Orthogonal Jacobi Rational Functions and Spectral Methods on the Half Line

Gu, Dong-qin; Wang, Zhongqing

doi:10.1007/s10915-021-01535-7

Cited by 7 publications

(5 citation statements)

References 22 publications

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“…Gu and Wang [17] introduced another family of Legendre rational basis functions on the half line, i.e., Rn (x) := (x + 1) −1 L n…”

Section: Discussion On Another Rational Interpolation Basis Functionsmentioning

confidence: 99%

“…Unfortunately, the approximation effects for algebraic decay functions are not good using this method, which may greatly limit the practical applications. In order to combine the respective advantages of the two methods, Gu and Wang [17] introduced a new family of Jacobi rational functions R (γ,α) n (x), which is mutually orthogonal with respect to the weight function x α on the half line, and its weight function is the same as that of the Laguerre functions. Using the new Jacobi rational basis functions, the symmetry and conservation of the schemes can be guaranteed by selecting the parameter α.…”

Section: Introductionmentioning

confidence: 99%

“…The purpose of this paper is to extend the work of [17,27,28], in order to obtain more suitable rational collocation methods based on Birkhoff interpolation for solving differential equations whose solutions decay at the infinity. To this end, we construct new interpolation basis functions based on Legendre rational basis functions on the unbounded intervals by Birkhoff interpolation problems.…”

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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“…Gu and Wang [17] introduced another family of Legendre rational basis functions on the half line, i.e., Rn (x) := (x + 1) −1 L n…”

Section: Discussion On Another Rational Interpolation Basis Functionsmentioning

confidence: 99%

Section: Introductionmentioning

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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“…The Falkner-Skan equation has been solved by using a variety of techniques, like the homotopy perturbation method [54], the homotopy analysis method [55], the Adomian decomposition method [56], the differential transformation method [57], the iterative transformation method [58], the Legendre rational polynomials method [59], the shifted Chebyshev collocation method [60], and the modified rational Bernoulli functions [61].…”

Section: The Falkner-skan Equationmentioning

confidence: 99%

Novel Approximate Solutions for Nonlinear Initial and Boundary Value Problems

Mahdi Salih,

A. AL-Jawary,

Mustafa Turkyilmazoglu

2023

IHJPAS

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This paper investigates an effective computational method (ECM) based on the standard polynomials used to solve some nonlinear initial and boundary value problems appeared in engineering and applied sciences. Moreover, the effective computational methods in this paper were improved by suitable orthogonal base functions, especially the Chebyshev, Bernoulli, and Laguerre polynomials, to obtain novel approximate solutions for some nonlinear problems. These base functions enable the nonlinear problem to be effectively converted into a nonlinear algebraic system of equations, which are then solved using Mathematica®12. The improved effective computational methods (I-ECMs) have been implemented to solve three applications involving nonlinear initial and boundary value problems: the Darcy-Brinkman-Forchheimer equation, the Blasius equation, and the Falkner-Skan equation, and a comparison between the proposed methods has been presented. Furthermore, the maximum error remainder () has been computed to prove the proposed methods' accuracy. The results convincingly prove that ECM and I-ECMs are effective and accurate in obtaining novel approximate solutions to the problems.

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“…We remark that alternative methods exist for high-order simulations on unbounded domains, but available methodologies are limited to the linear case without like-for-like efficiency comparisons, 26 concern elliptic problems, 27 or lack formal stability analyses. 28 Recent studies 20,29 introduced adaptive methods based on time-evolving distributions of Laguerre and Hermite collocation points for simulations on unbounded domains and applied them within spectrally adaptive neural network models.…”

Section: Introductionmentioning

confidence: 99%

Efficient hyperbolic–parabolic models on multi‐dimensional unbounded domains using an extended DG approach

Vismara,

Benacchio

2023

Numerical Methods in Fluids

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We introduce an extended discontinuous Galerkin discretization of hyperbolic–parabolic problems on multidimensional semi‐infinite domains. Building on previous work on the one‐dimensional case, we split the strip‐shaped computational domain into a bounded region, discretized by means of discontinuous finite elements using Legendre basis functions, and an unbounded subdomain, where scaled Laguerre functions are used as a basis. Numerical fluxes at the interface allow for a seamless coupling of the two regions. The resulting coupling strategy is shown to produce accurate numerical solutions in tests on both linear and nonlinear scalar and vectorial model problems. In addition, an efficient absorbing layer can be simulated in the semi‐infinite part of the domain in order to damp outgoing signals with negligible spurious reflections at the interface. By tuning the scaling parameter of the Laguerre basis functions, the extended DG scheme simulates transient dynamics over large spatial scales with a substantial reduction in computational cost at a given accuracy level compared to standard single‐domain discontinuous finite element techniques.

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Orthogonal Jacobi Rational Functions and Spectral Methods on the Half Line

Cited by 7 publications

References 22 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

Novel Approximate Solutions for Nonlinear Initial and Boundary Value Problems

Efficient hyperbolic–parabolic models on multi‐dimensional unbounded domains using an extended DG approach

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