Cubic spline quasi-interpolation operator to numerically solve integro-differential equations with weakly singular kernels

Abstract

In this paper we use cubic spline quasi-interpolant operator to numerically address a class of linear integro-differential equations with weakly singular kernel. As stated in Pedas and Tamme (2006), the exact solution of this equation lacks the desired level of smoothness and belongs to a particular function space. Then, in the first part of this paper, we analyze the approximation properties of the cubic spline quasi-interpolant operator in this particular space. Subsequently, we use these results in the analysis of the quasi-collocation method used to solve an integro-differential equation with weakly singular kernel. Also, some numerical tests are provided to confirm the theoretical results.