The Fredholm–Volterra integral equation of the second kind with continuous kernels with respect to position and time, is solved numerically, using the Collocation and Galerkin methods. Also the error, in each case, is estimated.

For second harmonic generation in two-dimensional wave-guiding structures composed of segments that are invariant in the longitudinal direction, we develop an efficient numerical method based on the Dirichlet-to-Neumann (DtN) maps of the segments and a marching scheme using two operators and two functions. A Chebyshev collocation method is used to discretize the longitudinal variable for comput...

We propose a modification of collocation methods extending the ‘averaged vector field method’ to high order. These new integrators exactly preserve energy for Hamiltonian systems, are of arbitrarily high order, and fall into the class of B-series integrators. We discuss their symmetry and conjugate-symplecticity, and we compare them to energypreserving composition methods. c © 2010 European Soc...

In this article we use discrete collocation method for solving Fredholm–Volterra integro– differential equations, because these kinds of integral equations are used in applied sciences and engineering such as models of epidemic diffusion, population dynamics, reaction–diffusion in small cells. Also the above integral equations with convolution kernel will be solved by discrete collocation metho...

This article investigates the stability of the collocation method for the Radiosity equation. We introduce graded meshes and trial spaces of piecewise poynomials and prove stability for some modified collocation method. Next we show that graded meshes are necessary, even for linear convergence. The generation of triangulations which allow higher order approximations leads to geometrical problem...

Syntactic configurations used in collocation extraction are highly divergent from one system to another, this questioning the validity of results and making comparative evaluation difficult. We describe a corpus-driven approach for inferring an exhaustive set of configurations from actual data by finding, with a parser, all the productive syntactic associations, then by appealing to human exper...

This paper presents a status quo of an ongoing research study of collocations – an essential linguistic phenomenon having a wide spectrum of applications in the field of natural language processing. The core of the work is an empirical evaluation of a comprehensive list of automatic collocation extraction methods using precision-recall measures and a proposal of a new approach integrating multi...

This paper reports on the development of a collocation extraction system that is designed within a commercial machine translation system in order to take advantage of the robust syntactic analysis that the system offers and to use this analysis to refine collocation extraction. Embedding the extraction system also addresses the need to provide information about the source language collocations ...

We develop spectral collocation methods for fractional differential equations with variable order with two end-point singularities. Specifically, we derive three-term recurrence relations for both integrals and derivatives of the weighted Jacobi polynomials of the form 1 x μ1 1 Æ x 2P a,b j x a, b, μ1, μ2 Æ1 , which leads to the desired differentiation matrices. We apply the new differentiation...

In this article, an implicit hybrid method of order six is developed for the direct solution of second order ordinary differential equations using collocation and interpolation approach. To derive this method, the approximate solution power series is interpolated at the first and off-step points and its second and third derivatives are collocated at all points in the given interval. Besides hav...