The current paper contributes a novel framework for solving a class of linear matrix differential equations. To do so, the operational matrix of the derivative based on the shifted Bernoulli polynomials together with the collocation method are exploited to reduce the main problem to system of linear matrix equations. An error estimation of presented method is provided. Numerical experiments are...

Collocations play a significant role in second language acquisition. In order to be able to offer efficient support to learners, an NLP-based CALL environment for learning collocations should be based on a representative collocation error annotated learner corpus. However, so far, no theoretically-motivated collocation error tag set is available. Existing learner corpora tag collocation errors ...

We discuss an a-posteriori error estimate for the numerical solution of boundary value problems for nonlinear systems of ordinary differential equations with a singularity of the first kind. The estimate for the global error of an approximation obtained by collocation with piecewise polynomial functions is based on the defect correction principle. We prove that for collocation methods which are...

A comparative study of weighted residual methods has been made on different types of advection diffusion equations. Both the linear and non-linear models have been discretized by orthogonal collocation method (OCM) and orthogonal collocation on finite elements (OCFE). Model equations have been solved by MATLAB ‘ode15s’ system solver. Numerical values have been compared with analytic ones and in...

Several optimal three-dimensional orbital transfer problems are solved for thrust-limited spacecrafts using collocation and nonlinear programming techniques. The solutions for full nonlinear equations of motion are obtained where the integrals of the free Keplerian motion in three dimensions are utilized for coasting arcs. In order to limit the solution space, interior-point constraints are use...

In this paper, collocation methods are used for detecting blow-up solutions of nonlinear homogeneous Volterra-Hammerstein integral equations. To do this, we introduce the concept of “blow-up collocation solution” and analyze numerically some blow-up time estimates using collocation methods in particular examples where previous results about existence and uniqueness can be applied. Finally, we d...

We consider a sparse grid collocation method in conjunction with a time discretization of the differential equations for computing expectations of functionals of solutions to differential equations perturbed by time-dependent white noise. We first analyze the error of Smolyak’s sparse grid collocation used to evaluate expectations of functionals of solutions to stochastic differential equations...

In this paper, we provide an analysis on the collocation methods(CM), which uses a large scale of admissible functions such as orthogonal polynomials, trigonometric functions, radial basis functions and particular solutions, etc. The admissible functions can be chosen to be piecewise, i.e., different functions are used in different subdomains. The key idea is that the collocation method can be ...

Collocations in the sense of idiosyncratic binary lexical co-occurrences are one of the biggest challenges for any language learner. Even advanced learners make collocation mistakes in that they literally translate collocation elements from their native tongue, create new words as collocation elements, choose a wrong subcategorization for one of the elements, etc. Therefore, automatic collocati...

In this paper, we investigate a stochastic spline-collocation approximation scheme for an optimal control problem governed by an elliptic PDE with random field coefficients. We obtain the necessary and sufficient optimality conditions for the optimal control problem and establish a scheme to approximate the optimality system through the discretization with respect to the spatial space by finite...