A Sinc-Collocation Method for Weakly Singular Volterra Integral Equations

Abstract

Let T^α:C[0,a] →C[0,a] denote the linear Volterra operator defined by

$${{\text{T}}^\alpha}{\text{f(x)= }}\int\limits_0^{\text{x}}{{{{\text{(x-t)}}}^{-\alpha}}}{\text{k(x,t) f(t)dt,0}}\leqslant{\text{x}}\leqslant{\text{a,}}$$

((1.1))

where 0 < α < 1 and where k is a smooth function on the domain {(x,t): 0 ≤ t ≤ x ≤ a}. Then, for smooth function g, the weakly singular linear Volterra integral equation is given by

$$y\left( x \right) = g\left( x \right) + {T^\alpha }y\left( x \right), 0 \leqslant x \leqslant a.$$

(1.2)