Imposing various boundary conditions on positive definite kernels

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 kernel based pseudospectral method for handling the differential equations with more complicated boundary conditions. In the proposed technique some new kernels are constructed using the positive definite kernels in a manner that they satisfy the required conditions. In addition, we prove the positive definiteness of the newly constructed kernel, and also the non-singularity of the collocation matrix is proved under some conditions. The proposed method is verified through the numerical solution of some benchmark problems such as a singularly perturbed steady-state convection–diffusion problem, two and three dimensional Poissons equations with various boundary conditions.