Optimal Collocation Nodes for Fractional Derivative Operators

Fatone, Lorella; Funaro, Daniele

doi:10.1137/140993697

Cited by 10 publications

(6 citation statements)

References 47 publications

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“…For this purpose, we take 17) and λ 1 , λ 2 to be the same as in (6.12). In Figure 6.3, we plot discrete L 2 -errors for various pairs of (µ, ν) of the B-COL schemes for both Caputo and Riemann-Liouville fractional boundary value problems (BVPs) (6.9) and (6.14) under the same setting.…”

Section: Well-conditioned Collocation Schemes and Numerical Resultsmentioning

confidence: 99%

“…For α, β > −1, the classical Jacobi polynomials are orthogonal with respect to the Jacobi weight function: 17) where δ nn ′ is the Dirac Delta symbol, and…”

Section: )mentioning

confidence: 99%

“…When it comes to FDEs, it is advantageous to use collocation methods, as the Galerkin approaches usually lead to full dense matrices as well. Recently, the development of collocation methods for FDEs has attracted much attention (see, e.g., [28,50,43,17]). It was numerically testified in [28,50] that for both Lagrange polynomial-based and JPF-based collocation methods, the condition number of the Caputo F-PSDM of order µ behaves like O(N 2µ ) which is consistent with the integer-order case.…”

Section: Introductionmentioning

confidence: 99%

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Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis

Jiao

Wang

Huang

2016

Journal of Computational Physics

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Abstract. The purpose of this paper is twofold. Firstly, we provide explicit and compact formulas for computing both Caputo and (modified) Riemann-Liouville (RL) fractional pseudospectral differentiation matrices (F-PSDMs) of any order at general Jacobi-Gauss-Lobatto (JGL) points. We show that in the Caputo case, it suffices to compute F-PSDM of order µ ∈ (0, 1) to compute that of any order k + µ with integer k ≥ 0, while in the modified RL case, it is only necessary to evaluate a fractional integral matrix of order µ ∈ (0, 1). Secondly, we introduce suitable fractional JGL Birkhoff interpolation problems leading to new interpolation polynomial basis functions with remarkable properties: (i) the matrix generated from the new basis yields the exact inverse of F-PSDM at "interior" JGL points; (ii) the matrix of the highest fractional derivative in a collocation scheme under the new basis is diagonal; and (iii) the resulted linear system is well-conditioned in the Caputo case, while in the modified RL case, the eigenvalues of the coefficient matrix are highly concentrated. In both cases, the linear systems of the collocation schemes using the new basis can solved by an iterative solver within a few iterations. Notably, the inverse can be computed in a very stable manner, so this offers optimal preconditioners for usual fractional collocation methods for fractional differential equations (FDEs). It is also noteworthy that the choice of certain special JGL points with parameters related to the order of the equations can ease the implementation. We highlight that the use of the Bateman's fractional integral formulas and fast transforms between Jacobi polynomials with different parameters, are essential for our algorithm development.

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Section: Well-conditioned Collocation Schemes and Numerical Resultsmentioning

confidence: 99%

“…For α, β > −1, the classical Jacobi polynomials are orthogonal with respect to the Jacobi weight function: 17) where δ nn ′ is the Dirac Delta symbol, and…”

Section: )mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis

Jiao

Wang

Huang

2016

Journal of Computational Physics

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“…Introduction. Fractional spectral collocation (FSC) methods [7,8,2] based on fractional Lagrange interpolation have recently been proposed to solve fractional differential equations. By a spectral theory developed in [6] for fractional Sturm-Liouville eigenproblems, the corresponding fractional differential matrices can be obtained with ease.…”

mentioning

confidence: 99%

“…In the Riemann-Liouville case, it is necessary to modify the fractional derivative operator in order to absorb singular fractional factors (see [3, §3]). In this paper, we extend the Birkhoff interpolation preconditioning techniques in [5,3] to the fractional spectral collocation methods [7,8,2] based on fractional Lagrange interpolation. Unlike that in [3], there are no singular fractional factors in the Riemann-Liouville case.…”

mentioning

confidence: 99%

Preconditioning fractional spectral collocation

Du¹

2015

Preprint

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Fractional spectral collocation (FSC) method based on fractional Lagrange interpolation has recently been proposed to solve fractional differential equations. Numerical experiments show that the linear systems in FSC become extremely ill-conditioned as the number of collocation points increases. By introducing suitable fractional Birkhoff interpolation problems, we present fractional integration preconditioning matrices for the ill-conditioned linear systems in FSC. The condition numbers of the resulting linear systems are independent of the number of collocation points. Numerical examples are given.

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Superconvergence Points for the Spectral Interpolation of Riesz Fractional Derivatives

2019

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In this paper, superconvergence points are located for the approximation of the Riesz derivative of order α using classical Lobatto-type polynomials when α ∈ (0, 1) and generalized Jacobi functions (GJF) for arbitrary α > 0, respectively. For the former, superconvergence points are zeros of the Riesz fractional derivative of the leading term in the truncated Legendre-Lobatto expansion. It is observed that the convergence rate for different α at the superconvergence points is at least O(N −2 ) better than the optimal global convergence rate. Furthermore, the interpolation is generalized to the Riesz derivative of order α > 1 with the help of GJF, which deal well with the singularities. The well-posedness, convergence and superconvergence properties are theoretically analyzed. The gain of the convergence rate at the superconvergence points is analyzed to be O(N −(α+3)/2 ) for α ∈ (0, 1) and O(N −2 ) for α > 1. Finally, we apply our findings in solving model FDEs and observe that the convergence rates are indeed much better at the predicted superconvergence points.

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Optimal Collocation Nodes for Fractional Derivative Operators

Cited by 10 publications

References 47 publications

Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis

Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis

Preconditioning fractional spectral collocation

Superconvergence Points for the Spectral Interpolation of Riesz Fractional Derivatives

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