Imposing various boundary conditions on positive definite kernels

Abstract

This paper presents a new approach for the imposing various boundary conditions on radial basis functions and their application in pseudospectral radial basis function method. The various boundary conditions such as Dirichlet, Neumann, Robin, mixed and multi-point boundary conditions, have been considered. Here we propose a new technique to force the radial basis functions to satisfy the boundary conditions exactly. It can improve the applications of existing methods based on radial basis functions especially the pseudospectral radial basis function method to handling the differential equations with more complicated boundary conditions. Several examples of one, two, and three dimensional problems with various boundary conditions have been considered to show the efficacy and versatility of the proposed method.

Numerical approximation of the first-passage time distribution of time-varying diffusion decision models: A mesh-free approach

“…Additional to the modal functions technique, there exist some methods for imposing the boundary conditions in the finite element and spectral element approaches such as the Nitsche method [101] or extended finite element method [102]. Recently, Azarnavid and his colleagues proposed a general imposing algorithm for positive definite kernel functions [103] which can be used for various types of boundary conditions such as Dirichlet, Neumann, and Robin. Since the considered problem in this paper is FPE with homogeneous Dirichlet boundary conditions, we just discuss imposing the Dirichlet conditions in the positive definite radial basis functions.…”

“…Theorem 2. [103] If ρ(x, y) be the reproducing kernel of reproducing kernel Hilbert space H defined on a region Ω ⊂ R and H : H −→ R be such that Hu = 0 for u ∈ H result that u = 0. Then the operator matrix A H is nonsingular.…”