An adaptive method for Volterra–Fredholm integral equations on the half line

In this paper we develop a direct quadrature method for solving Volterra–Fredholm integral equations on an unbounded spatial domain. These problems, when related to some important physical and biological phenomena, are characterized by kernels that present variable peaks along space. The method we propose is adaptive in the sense that the number of spatial nodes of the quadrature formula varies with the position of the peaks. The convergence of the method is studied and its performances are illustrated by means of a few significative examples. The parallel algorithm which implements the method and its performances are described.