We consider numerical solutions of a class of nonlinear (nonstandard) Volterra integral equations. We first prove the existence and uniqueness of the solution of the Volterra integral equation in the context of the space of continuous funtions over a closed interval. We then use one point collocation methods and quadrature methods with a uniform mesh to construct solutions of the nonlinear VIE....

Pseudospectral collocation is employed for the numerical solution of nonlinear two-point boundary value problems with separated end conditions. Second-order finite difference schemes are used as preconditioners for the spectral calculation, and a solution of the discretized equations is obtained using versions of the defect correction principle. The method and a variant based on an adaptive gri...

A uniform bound of the 1−norm is given for the inverse of lower triangular Toeplitz matrices with nonnegative monotonically decreasing entries whose limit is zero. The new bound is sharp under certain specified constraints. This result is then employed to throw light upon a long standing open problem posed by Brun-ner concerning the convergence of the one-point collocation method for the Abel's...

In this paper, we analyse the iterated collocation method for Hammerstein equations with smooth and weakly singular kernels. The paper expands the study which began in [14] concerning the superconvergence of the iterated Galerkin method for Hammerstein equations. We obtain in this paper a similar superconvergence result for the iterated collocation method for Hammerstein equations. We also disc...

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...