# A reproducing kernel Hilbert space approach in meshless collocation method

## TL;DR

This paper introduces a meshless collocation method using reproducing kernel Hilbert spaces to efficiently solve high-order and multidimensional boundary value problems with proven accuracy and stability.

## Contribution

It develops a new algorithm for cardinal functions in RKHS, constructs differentiation matrices, and proves nonsingularity of the collocation matrix, enhancing meshless solution techniques.

## Key findings

- High accuracy in solving boundary value problems
- Effective for multidimensional and high-order problems
- Numerical results outperform existing methods

## Abstract

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