On approximate inverse of Hermite and Laguerre collocation differentiation matrices and new collocation schemes in unbounded domains

Zhang, Chao; Wang, Li-Lian; Gu, Dong-qin; Liu, Wenjie

doi:10.1016/j.cam.2018.05.061

Cited by 3 publications

(7 citation statements)

References 37 publications

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“…Let {S j (x)} N j=0 be the Laguerre-Birkhoff interpolation basis functions (see [34]), which is based on the Laguerre function L (α,β) k (x), and write…”

Section: Collocation Methods Using Laguerre-birkhoff Interpolation Basismentioning

confidence: 99%

“…Let {S j (x)} N j=0 be the Laguerre-Birkhoff interpolation basis functions (see [34]) and expand the numerical solution as…”

Section: Rational Collocation Methods Using Rational-lagrange Basismentioning

confidence: 99%

“…and {y i } N i=0 be the LGR points as before. The collocation scheme for (2.48) is to find u N ∈ R N (Λ) (or u N ∈ S N (Λ) which can be seen in [34]) such that…”

Section: One-dimensional Elliptic Equationmentioning

confidence: 99%

“…Effective attempts are available, such as low-order finite difference or finite elements [4,6,10,11,22,23], and preconditioning differentiation by integration [7-9, 12, 13, 16, 20, 28, 29, 31]. In addition, suitable basis functions can be built from Birkhoff interpolation, and usually provide good appropriate preconditioners for usual collocation schemes, see [18,28,33,34].…”

Section: Introductionmentioning

confidence: 99%

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New Rational Interpolation Basis Functions on the Unbounded Intervals and Their Applications

Gu¹,

null²,

Zhang³

2023

NMTMA

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Based on the rational system of Legendre rational functions, we construct two set of new interpolation basis functions on the unbounded intervals. Their explicit expressions are derived, and fast and stable algorithms are provided for computing the new basis functions. As applications, new rational collocation methods based on these new basis functions are proposed for solving various second-order differential equations on the unbounded domains. Numerical experiments illustrate that our new methods are more effective and stable than the existing collocation methods.

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“…Let {S j (x)} N j=0 be the Laguerre-Birkhoff interpolation basis functions (see [34]), which is based on the Laguerre function L (α,β) k (x), and write…”

Section: Collocation Methods Using Laguerre-birkhoff Interpolation Basismentioning

confidence: 99%

“…Let {S j (x)} N j=0 be the Laguerre-Birkhoff interpolation basis functions (see [34]) and expand the numerical solution as…”

Section: Rational Collocation Methods Using Rational-lagrange Basismentioning

confidence: 99%

“…and {y i } N i=0 be the LGR points as before. The collocation scheme for (2.48) is to find u N ∈ R N (Λ) (or u N ∈ S N (Λ) which can be seen in [34]) such that…”

Section: One-dimensional Elliptic Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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New Rational Interpolation Basis Functions on the Unbounded Intervals and Their Applications

Gu¹,

null²,

Zhang³

2023

NMTMA

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show abstract

“…It is noteworthy that the main idea for such a construction is inspired from the design of well-conditioned collocation methods using the notion of the Birkhoff interpolation (cf. [7,19,32,48,56,57]). Here, we also estimate the interpolation error of this type of the generalised Birkhoff interpolation and demonstrate that the new approach can provide very accurate approximation to a class of time-fractional PDEs with smooth source terms.…”

Section: Introductionmentioning

confidence: 99%

A New Spectral Method Using Nonstandard Singular Basis Functions for Time-Fractional Differential Equations

Liu

2019

Commun. Appl. Math. Comput.

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In this paper, we introduce new non-polynomial basis functions for spectral approximation of time-fractional partial differential equations (PDEs). Different from many other approaches, the nonstandard singular basis functions are defined from some generalised Birkhoff interpolation problems through explicit inversion of some prototypical fractional initial value problem (FIVP) with a smooth source term. As such, the singularity of the new basis can be tailored to that of the singular solutions to a class of time-fractional PDEs, leading to spectrally accurate approximation. It also provides the acceptable solution to more general singular problems.

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Well-Conditioned Spectral Collocation Methods for Problems in Unbounded Domains

sci¹

2020

NMTMA

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On approximate inverse of Hermite and Laguerre collocation differentiation matrices and new collocation schemes in unbounded domains

Cited by 3 publications

References 37 publications

New Rational Interpolation Basis Functions on the Unbounded Intervals and Their Applications

New Rational Interpolation Basis Functions on the Unbounded Intervals and Their Applications

A New Spectral Method Using Nonstandard Singular Basis Functions for Time-Fractional Differential Equations

Well-Conditioned Spectral Collocation Methods for Problems in Unbounded Domains

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