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).

We present an iterative method based on the Hermitian and skew-Hermitian splitting for solving the continuous Sylvester equation. By using the Hermitian and skew-Hermitian splitting of the coefficient matrices A and B, we establish a method which is practically inner/outer iterations, by employing a CGNR-like method as inner iteration to approximate each outer iterate, while each outer iteratio...

in the present article, a numerical method is proposed for the numerical solution of thekdv equation by using a new approach by combining cubic b-spline functions. in this paper we convert the kdv equation to system of two equations. the method is shown to be unconditionally stable using von-neumann technique. to test accuracy the error norms2l, ∞l are computed. three invariants of motion are p...

‎in this paper we introduce a numerical approach that solves optimal control problems (ocps)‎‎using collocation methods‎. ‎this approach is based upon b-spline functions‎.‎the derivative matrices between any two families of b-spline functions are utilized to‎‎reduce the solution of ocps to the solution of nonlinear optimization problems‎.‎numerical experiments confirm our theoretical findings‎.

we develope a numerical method based on b-spline collocation method to solve linear klein-gordon equation. the proposed scheme is unconditionally stable. the results of numerical experiments have been compared with the exact solution to show the efficiency of the method computationally. easy and economical implementation is the strength of this approach.

the numerical methods are of great importance for approximating the solutions of nonlinear ordinary or partial differential equations, especially when the nonlinear differential equation under consideration faces difficulties in obtaining its exact solution. in this latter case, we usually resort to one of the efficient numerical methods. in this paper, the chebyshev collocation method is sugge...

and Applied Analysis 3 2. Preliminaries Let w α,β x 1 − x α 1 x β be a weight function in the usual sense for α, β > −1. The set of Jacobi polynomials {P α,β k x } k 0 forms a complete L 2 w α,β −1, 1 -orthogonal system, and ∥ ∥ ∥P α,β k ∥ ∥ ∥ 2 w α,β h α,β k 2 β 1Γ k α 1 Γ ( k β 1 ) ( 2k α β 1 ) Γ k 1 Γ ( k α β 1 ) . 2.1 Here, L2 w α,β −1, 1 is a weighted space defined by L2 w α,β −1, 1 {v : v...