Collocation Method For Integro-differential Equations Research Articles

Spline collocation method for integro-differential equations with weakly singular kernels

In the first part of this paper we study the regularity properties of solutions to initial or boundary-value problems of Fredholm integro-differential equations with weakly singular or other nonsmooth kernels. We then use these results in the analysis of a piecewise polynomial collocation method for solving such problems numerically. Presented numerical examples display that theoretical results are in good accordance with actual convergence rates of proposed algorithms.

Collocation Methods for Integro-Differential Equations

In this note we extend the work of de Boor and Swartz (SIAM J. Numer. Anal., 10 (1973), pp. 582-606) on the solution of two-point boundary value problems by collocation. In particular, we are concerned with boundary value problems described by integro-differential equations involving derivatives of order up to and including m with m boundary conditions. We study the approximation of (isolated) solutions by means of piecewise polynomial functions of degree less than m + k possessing m -1 continuous derivatives. If the problem is sufficiently smooth and the solution has m +2k continuous derivatives, then one can achieve O(IAlk+m) global convergence by collocating at the zeros of the kth Legendre polynomial relative to each subinterval. At the knots, the approximation and its first m - 1 derivatives are O(IA12k) accurate.