Imposing various boundary conditions on positive definite kernels

TL;DR

This paper introduces a new technique to modify positive definite kernels so they satisfy boundary conditions exactly, enhancing the accuracy of kernel-based methods for complex boundary value problems in engineering and physics.

Contribution

A novel method to impose boundary conditions on positive definite kernels, improving their application in solving differential equations with complex boundaries.

Findings

Successfully applied to singularly perturbed convection-diffusion problems

Enhanced the accuracy of radial basis function methods for boundary conditions

Validated on 2D and 3D Poisson equations

Abstract

This work is motivated by the frequent occurrence of boundary value problems with various boundary conditions in the modeling of some problems in engineering and physical science. Here we propose a new technique to force the positive definite kernels such as some radial basis functions to satisfy the boundary conditions exactly. It can improve the applications of existing methods based on positive definite kernels and radial basis functions especially the pseudospectral radial basis function method for handling the differential equations with more complicated boundary conditions. The proposed method is applied to a singularly perturbed steady-state convection-diffusion problem, two and three dimensional Poisson's equations with various boundary conditions.