Numerical solution of Volterra integral equations with weak singularities M. Kolk and A. Pedas Abstract We propose a piecewise polynomial collocation method for solving linear Volterra integral equations of the second kind with kernels which, in addition to a weak diagonal singularity, may have a weak boundary singularity. The attainable order of global and local convergence of proposed algorithms is discussed and a collection of numerical results is given. Key words: Volterra integral equation, weakly singular kernel, boundary singular- ity, collocation method. 1 Introduction Let C k (Ω ) be the set of all k times continuously differentiable functions on Ω , C 0 (Ω )= C(Ω ). By C[a, b] we denote the Banach space of continuous functions f on [a, b] with the usual norm ‖ f ‖ = max{| f (x)| : x ∈ [a, b]}. Let D b = {(x, y) :0 < y < x ≤ b}, D b = {(x, y) :0 ≤ y ≤ x ≤ b}. In many practical applications there arise integral equations of the form u(x)=  x 0 K(x, y)u(y)dy + f (x), 0 ≤ x ≤ b, (1) with f ∈ C m [0, b], K(x, y)= g(x, y)(x − y) −ν ,0 < ν < 1, g ∈ C m ( D b ), m ∈ N = {1, 2, ...}. The solution u(x) to (1) is typically non-smooth at x = 0 where its deriva- Marek Kolk Institute of Mathematics, University of Tartu, Estonia, e-mail: marek.kolk@ut.ee Arvet Pedas Institute of Mathematics, University of Tartu, Estonia, e-mail: arvet.pedas@ut.ee 1