Collocation Solutions of a Weakly Singular Volterra Integral Equation
DOI:
https://doi.org/10.5540/tema.2007.08.02.0229Abstract
The discrete superconvergence properties of spline collocation solutions for a certain Volterra integral equation with weakly singular kernel are analyzed. In particular, the attainable convergence orders at the collocation points are examined for certain choices of the collocation parameters.References
[1] P. Baratella, A.P. Orsi, A new approach to the numerical solution of weakly singular volterra integral equations, J. Comput. Appl. Math., 163 (2004), 401-418.
M.A. Bartoshevich, On a heat conduction problem, In˘z.- Fiz. ˘Z., 28 (1975), 340-346. (In russian)
H. Brunner, The numerical solution of weakly singular Volterra integral equations by collocation on graded meshes, Math. Comp., 45 (1985), 417-437.
H. Brunner, “Collocation methods for Volterra integral and related functional differential equations”, Cambridge University Press, 2004.
H. Brunner, On discrete superconvergence properties of spline collocation methods for nonlinear Volterra integral equations, J. Comput. Appl. Math., 10 (1992), 348-357
H. Brunner, P.J. van der Houwen, “The Numerical Solution of Volterra Equations”, North Holland, 1986.
H. Brunner, S. Nørsett, Superconvergence of collocation methods for Volterra and Abel integral equations of the second kind, Numer. Math., 36 (1981), 347-358.
H. Brunner, A. Pedas, G. Vainikko, The piecewise polynomial collocation method for nonlinear weakly singular Volterra equations, Math. Comp., 68 (1999), 1079-1095.
T. Diogo, A note on collocation and iterated collocation methods for a class of weakly singular Volterra integral equations, submitted.
T. Diogo, N.B. Franco, P. Lima, High order product integration methods for a Volterra integral equation with logarithmic singular kernel, Commun. Pure Appl. Anal., 3 (2004), 217-235.
T. Diogo, P. Lima, Superconvergence properties of collocation methods for a Volterra integral equation with weakly singular kernel, to appear in J. Comput. Appl. Math.
T. Diogo, S. McKee, T. Tang, A Hermite-type collocation method for the solution of an integral equation with a certain weakly singular kernel, IMA J. Numer. Anal., 11 (1991), 595-605.
T. Diogo, S. Mckee, T. Tang, Collocation methods for second-kind Volterra integral equations with weakly singular kernels, Proc. Roy. Soc. Edinburgh, 124A (1994), 199-210.
E.A. Galperin, E.J. Kansa, A. Makroglou, S.A. Nelson, Variable transformations in the numerical solution of the second kind Volterra integral equations with continous and weakly singular kernels; extension to Fredholm integral equations, J. Comput. Appl. Math., 115 (2000), 193-211.
R. Gorenflo, S. Vessella, “Abel Integral Equations, a Historico-bibliographical Survey”, IAGA, Firenze, 1984.
W. Han, Existence, uniqueness and smoothness results for second-kind Volterra equations with weakly singular kernels, J. Integral Equations Appl., 6 (1994), 365-384.
P.M. Lima, T. Diogo, An extrapolation method for a Volterra integral equation with weakly singular kernel, Appl. Numer. Math. 24 (1997), 131-148.
A. Pedas, G. Vainikko, Smoothing transformation and piecewise polynomial collocation for weakly singular Volterra integral equations, Computing, 73 (2004), 271-293.
T. Tang, A note on collocation methods for Volterra integro-differential equations with weakly singular kernels, IMA J. Numer. Math., 13 (1993), 93-99.
Downloads
Published
How to Cite
Issue
Section
License
Authors who publish in this journal agree to the following terms:
Authors retain copyright and grant the journal the right of first publication, with the work simultaneously licensed under the Creative Commons Attribution License that allows the sharing of the work with acknowledgment of authorship and initial publication in this journal.
Authors are authorized to assume additional contracts separately, for non-exclusive distribution of the version of the work published in this journal (eg, publish in an institutional repository or as a book chapter), with acknowledgment of authorship and initial publication in this journal.
Authors are allowed and encouraged to publish and distribute their work online (eg, in institutional repositories or on their personal page) at any point before or during the editorial process, as this can generate productive changes as well as increase impact and the citation of the published work (See The effect of open access).
This is an open access journal which means that all content is freely available without charge to the user or his/her institution. Users are allowed to read, download, copy, distribute, print, search, or link to the full texts of the articles, or use them for any other lawful purpose, without asking prior permission from the publisher or the
author. This is in accordance with the BOAI definition of open access
Intellectual Property
All the contents of this journal, except where otherwise noted, is licensed under a Creative Commons Attribution License under attribution BY.