In this paper, we introduce a generalized Chebyshev collocation method (GCCM) based on the generalized Chebyshev polynomials for solving stiff systems. For employing a technique of the embedded Runge-Kutta method used in explicit schemes, the property of the generalized Chebyshev polynomials is used, in which the nodes for the higher degree polynomial are overlapped with those for the lower deg...

We develop an exponentially accurate fractional spectral collocation method for solving steady-state and time-dependent fractional PDEs (FPDEs). We first introduce a new family of interpolants, called fractional Lagrange interpolants, which satisfy the Kronecker delta property at collocation points. We perform such a construction following a spectral theory recently developed in [M. Zayernouri ...

We present in this paper the convergence properties of Jacobi spectral collocation method when used to approximate the solution of multidimensional nonlinear Volterra integral equation. The solution is sufficiently smooth while the source function and the kernel function are smooth. We choose the Jacobi-Gauss points associated with the multidimensional Jacobi weight function [Formula: see text]...

The sparse grid stochastic collocation method is a new method for solving partial differential equations with random coefficients. However, when the probability space has high dimensionality, the number of points required for accurate collocation solutions can be large, and it may be costly to construct the solution. We show that this process can be made more efficient by combining collocation ...

The commonly used graded piecewise polynomial collocation method for weakly singular Volterra integral equations may cause serious round-off error problems due to its use of extremely nonuniform partitions and the sensitivity of such time-dependent equations to round-off errors. The singularity preserving (nonpolynomial) collocation method is known to have only local convergence. To overcome th...

In this paper, we construct numerical schemes for spectral collocation methods and spectral variational integrators which converge geometrically. We present a systematic comparison of how spectral collocation methods and Galerkin spectral variational integrators perform in terms of their ability to reproduce accurate trajectories in configuration and phase space, their ability to conserve momen...

The paper describes ongoing work on the evaluation of methods for extracting collocation candidates from large text corpora. Our research is based on a German treebank corpus used as gold standard. Results are available for adjective+noun pairs, which proved to be a comparatively easy extraction task. We plan to extend the evaluation to other types of collocations (e.g., PP+verb pairs).