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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.

A reproducing kernel Hilbert space approach in meshless collocation method