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Numerical Caputo Differentiation by Radial Basis Functions

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Abstract

Previously, based on the method of (radial powers) radial basis functions, we proposed a procedure for approximating derivative values from one-dimensional scattered noisy data. In this work, we show that the same approach also allows us to approximate the values of (Caputo) fractional derivatives (for orders between 0 and 1). With either an a priori or a posteriori strategy of choosing the regularization parameter, our convergence analysis shows that the approximated fractional derivative values converge at the same rate as in the case of integer order 1.

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Notes

  1. More precisely, radial power RBF is \((-1)^\beta \Vert x\Vert ^{2\beta -1}\). As the interpolation matrix is not required in this work, we drop the term \((-1)^k\) for the sake of simplicity.

  2. By the Morozov’s discrepancy principle, we select \(\sigma \) that satisfies

    $$\begin{aligned} \frac{1}{n}\sum _{i=1}^{n}\big (y_{\beta ,\sigma }(x_{i})-{y}_{i}^\delta \big )^{2}=\delta ^{2}. \end{aligned}$$

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Acknowledgments

This project was partially supported by a CERG Grant of the Hong Kong Research Grant Council, a FRG Grant of the Hong Kong Baptist University, a grant from the National Natural Science Foundation of China (No. 11126126), the Natural Science Foundation of Shanxi (No. 2012021002-2), and the Shanxi Scholarship Council of China (Project No. 2011-025).

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Authors and Affiliations

  1. Department of Mathematics, Taiyuan University of Technology, Taiyuan, China

    Ming Li & Yujiao Wang

  2. Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong

    Leevan Ling

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  1. Ming Li
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  2. Yujiao Wang
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  3. Leevan Ling
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Correspondence to Ming Li or Leevan Ling.

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Li, M., Wang, Y. & Ling, L. Numerical Caputo Differentiation by Radial Basis Functions. J Sci Comput 62, 300–315 (2015). https://doi.org/10.1007/s10915-014-9857-6

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  • DOI: https://doi.org/10.1007/s10915-014-9857-6

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