Stieltjes Derivatives and $\beta $-Polynomial Spline Collocation for Volterra Integrodifferential Equations with Singularities

Abstract

In this paper, we introduce a new tool, the Stieltjes Derivative, and new kinds of meshes which are more practical than the graded meshes. Based on these, we discuss a nonpolynomial spline collocation method for Volterra integrodifferential equations with weakly singular kernels. Under very weak hypotheses which allow the nonsmooth behaviour of the given functions in these equations, collocation on such meshes is shown to yield optimal convergence rates. An ideal super-convergence result is also obtained.