Piecewise Polynomial Collocation Methods for Linear Volterra Integro-Differential Equations with Weakly Singular Kernels

Abstract

In the first part of this paper we study the regularity properties of solutions of linear Volterra integro-differential equations with weakly singular or other nonsmooth kernels. We then use these results in the analysis of two piecewise polynomial collocation methods for solving such equations numerically. The main purpose of the paper is the derivation of optimal global convergence estimates and the analysis of the attainable order of local superconvergence at the collocation points.

An efficient collocation method for a Caputo two-point boundary value problem

“…When investigating solutions of (1.5), the natural setting is a weighted normed linear space C q,δ with q ∈ N, which we now define and which is a particular case of the more general Banach spaces C q,ν for −∞ < ν < 1 that are considered in [4,13]. Let C q,δ (0, 1] be the space of all q-times continuously differentiable functions y :…”

Numerical investigation of the weakly singular Volterra integro-differential equations using He's Homotopy perturbation method

“…Due to the wide application of these equations, they must be solved successfully with efficient numerical methods. Brunner [3] introduced polynomial spline collocation methods for Volterra integro-differential equations with weakly singular kernels. Brunner et al [4] introduced piecewise polynomial collocation methods for linear Volterra integrodifferential equations with weakly singular kernels.…”

Jacobi-spectral method for integro-delay differential equations with weakly singular kernels

“…The numerical solution of (1.1) with smooth kernels was investigated by spectral methods, which leads to a higher (exponential) convergence order [3], where it was shown that the results on the exponential order of convergence of the spectral method for the pantograph DDE [1,2] remain valid for pantograph-type integro-differential equation (1.1) with smooth kernels. Several numerical methods have been proposed for (1.1) without a delay (see, e.g., [6][7][8][9][10][11][12][13][14][15][16][24][25][26][27][28]), where they show that its numerical treatment is not simple, due to the fact that the solution of (1.1) usually has a * Correspondence: ishtiaqali@comsats.edu.pk 2010 AMS Mathematics Subject Classification: 45J05, 65R20, 34K06, 34K28.…”