A reproducing kernel Hilbert space approach in meshless collocation method

In this paper, we combine the theory of the reproducing kernel Hilbert spaces with the field of collocation methods to solve boundary value problems with a special emphasis on the reproducing property of kernels. Using the reproducing property of the kernels, a new efficient algorithm is proposed to obtain the cardinal functions of a reproducing kernel Hilbert space, which can be applied conveniently for multi-dimensional domains. The differentiation matrices are constructed and also a pointwise error estimate of applying them is derived. In addition, we prove the non-singularity of the collocation matrix. The proposed method is truly meshless, and can be applied conveniently and accurately for high order and also multi-dimensional problems. Numerical results are presented for the several problems such as second- and fifth-order two-point boundary value problems, one- and two-dimensional unsteady Burgers' equations, and a three-dimensional parabolic partial differential equation. In addition, we compare the numerical results with the best-reported results in the literature to show the high accuracy and efficiency of the proposed method.