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Weight calculation and convergence analysis of polyharmonic spline (PHS) with polynomials for different stencils

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Abstract

Recent developments in the field of the radial basis function-finite difference (RBF-FD) framework have been focused on conditionally positive definite polyharmonic splines (PHS). Within this context, our research focuses on deriving analytical weights for the RBF-FD+polynomials method within the framework of PHS. We provide convergence analyses for various stencils. To validate the accuracy of our derived formulations, we conduct a series of computational experiments across a range of test problems.

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Acknowledgements

The authors are grateful to an anonymous referee for several suggestions on the improvement of this work.

Funding

The study was supported by Key Scientific Research Projects of Colleges and Universities in Henan Province (No. 24B110018).

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Authors and Affiliations

  1. Institute of Statistics and Big Data, Zhengzhou College of Finance and Economics, Zhengzhou, 450000, Henan, China

    Zhihong Liu

  2. Department of Mathematics and Computer Science, Sirjan University of Technology, Sirjan, Iran

    Mahdiar Barfeie

  3. Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan, 45137-66731, Iran

    Fazlollah Soleymani

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  1. Zhihong Liu
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  3. Fazlollah Soleymani
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Correspondence to Mahdiar Barfeie.

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Appendix

Appendix

For cases where \(k = 3\) and using a stencil (12), the RBF-FD matrix becomes singular, necessitating the use of a pseudo-inverse for weight computation; analytical weights are provided exclusively for this unstructured three-point case in Table 6.

Table 6 The three-point weights for the approximating derivatives under singular cases for the linear system
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Liu, Z., Barfeie, M. & Soleymani, F. Weight calculation and convergence analysis of polyharmonic spline (PHS) with polynomials for different stencils. Calcolo 61, 22 (2024). https://doi.org/10.1007/s10092-024-00570-8

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