A reproducing kernel Hilbert space approach in meshless collocation method
For researchers in numerical methods, it offers a meshless approach for high-order and multidimensional problems, but the novelty is incremental as it builds on existing RKHS and collocation techniques.
The paper combines reproducing kernel Hilbert space theory with meshless collocation to solve boundary value problems, proposing an efficient algorithm for cardinal functions and differentiation matrices. Numerical results show high accuracy for 1D, 2D, and 3D problems, including Burgers equations.
In this paper we combine the theory of reproducing kernel Hilbert spaces with the field of collocation methods to solve boundary value problems with special emphasis on reproducing property of kernels. From the reproducing property of kernels we proposed a new efficient algorithm to obtain the cardinal functions of a reproducing kernel Hilbert space which can be apply conveniently for multidimensional domains. The differentiation matrices are constructed and also we drive pointwise error estimate of applying them. In addition we prove the nonsingularity of collocation matrix. The proposed method is truly meshless and can be applied conveniently and accurately for high order and also multidimensional 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 parabolic partial differential equation in three dimensions. Also we compare the numerical results with those reported in the literature to show the high accuracy and efficiency of the proposed method