In this paper, we propose a natural collocation method with exact imposition of mixed boundary conditions based on a generalized Gauss-Lobatto-Legendre-Birhoff quadrature rule that builds in the underlying boundary data. We provide a direct construction of the quadrature rule, and show that the collocation method can be implemented as efficiently as the usual collocation scheme for PDEs with Di...

This paper presents an annotated Chinese collocation bank developed at the Hong Kong Polytechnic University. The definition of collocation with good linguistic consistency and good computational operability is first discussed and the properties of collocations are then presented. Secondly, based on the combination of different properties, collocations are classified into four types. Thirdly, th...

In recent years, collocation has been widely acknowledged as an essential characteristic to distinguish native speakers from non-native speakers. Research on academic writing has also shown that collocations are not only common but serve a particularly important discourse function within the academic community. In our study, we propose a machine learning approach to implementing an online collo...

A numerical approximation of the initial-boundary system of nonlinear hyperbolic equations based on spectral collocation method is presented in this article. A Chebyshev-Gauss-Radau collocation (C-GR-C) method in combination with the implicit RungeKutta scheme are employed to obtain highly accurate approximations to the mentioned problem. The collocation points are the Chebyshev interpolation n...

Collocation methods based on quartic splines are presented for second-order two-point boundary value problems. In order to obtain a uniquely solvable linear system for the degrees of freedom of the quartic spline collocation approximation, in addition to the boundary conditions specified by the problem, extra boundary or near-boundary conditions are introduced. Non-optimal (fourth-order) and op...

Current educational reforms and initiatives are addressing numeracy outcomes for school-age students as a response to concerns regarding their mathematical achievements. In Australia, the Ministerial Council on Education stated, numeracy remains one of the cornerstones of schooling for young Australians (2008). Ultimately, the responsibility for attending to these educational priorities is plac...

In this column, we take break from our ongoing discussion of the CORBA Messaging specification to cover collocation, which is an important topic for component-based application developers. Collocation is a technique for transparently optimizing communication overhead when clients and servants are configured into the same address space. For instance, Microsoft COM [1] applications frequently use...

We present a collocatIon method based on piecewise cubic polynomials I In C for solving various Integral and differential equatIons. Two new methods for handling curved boundaries are discussed based on collocation and discrete least-squares. Finally, a summary of the properties of the collocation method Is given which IndIcates the simplicity, flexlbl1fty, efficiency, and generality of the met...