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.
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