A Review of Prolate Spheroidal Wave Functions from the Perspective of Spectral Methods

Wang, Li Lian

doi:10.4208/jms.v50n2.17.01

Cited by 40 publications

(23 citation statements)

References 90 publications

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“…Theoretical aspects of the Mathieu functions have been studied by many authors, including Stratton [42], McLachlan [33], Sips [41], Meixner & Schäfke [34] and Wang and Zhang [45] (cf. also [44]).…”

Section: Introductionmentioning

confidence: 87%

Highly Efficient and Accurate Spectral Approximation of the Angular Mathieu Equation for any Parameter Values q

Shen¹

2018

JMS

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The eigenpairs of the angular Mathieu equation under the periodicity condition are accurately approximated by the Jacobi polynomials in a spectral-Galerkin scheme for small and moderate values of the parameter q. On the other hand, the periodic Mathieu functions are related with the spheroidal functions of order ±1/2. It is well-known that for very large values of the bandwidth parameter, spheroidal functions can be accurately approximated by the Hermite or Laguerre functions scaled by the square root of the bandwidth parameter. This led us to employ the Laguerre polynomials in a pseudospectral manner to approximate the periodic Mathieu functions and the corresponding characteristic values for very large values of q.

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Section: Introductionmentioning

confidence: 87%

Highly Efficient and Accurate Spectral Approximation of the Angular Mathieu Equation for any Parameter Values q

Shen¹

2018

JMS

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“…As recommended in [30,32], in practice, a quite safe choice is c = N 2 . With this in mind, we choose c = N 2 .…”

Section: Preliminariesmentioning

confidence: 99%

“…As we can see, the convergence is exponential and is almost indistinguishable for different c. Moreover, it is shown that the optimal c depends on the function being approximated [7]. In the following, we will take c = n/2 for general functions, which is recommended in [30,32]. 3 and 4.…”

Section: Numerical Testsmentioning

confidence: 99%

Barycentric prolate interpolation and pseudospectral differentiation

Tian

2021

Numer Algor

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In this paper, we provide further illustrations of prolate interpolation and pseudospectral differentiation based on the barycentric perspectives. The convergence rates of the barycentric prolate interpolation and pseudospectral differentiation are derived. Furthermore, we propose the new preconditioner, which leads to the well-conditioned prolate collocation scheme. Numerical examples are included to show the high accuracy of the new method. We apply this approach to solve the second-order boundary value problem and Helmholtz problem.

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“…To the best of our knowledge, there are few reports for the spectral method of fourth‐order problems in circular domains. The main reason is that the polar transformation introduces singularity at the pole and the variable coefficients of the form r ± k [18, 19, 20], which brings some difficulties to implementation of the algorithm and error analysis.…”

Section: Introductionmentioning

confidence: 99%

An efficient spectral method and rigorous error analysis based on dimension reduction scheme for fourth order problems

2020

Numerical Methods Partial

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In this paper, we propose an effective spectral method based on dimension reduction scheme for fourth order problems in polar geometric domains. First, the original problem is decomposed into a series of one-dimensional fourth order problems by polar coordinate transformation and the orthogonal properties of Fourier basis function. Then the weak form and the corresponding discrete scheme of each one-dimensional fourth order problem are derived by introducing polar conditions and appropriate weighted Sobolev spaces. In addition, we define the projection operators in the weighted Sobolev space and give its approximation properties, and further prove the error estimation of each one-dimensional fourth order problem. Finally, we provide some numerical examples, and the numerical results show the effectiveness of our algorithm and the correctness of the theoretical results.

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A Review of Prolate Spheroidal Wave Functions from the Perspective of Spectral Methods

Cited by 40 publications

References 90 publications

Highly Efficient and Accurate Spectral Approximation of the Angular Mathieu Equation for any Parameter Values q

Highly Efficient and Accurate Spectral Approximation of the Angular Mathieu Equation for any Parameter Values q

Barycentric prolate interpolation and pseudospectral differentiation

An efficient spectral method and rigorous error analysis based on dimension reduction scheme for fourth order problems

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