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Galerkin method with trigonometric basis on stable numerical differentiation

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  • Luo, Yidong

Abstract

This paper considers the p (p=1,2,3) order numerical differentiation on function y in (0, 2π). They are transformed into corresponding Fredholm integral equation of the first kind. Computational schemes with analytic solution formulas are designed using Galerkin method on trigonometric basis. Convergence and divergence are all analysed in Corollaries 5.1, 5.2, and a-priori error estimate is uniformly obtained in Theorem 6.1, 7.1, 7.2. Therefore, the algorithm achieves the optimal convergence rate O(δ2μ2μ+1)(μ=12or1) with periodic Sobolev source condition of order 2µp. Besides, we indicate a noise-independent a-priori parameter choice when the function y possesses the form of∑k=0p−1aktk+∑k=1N1bkcoskt+∑k=1N2cksinkt,bN1,cN2≠0,In particular, in numerical differentiations for functions above, good filtering effect (error approaches 0) is displayed with corresponding parameter choice. In addition, several numerical examples are given to show that even derivatives with discontinuity can be recovered well.

Suggested Citation

  • Luo, Yidong, 2020. "Galerkin method with trigonometric basis on stable numerical differentiation," Applied Mathematics and Computation, Elsevier, vol. 370(C).
    • Handle: RePEc:eee:apmaco:v:370:y:2020:i:c:s009630031930904x
    DOI: 10.1016/j.amc.2019.124912
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    References listed on IDEAS

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    1. Dehghan, Mehdi, 2006. "Finite difference procedures for solving a problem arising in modeling and design of certain optoelectronic devices," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 71(1), pages 16-30.
    2. Assari, Pouria & Dehghan, Mehdi, 2019. "A meshless local discrete Galerkin (MLDG) scheme for numerically solving two-dimensional nonlinear Volterra integral equations," Applied Mathematics and Computation, Elsevier, vol. 350(C), pages 249-265.
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    Cited by:

    1. Egidi, Nadaniela & Giacomini, Josephin & Maponi, Pierluigi & Youssef, Michael, 2023. "An FFT method for the numerical differentiation," Applied Mathematics and Computation, Elsevier, vol. 445(C).

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