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A reproducing kernel Hilbert space approach in meshless collocation method

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics, Imam Khomeini International University, Ghazvin, 34149-16818, Iran

    Babak Azarnavid, Mahdi Emamjome & Saeid Abbasbandy

  2. Department of Basic Sciences, Abadan Faculty of Petroleum Engineering, Petroleum University of Technology, Abadan, Iran

    Mohammad Nabati

Authors
  1. Babak Azarnavid
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  2. Mahdi Emamjome
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  3. Mohammad Nabati
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  4. Saeid Abbasbandy
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Correspondence to Babak Azarnavid.

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Communicated by Frederic Valentin.

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Azarnavid, B., Emamjome, M., Nabati, M. et al. A reproducing kernel Hilbert space approach in meshless collocation method. Comp. Appl. Math. 38, 72 (2019). https://doi.org/10.1007/s40314-019-0838-0

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  • DOI: https://doi.org/10.1007/s40314-019-0838-0

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