Steklov approximations of Green’s functions for Laplace equations

Purpose This paper aims to present a meshless technique to find the Green’s functions for solutions of Laplacian boundary value problems on rectangular domains. This paper also investigates a theoretical basis for the Steklov series expansion methods to reduce and estimate the error of numerical approaches for the boundary correction kernel of the Laplace operator. Design/methodology/approach The main interest is how the Green's functions differ from the fundamental solution of the Laplace operator. Steklov expansion methods for finding the correction term are supported by the analysis that bases of the class of all finite harmonic functions can be formed using harmonic Steklov eigenfunctions. These functions construct a basis of the space of solutions of harmonic boundary value problems and their boundary traces generate an orthogonal basis of the trace space of solutions on the boundary. Findings The main conclusion is that the boundary correction term for the Green's functions is well-approximated by Steklov expansions with a few Steklov eigenfunctions. The error estimates for the Steklov approximations of the boundary correction term involved in Dirichlet or Robin boundary value problems are found. They appear to provide very good approximations in the interior of the region and become quite oscillatory close to the boundary. Originality/value This paper concentrates to document the first attempt to find the Green's function for various harmonic boundary value problems with the explicit Steklov eigenfunctions without concerns regarding discretizations when the region is a rectangle.

Estimating a Distribution Function at the Boundary

Estimation of distribution functions has many real-world applications. We study kernel estimation of a distribution function when the density function has compact support. We show that, for densities taking value zero at the endpoints of the support, the kernel distribution estimator does not need boundary correction. Otherwise, boundary correction is necessary. In this paper, we propose a boundary distribution kernel estimator which is free of boundary problem and provides non-negative and non-decreasing distribution estimates between zero and one. Extensive simulation results show that boundary distribution kernel estimator provides better distribution estimates than the existing boundary correction methods. For practical application of the proposed methods, a data-dependent method for choosing the bandwidth is also proposed.

Improving Water Level Forecast of an Oceanographic Model in Malacca Strait Based on Data-Driven Open Boundary Correction

Numerical model is an indispensable tool for understanding oceanographic phenomena and resolving associated physical processes. However, model error cannot be avoided due to limitations such as underlying assumption, insufficient information of bathymetry or boundary condition and so on. Data assimilation technique thus becomes an effective and essential tool to improve prediction accuracy. Updating of output is an efficient way to correct the model, but it is often carried out locally at specific locations in the model domain where measurement is available. In this study, instead of correcting output of numerical model locally, we propose to combine local correction and input correction to update open boundary of numerical model. The open boundary condition is corrected through spatial interpolation algorithm based on nearby observation in the hindcast period. Then the local forecast at measured location is distributed using the same interpolation scheme to update the boundary in the forecast period. Such boundary correction not only explores the variation in the future time step from the input updating but also allows the backbone physics embedded in numerical model to resolve the hydrodynamics in the entire computational domain.