Iterated continuous extensions (ICEs) are continuous explicit Runge-Kutta methods developed for the numerical solution of evolutionary problems in ordinary and delay diierential equations (DDEs). ICEs have a particular r^ ole in the explicit solution of DDEs with vanishing lags. They may be regarded as parallel continuous explicit Runge-Kutta (PCERK) methods, as they allow advantage to be taken...

We present a meshless technique which can be seen as an alternative to the Method of Fundamental Solutions (MFS). It calculates homogeneous solutions of the Laplacian (i.e. harmonic functions) for given boundary data by a direct collocation technique on the boundary using kernels which are harmonic in two variables. In contrast to the MFS, there is no artifical boundary needed, and there is a f...

A method for the shallow water equations governing wave motions in the nearshore environment is presented. Spatial derivatives contained in these equations are computed using spectral collocation methods. A high-order time integration scheme is used to compute the time evolution of the velocities and water surface elevation given initial conditions. The model domain extends from the shoreline t...

An integration algorithm that conjugates a Method of Lines (MOL) strategy based on finite differences space discretizations, with a collocation strategy based on increasing level dyadic grids is presented. It reveals potential either as a grid generation procedure and a Partial Differential Equation (PDE) integration scheme. It copes satisfactorily with a example characterized by a steep travel...

A new type of variable coefficient Runge-Kutta-Nyström methods is proposed for solving the initial value problems of the special form y(t) = f(t, y(t)). The method is based on the exact integration of some given functions in order to solve the problem exactly when the solution is the linear combination of these functions. If this is not the case, the algebraic order (order of accuracy) of the m...

The work presented provides computational linguistics methods and tools for collocation identiication from arbitrary text, and methods and tools for representing collocations in a relational database integrating competence (collocation-type-speciic linguistic analysis) and performance information (corpus sentences). The work diiers from existing approaches to collo-cation identiication in syste...

‎In this paper‎, ‎an approach based on statistical spline model (SSM) and collocation method is proposed to solve Volterra-Fredholm integral equations‎. ‎The set of collocation nodes is chosen so that the points yield minimal error in the nodal polynomials‎. ‎Under some standard assumptions‎, ‎we establish the convergence property of this approach‎. ‎Numerical results on some problems are given...

We have suggested the numerical schemes of collocation methods and mechanical quadrature methods for approximative solution of singular integro-differential equations with kernels of Cauchy type. The equations are defined on the arbitrary smooth closed contours of complex plane. The researched methods are based on the descritization by Fejér points. Theoretical background for collocation method...

Fast direct methods are presented for the solution of linear systems arising in highorder, tensor-product orthogonal spline collocation applied to separable, second order, linear, elliptic partial di erential equations on rectangles. The methods, which are based on a matrix decomposition approach, involve the solution of a generalized eigenvalue problem corresponding to the orthogonal spline co...

The application of collocation methods using spline basis functions to solve differential model equations has been in use for a few decades. However, the application of spline collocation to the solution of the nonlinear, coupled, partial differential equations (in primitive variables) that define the motion of fluids has only recently received much attention. The issues that affect the effecti...