66 research outputs found
The Numerical Solution of Two-Dimensional Volterra Integral Equations by Collocation and Iterated Collocation
While the numerical solution of one-dimensional Volterra integral equations of the second kind with regular kernels is now well understood there exist no systematic studies of the approximate solution of their two-dimensional counterparts. In the present paper we analyse the numerical solution of such equations by methods based on collocation and iterated collocation techniques in certain polynomial spline spaces. The analysis focuses on the global convergence and local superconvergence properties of the approximating spline function
Multistep collocation methods for Volterra Integral Equations
We introduce multistep collocation methods for the numerical integration of Volterra Integral Equations, which depend on the numerical solution in a fixed number of previous time steps. We describe the constructive technique, analyze the order of the
resulting methods and their linear stability properties. Numerical experiments confirm the theoretical expectations
An improvement of the product integration method for a weakly singular Hammerstein equation
We present a new method to solve nonlinear Hammerstein equations with weakly
singular kernels. The process to approximate the solution, followed usually,
consists in adapting the discretization scheme from the linear case in order to
obtain a nonlinear system in a finite dimensional space and solve it by any
linearization method. In this paper, we propose to first linearize, via Newton
method, the nonlinear operator equation and only then to discretize the
obtained linear equations by the product integration method. We prove that the
iterates, issued from our method, tends to the exact solution of the nonlinear
Hammerstein equation when the number of Newton iterations tends to infinity,
whatever the discretization parameter can be. This is not the case when the
discretization is done first: in this case, the accuracy of the approximation
is limited by the mesh size discretization. A Numerical example is given to
confirm the theorical result
Superconvergent Nyström and Degenerate Kernel Methods for Integro-Differential Equations
This research received no external funding and APC was funded by University of Granada.The aim of this paper is to carry out an improved analysis of the convergence of the Nystrom and degenerate kernel methods and their superconvergent versions for the numerical solution of a class of linear Fredholm integro-differential equations of the second kind. By using an interpolatory projection at Gauss points onto the space of (discontinuous) piecewise polynomial functions of degree <= r - 1, we obtain convergence order 2r for degenerate kernel and Nystrom methods, while, for the superconvergent and the iterated versions of theses methods, the obtained convergence orders are 3r + 1 and 4r, respectively. Moreover, we show that the optimal convergence order 4r is restored at the partition knots for the approximate solutions. The obtained theoretical results are illustrated by some numerical examples.University of Granad
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