly singular Fredholm integral equations, we present a complete discussion of the optimal (global and local) order of convergence of piecewise polynomial collocation methods on graded grids for nonlinear Volterra integral equations with algebraic or logarithmic singularities in their kernels.

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Additional Information

Hermann Brunner
Affiliation: Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, Nfld., Canada A1C 5S7

Arvet Pedas
Affiliation: Department of Applied Mathematics, University of Tartu, 0000 Liivi 2–206, Tartu EE2400, Estonia

Gennadi Vainikko
Affiliation: Institute of Mathematics, Helsinki University of Technology, P.O.Box 1100, FIN–02015 HUT, Finland

Keywords: Nonlinear weakly singular Volterra and Fredholm integral equations, piecewise polynomial collocation, graded grids, optimal order of convergence
Received by editor(s): September 2, 1997
Published electronically: February 8, 1999
Article copyright: © Copyright 1999 American Mathematical Society