The Numerical Solution of Weakly Singular Volterra Integral Equations by Collocation on Graded Meshes

1 October 1985

journal article
Published by JSTOR in Mathematics of Computation

Vol. 45 (172) , 417-437
https://doi.org/10.2307/2008134

Abstract

Since the solution of a second-kind Volterra integral equation with weakly singular kernel has, in general, unbounded derivatives at the left endpoint of the interval of integration, its numerical solution by polynomial spline collocation on uniform meshes will lead to poor convergence rates. In this paper we investigate the convergence rates with respect to graded meshes, and we discuss the problem of how to select the quadrature formulas to obtain the fully discretized collocation equation.

Keywords

This publication has 16 references indexed in Scilit:

More Generalised Discrete Gronwall Inequalities
ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 1985
The numerical solution of integral equations with weakly singular kernels
Published by Springer Nature ,1984
Nonpolynomial Spline Collocation for Volterra Equations with Weakly Singular Kernels
SIAM Journal on Numerical Analysis, 1983
A Sinc Approximation for the Indefinite Integral
Mathematics of Computation, 1983
Runge-Kutta Theory for Volterra and Abel Integral Equations of the Second Kind
Mathematics of Computation, 1983
Galerkin Methods for Second Kind Integral Equations with Singularities
Mathematics of Computation, 1982
Generalised Discrete Gronwall Lemmas
ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 1982
On the Degree of Convergence of Piecewise Polynomial Approximation on Optimal Meshes
Transactions of the American Mathematical Society, 1977
On the Solution of a Volterra Integral Equation with a Weakly Singular Kernel
SIAM Journal on Mathematical Analysis, 1973
Smoothness of Solutions of Volterra Integral Equations with Weakly Singular Kernels
SIAM Journal on Mathematical Analysis, 1971