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A sparse fractional Jacobi–Galerkin–Levin quadrature rule for highly oscillatory integrals

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  • Ma, Junjie
  • Liu, Huilan

Abstract

Levin’s quadrature rule is well-known for its good performance on numerical treatment with highly oscillatory integrals. However, most existing work treats with sufficiently smooth integrands. In this paper, we study Levin’s quadrature rule for a class of singular and oscillatory integrals. Based on fractional Jacobi polynomials, a class of differential equation solvers for Levin’s equation are developed, which leads to the fractional Jacobi–Galerkin–Levin method. The discretized equation is turned into a sparse linear system by properly choosing Jacobi polynomials and the inner product. Furthermore, convergence analysis with respect to the oscillation is presented by studying coefficients of the fractional Jacobi expansion of the error function. Numerical experiments indicate that in contrast to existing quadrature rules, the new method is efficient for computing oscillatory integrals with stationary points and unknown singular parameters.

Suggested Citation

  • Ma, Junjie & Liu, Huilan, 2020. "A sparse fractional Jacobi–Galerkin–Levin quadrature rule for highly oscillatory integrals," Applied Mathematics and Computation, Elsevier, vol. 367(C).
    • Handle: RePEc:eee:apmaco:v:367:y:2020:i:c:s0096300319307672
    DOI: 10.1016/j.amc.2019.124775
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    References listed on IDEAS

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    1. Singh, Jagdev & Kumar, Devendra & Baleanu, Dumitru & Rathore, Sushila, 2018. "An efficient numerical algorithm for the fractional Drinfeld–Sokolov–Wilson equation," Applied Mathematics and Computation, Elsevier, vol. 335(C), pages 12-24.
    2. Kumar, Devendra & Singh, Jagdev & Baleanu, Dumitru & Sushila,, 2018. "Analysis of regularized long-wave equation associated with a new fractional operator with Mittag-Leffler type kernel," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 492(C), pages 155-167.
    3. Goswami, Amit & Singh, Jagdev & Kumar, Devendra & Sushila,, 2019. "An efficient analytical approach for fractional equal width equations describing hydro-magnetic waves in cold plasma," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 524(C), pages 563-575.
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    Cited by:

    1. Zhen Yang & Junjie Ma, 2020. "Efficient Computation of Highly Oscillatory Fourier Transforms with Nearly Singular Amplitudes over Rectangle Domains," Mathematics, MDPI, vol. 8(11), pages 1-21, November.

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