Pith. sign in

REVIEW 2 major objections 4 minor 52 references

Fast Fourier-like Mapped Chebyshev Spectral-Galerkin Methods for PDEs with Integral Fractional Laplacian in Unbounded Domains

T0 review · 2 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The integral fractional Laplacian on $\mathbb{R}^d$ is fully diagonalized by Fourier-like mapped Chebyshev functions, giving an $O((N\log_2 N)^d)$ spectral-Galerkin solve.

desk verdict Genuinely neat diagonalization result; the abstract's quasi-optimal complexity is not supported by the algorithm as described. read the letter →

arxiv 1908.10029 v1 pith:ZIVQJOLV submitted 2019-08-27 math.NA cs.NA

Changtao Sheng , Jie Shen , Tao Tang , Li-Lian Wang , Huifang Yuan This is my paper · ORCID
classification math.NAcs.NA MSC 65N3565M7041A0541A25
keywords IntegralfractionalLaplacianDunford-TaylorformulaMappedChebyshevfunctionsbi-orthogonalbasisspectral-GalerkinmethodunboundeddomainsnonlinearSchrödingerequationnonlocalsingularoperators
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper is trying to establish that PDEs driven by the integral fractional Laplacian on $\mathbb{R}^d$ can be solved by a spectral-Galerkin method whose operator assembly is exact and whose cost is near-optimal. The key idea is to combine the Dunford-Taylor integral representation of the fractional Laplacian with a purpose-built basis: the tensorial Fourier-like bi-orthogonal mapped Chebyshev functions. Under this basis the resolvent of the ordinary Laplacian is diagonal, the auxiliary integration in $t$ is done in closed form, and the fractional Laplacian itself becomes a diagonal operator. If correct, elliptic fractional problems in any dimension are reduced to fast Chebyshev transforms, and the same diagonal structure makes time-splitting schemes for fractional nonlinear Schrödinger equations immediate. The paper backs this with convergence estimates and numerical tests in one, two, and three dimensions.

What carries the argument

The machinery is the pair consisting of the Dunford-Taylor formula and the Fourier-like bi-orthogonal mapped Chebyshev functions (MCFs). The Dunford-Taylor formula writes the fractional Laplacian's energy as $C_s\int_0^\infty t^{1-2s}((-\Delta)(I-t^2\Delta)^{-1}u,v)\,dt$, turning the nonlocal operator into a weighted integral of ordinary resolvents. The MCFs are obtained by diagonalizing the one-dimensional stiffness matrix of mapped Chebyshev functions, so the resulting basis is orthonormal in $L^2(\mathbb{R})$ and diagonal in $H^1(\mathbb{R})$; in $d$ dimensions the tensor-product basis has gradient Gram matrix $|\lambda_p|_1=\lambda_{p_1}+\cdots+\lambda_{p_d}$. Substituting this basis into the Dunford-Taylor integral produces a rational expression in $|\lambda_p|_1$ whose $t$-integral is exactly $\pi/(2\sin\pi s)$, the inverse of $C_s$. This cancellation is the mechanism that diagonalizes the fractional Laplacian.

What would settle it

Run the MCF scheme in one dimension with $s=0.9$ and the exact solution $u=(1+x^2)^{-2.3}$ at $N=200$ and $N=400$, while solving the inner elliptic problem (3.19) at a fixed coarse tolerance; if the $H^s$ error stops improving once that tolerance is reached instead of following the predicted $O(N^{s-m})$ rate, then the exact-solve assumption in Proposition 4.1 is the active limitation and the convergence claim is conditional on it.

Watch

Extended reading notes

Core claim

The central discovery is stated as Theorem 3.2. With the tensorial Fourier-like MCF basis $\{\hat T_p\}_{p\in\Upsilon_N}$, the discrete solution of $(-\Delta)^s u+\gamma u=f$ in $\mathbb{R}^d$ has the explicit expansion $$u_N(x)=\sum_{p\in\Upsilon_N}\frac{\hat f_p}{\gamma+|\lambda_p|_1^s}\,\hat T_p(x),$$ where $\hat f_p$ is the $L^2$ inner product of the interpolated source with $\hat T_p$ and $|\lambda_p|_1$ is the sum of the one-dimensional Laplace eigenvalues attached to $p$. The proof is constructive: the auxiliary function $w_N=(I-t^2\Delta)^{-1}u_N$ has coefficients $\hat u_p/(1+t^2|\lambda_p|_1)$, and the $t$-integral in the Dunford-Taylor formula evaluates exactly to the reciprocal of the constant $C_s=2\sin(\pi s)/\pi$. Hence no quadrature in $t$ is needed and the stiffness matrix is the diagonal operator with entries $\gamma+|\lambda_p|_1^s$.

Load-bearing premise

The load-bearing premise is that the auxiliary elliptic problem (3.19) is solved exactly; the paper states that the size of that error is largely open in $\mathbb{R}^d$ because no usable spectrum estimate for the integral fractional Laplacian is available, so if the inner solve contributes noticeable error, the promised convergence order is not established.

Editorial extensions

If this is right

  • The elliptic solve for $(-\Delta)^s u+\gamma u=f$ requires no linear solver: the coefficients are obtained by division, and the dominant cost is the FFT-based MCF transform, giving $O((N\log_2 N)^d)$ complexity.
  • Multi-term fractional equations $\sum_j\rho_j(-\Delta)^{s_j}u=f$ are solved with the same diagonal matrix, with $\gamma+|\lambda_p|_1^s$ replaced by $\sum_j\rho_j|\lambda_p|_1^{s_j}$.
  • For the fractional nonlinear Schrödinger equation, the linear half-step of the time splitter is multiplication by $e^{-\frac{i}{2}|\lambda_k|_1^s\Delta t}$ in coefficient space, so the fourth-order splitting scheme inherits exact mass conservation.
  • The expected convergence for algebraically decaying solutions is algebraic, with orders set by the decay rate and dimension; the reported tests in $d=1,2,3$ match those predicted orders.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The diagonalization argument rests on one scalar integral, so the same construction should extend to operators whose symbol is $a(|\xi|)|\xi|^{2s}$ for smooth radial $a$; the paper notes the particular extension to $(\gamma I-\Delta)^s$ but does not develop it numerically.
  • A practical test of the exact-solve assumption is to vary the tolerance used for the inner elliptic problem (3.19) and watch whether the $H^s$ error stalls at that tolerance; this would show whether Proposition 4.1's convergence order is contingent on the inner solve.
  • Because the diagonal entries depend only on $|\lambda_p|_1$, hyperbolic-cross or sparse-grid versions of the MCF basis should retain the diagonalization while using fewer degrees of freedom in high dimensions; the paper mentions this possibility in its concluding remarks without analysis.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper develops a spectral-Galerkin method for elliptic PDEs with the integral fractional Laplacian on R^d. The method combines the Dunford-Taylor representation of the fractional Laplacian with mapped Chebyshev functions that are re-orthogonalized, via an eigenvector matrix E, into a 'Fourier-like' basis. The main theoretical result, Theorem 3.2, shows that in this basis the discrete Dunford-Taylor formulation (3.18)-(3.19) is diagonal, with the exact t-integration reducing to a standard Beta-function identity and yielding the solution coefficients u_p = f_p/(γ+|λ_p|_1^s). The paper also proves mapped-Chebyshev approximation and interpolation estimates, gives a conditional a priori error estimate, derives exact fractional-Laplacian formulas for Gaussian and rational test functions, and applies the solver to the fractional nonlinear Schrödinger equation with a fourth-order time-splitting method. Numerical experiments in one, two, and three dimensions show algebraic convergence rates consistent with the stated estimates.

Significance. The central diagonalization idea is elegant and, as presented in Theorem 3.2, the derivation is self-contained and correct: bi-orthogonality (3.23) plus the exact integral (3.29) rigorously yields the closed-form solution (3.25). The analytic formulas in Propositions 4.2 and 4.3 for the fractional Laplacian of Gaussian and rational functions are useful benchmarking tools, and the numerical tests are extensive. If the complexity and error-analysis claims were fully established, the method would be a valuable contribution to spectral methods for fractional PDEs on unbounded domains. However, the quasi-optimal complexity claim in the abstract and Remark 3.2 is not supported by the described algorithm, and the a priori error estimate in Proposition 4.1 is explicitly conditional on an assumption whose analysis the authors state is largely open. These two points affect the paper's central claims.

major comments (2)
  1. [§2.2, Eq. (2.11); §3.3, Remark 3.2; abstract] The claimed quasi-optimal complexity O((N log_2 N)^d) is not justified by the algorithm as written. The Fourier-like basis is defined in (2.11) as \widehat{T}_p = \sum_j e_{jp} T_j, where E is the dense eigenvector matrix from (2.10). Computing the coefficients \widehat{f}_p in (3.26) therefore requires applying E^T to the vector of Chebyshev coefficients, and no fast structure for E (Toeplitz, low displacement rank, or similar) is established or cited. For a d-dimensional tensor grid, applying E^T along each dimension costs O(N^{d+1}) operations, not O((N log_2 N)^d); even in one dimension the transform costs O(N^2). The same dense transform is required in every application of the operator T_N in the time-splitting scheme (5.12)-(5.13). In addition, forming E itself is a nontrivial O(N^2) per-dimension computation that is not counted. Thus the central 'fast' and 'quasi-optimal' claims are unsupported by the described algorithm.
  2. [§4.1, before Proposition 4.1] The a priori error estimate (4.21) assumes that the elliptic subproblem (3.19) is solved exactly, i.e., that w_N equals the continuous solution (I - t^2Δ)^{-1} u_N. The paper explicitly states that the analysis of this error is 'largely open' because a spectrum estimate for the fractional Laplacian on R^d is unavailable. Since the actual scheme uses the V_N-based solution of (3.19), the consistency error between the discrete bilinear form B_N in (3.18) and the continuous form B is not controlled, so the stated convergence rate in Proposition 4.1 is not proved for the computed method. The numerical experiments in §4.3 are encouraging but do not replace the missing analysis.
minor comments (4)
  1. [Remark 2.1] The text contains the typo 'MFCs' where 'MCFs' is meant; please correct it.
  2. [Eqs. (3.20)-(3.23)] The bold-index notation for \widehat{T}_n and λ_n is introduced, but in (3.21)-(3.23) the subscripts are not consistently printed in bold; please unify the notation so that p,q,n are clearly multi-indices.
  3. [§5.1, Eq. (5.12)] The map T_N[ω;Ψ_p] is defined using Ψ_p on the left and Ψ on the right; the notation is confusing because p is used both as a grid index and as the argument of the coefficient map. Please clarify.
  4. [Figure 4.2(b)] The caption says 'Hermite approach' while the text refers to the 'Hermite spectral method in [35]'; please make the terminology consistent.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the diagonalization of the fractional Laplacian is a direct spectral calculation from an explicitly constructed eigenbasis; the acknowledged exact-solve assumption and the disputed complexity claim are correctness concerns, not circular reasoning.

full rationale

The central claim, Theorem 3.2 with formula (3.25), is derived from three independent ingredients: the bi-orthogonality of the Fourier-like MCF basis (Lemma 2.1 and Theorem 3.1), the Dunford--Taylor representation of the fractional Laplacian (Lemma 3.2, attributed to Bonito--Lei--Pasciak), and the closed-form t-integral identity (3.29), which is evaluated by a standard Beta/Gamma formula. The basis is not defined in terms of the fractional Laplacian: it is obtained from the eigenvectors of the H1 stiffness matrix S of the standard Laplacian in (2.10), and the fractional operator enters only through the known Dunford--Taylor identity. Consequently, the diagonal coefficient formula is a proved algebraic consequence rather than a restatement of an input. No parameter is fitted to data, and no subset of solutions is used to infer the formula. The error estimates in Proposition 4.1 cite the authors' own works [45] and [43], but those are externally published, parameter-free approximation results whose assumptions (B^m regularity) do not include the target diagonalization formula; they therefore provide genuine support rather than circularity. The paper itself explicitly flags a limitation before Proposition 4.1: 'we assume that the error for solving the elliptic problem (3.19) is negligible' and notes the analysis 'is largely open in this situation, mainly because the spectrum estimate of the fractional Laplacian operator in R^d appears unavailable.' This is an admitted gap in the convergence proof, not a circular step. Likewise, the concern that Remark 3.2's O((N log_2 N)^d) cost claim omits the dense eigenvector transform needed to pass from Chebyshev coefficients to Fourier-like coefficients is an unverified complexity assertion, not a derivation whose conclusion equals its premise. The diagonalization mathematics itself is self-contained, so the circularity score is 0.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central diagonalization rests on the Dunford-Taylor identity and the Beta integral, both quoted from the literature. Error estimates rest on previously established MCF approximation theorems and on an explicitly unproved exact-solve assumption for the auxiliary elliptic problem. No new physical entities are introduced.

free parameters (1)
  • scaling factor ν in the mapped Chebyshev functions = ν = 2.5 (default), ν = 1 or 4 in Example 4.2
    The MCF basis includes a scaling/map parameter ν chosen by hand in numerical experiments. It affects accuracy but not the asymptotic convergence rates or the diagonalization claim, and it is not part of the theoretical error bounds.
assumptions (5)
  • domain assumption Dunford-Taylor formulation: ((-Δ)^{s/2}u, (-Δ)^{s/2}v) = C_s ∫_0∞ t^{1-2s} ((-Δ)(I - t^2 Δ)^{-1}u, v) dt
    Invoked as Lemma 3.2 and quoted from [11, Theorem 4.1]; not proved in this paper and it is the starting point of the algorithm.
  • domain assumption MCF bi-orthogonality and projection/interpolation error estimates from [45] and [43]
    The Fourier-like basis is built from the eigen-decomposition of the MCF stiffness matrix, and the L2/H1 projection and interpolation bounds used in Theorems 4.1-4.2 are cited from prior work by the same group.
  • standard math Sobolev space characterization and interpolation inequality (3.6)
    Used in Theorem 4.1 to interpolate between L2 and H1 projection errors; taken from [40] and [3].
  • standard math Integral identity ∫_0∞ x^{μ-1}/(1+x^ν) dx = π/(ν sin(πμ/ν))
    Used as identity (3.30) from Gradshteyn-Ryzhik; the exact t-integration in Theorem 3.2 depends on it.
  • standard math Asymptotic formulas for the hypergeometric functions 1F1 and 2F1
    Needed for Propositions 4.2-4.3 and the predicted algebraic convergence rates; taken from [8].

how reviews work

0 comments
Cite this review

Pith. "Pith review of Fast Fourier-like Mapped Chebyshev Spectral-Galerkin Methods for PDEs with Integral Fractional Laplacian in Unbounded Domains." pith.science (2026). https://pith.science/paper/ZIVQJOLV

@misc{pith2026190810029,
  author       = {Pith},
  title        = {Pith review of: Fast Fourier-like Mapped Chebyshev Spectral-Galerkin Methods for PDEs with Integral Fractional Laplacian in Unbounded Domains},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZIVQJOLV}},
  note         = {Machine review of arXiv:1908.10029}
}
abstract

In this paper, we propose a fast spectral-Galerkin method for solving PDEs involving integral fractional Laplacian in $\mathbb{R}^d$, which is built upon two essential components: (i) the Dunford-Taylor formulation of the fractional Laplacian; and (ii) Fourier-like bi-orthogonal mapped Chebyshev functions (MCFs) as basis functions. As a result, the fractional Laplacian can be fully diagonalised, and the complexity of solving an elliptic fractional PDE is quasi-optimal, i.e., $O((N\log_2N)^d)$ with $N$ being the number of modes in each spatial direction. Ample numerical tests for various decaying exact solutions show that the convergence of the fast solver perfectly matches the order of theoretical error estimates. With a suitable time-discretization, the fast solver can be directly applied to a large class of nonlinear fractional PDEs. As an example, we solve the fractional nonlinear Schr{\"o}dinger equation by using the fourth-order time-splitting method together with the proposed MCF-spectral-Galerkin method.

Figures

Figures reproduced from arXiv: 1908.10029 by the authors.

Figure 4.1
Figure 4.1. Decay of Hs -errors of the MCF scheme with γ = 1 and the scaling factor ν = 2.5 for Example 4.1 with exact solutions in (4.27). Here s = 0.3, 0.7 and r = 2.3. The dashed reference lines are expected orders predicted by Proposition 4.1. We now turn to the second case with the exact solution ua(x) = (1 + |x| 2 ) −r , where we take r = 2.3 in the numerical tests. As r > d/2, we derive from Proposition 4.3 that fa(x) ∼ … view at source ↗
Figure 4.2
Figure 4.2. (a). Maximum error for the exact solution u(x) = (1 + x 2 ) −2.3 with different scaling factor ν, and s = 0.3, 0.7; (b). A comparison of maximum error between our method and Hermite-Galerkin method [35] for exact solution u(x) = (1 + x 2 ) −2.3 , with s = 0.3, 0.7, ν = 1. Example 4.2. (Effect of the scaling factor). In this example, we first show the influence of the scaling factor ν to the accuracy. It is known tha… view at source ↗
Figure 4
Figure 4. (a)-(c), we compare the [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figures from the paper (4 more)
Figure 4.3
Figure 4.3. Figure 4.3: (a)-(c): A comparison of L 2 -errors between our method and Hermite-Galerkin method in [35] for different source function f(x); (d)-(f): The maximum errors for different source function f(x) with d = 2, 3. In the tests, we take and γ = 1, ν = 2.5. In [PITH_FULL_IMAG…
Figure 4.4
Figure 4.4. Figure 4.4: (a). The maximum error for (4.35) with u(x) = (1 + |x| 2 ) − 3π 4 and ν = 2.5; (b). The maximum error for (4.35) with f(x) = (1 + x1 + 2x 2 2 + 3x 2 3)e − |x| 2 2 and ν = 2.5. 5. MCF approximation of nonlinear fractional Schrodinger equations ¨ In this section, we ap…
Figure 5.1
Figure 5.1. Figure 5.1: (a). The numerical error of (5.17) with s = 0.7, γ = −1, T = 2; (b). The numerical error of (5.17) with s = 0.7, γ = −1, T = 2. (a) T = 1, s = 0.3 and γ = 1 (b) T = 2, s = 0.3 and γ = 1 (c) T = 1, s = 0.3 and γ = −1 (d) T = 1, s = 0.7 and γ = −1 [PITH_FULL_IMAGE:fig…
Figure 5.2
Figure 5.2. Figure 5.2: Profiles of the modulus square of the numerical solutions at different time and with different fractional orders. 5.3. Concluding Remarks. We developed a fast MCF-spectral-Galerkin method for PDEs involving integral fractional Laplacian in R d . The fast solver is in…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

52 extracted references · 51 canonical work pages

  1. [50]

    Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains

    T. Tang, L.-L. Wang, H. Yuan, and T. Zhou, Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains, arXiv preprint arXiv:1905.02476, (2019)

    work page Pith review arXiv 2019

  2. [1]

    Acosta, F

    G. Acosta, F. M. Bersetche, and J. P. Borthagaray, A short FE implementation for a 2d homogeneous Dirichlet problem of a fractional Laplacian , Comput. Math. Appl., 74 (2017), pp. 784–816

    work page 2017

  3. [2]

    Acosta and J

    G. Acosta and J. P. Borthagaray, A fractional Laplace equation: regularity of solutions and finite element approx- imations, SIAM J. Numer. Anal., 55 (2017), pp. 472–495

    work page 2017

  4. [3]

    Agranovich, Sobolev spaces, their generalizations and elliptic problems in smooth and Lipschitz domains, Springer, 2015

    M. Agranovich, Sobolev spaces, their generalizations and elliptic problems in smooth and Lipschitz domains, Springer, 2015

    work page 2015

  5. [4]

    Ainsworth and C

    M. Ainsworth and C. Glusa, Aspects of an adaptive finite element method for the fractional laplacian: a priori and a posteriori error estimates, efficient implementation and multigrid solver , Comput. Methods Appl. Mech. Engrg., 327 (2017), pp. 4–35

    work page 2017

  6. [5]

    , Hybrid finite element–spectral method for the fractional Laplacian: approximation theory and efficient solver , SIAM J. Sci. Comput., 40 (2018), pp. A2383–A2405

    work page 2018

  7. [6]

    I. Babuska, Survey lectures on the mathematical foundations of the finite element method , The Mathematical Foun- dations of the Finite Element Method with Applicaions to Partial Differential Equations, (1972), pp. 3–359

    work page 1972

  8. [7]

    Bao and J

    W. Bao and J. Shen , A fourth-order time-splitting Laguerre–Hermite pseudospectral method for Bose–Einstein con- densates, SIAM J. Sci. Comput., 26 (2005), pp. 2010–2028

    work page 2005

Show all 52 references
  1. [8]

    Bateman, Higher transcendental functions [volumes i-iii] , 1953

    H. Bateman, Higher transcendental functions [volumes i-iii] , 1953

    1953

  2. [9]

    D. A. Benson, S. W. Wheatcraft, and M. M. Meerschaert , Application of a fractional advection-dispersion equation, Water Resour. Res., 36 (2000), pp. 1403–1412

    2000

  3. [10]

    Bonito, J

    A. Bonito, J. P. Borthagaray, R. H. Nochetto, E. Ot´arola, and A. J. Salgado, Numerical methods for fractional diffusion, Comput. Vis. Sci., 19 (2018), pp. 19–46

    2018

  4. [11]

    Bonito, W

    A. Bonito, W. Lei, and J. E. Pasciak, Numerical approximation of the integral fractional Laplacian , Numer. Math., 142 (2019), pp. 235–278

    2019

  5. [12]

    , On sinc quadrature approximations of fractional powers of regularly accretive operators , J. Numer. Math., 27 (2019), pp. 57–68

    2019

  6. [13]

    Brockmann, L

    D. Brockmann, L. Hufnagel, and T. Geisel , The scaling laws of human travel , Nature, 439 (2006), p. 462

    2006

  7. [14]

    Caffarelli and L

    L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), pp. 1245–1260

    2007

  8. [15]

    Carmichael, H

    B. Carmichael, H. Babahosseini, S. Mahmoodi, and M. Agah, The fractional viscoelastic response of human breast tissue cells, Phys. Biol., 12 (2015), p. 046001

    2015

  9. [16]

    L. Chen, Z. Mao, and H. Li , Jacobi-Galerkin spectral method for eigenvalue problems of Riesz fractional differential equations, arXiv preprint arXiv:1803.03556, (2018)

    2018 arXiv

  10. [17]

    S. Chen, J. Shen, and L.-L. Wang , Laguerre functions and their applications to tempered fractional differential equations on infinite intervals , J. Sci. Comput., 74 (2018), pp. 1286–1313

    2018

  11. [18]

    J. H. Cushman and T. Ginn, Nonlocal dispersion in media with continuously evolving scales of heterogeneity , Transp. Porous Media, 13 (1993), pp. 123–138

    1993

  12. [19]

    Deng, Finite element method for the space and time fractional Fokker–Planck equation , SIAM J

    W. Deng, Finite element method for the space and time fractional Fokker–Planck equation , SIAM J. Numer. Anal., 47 (2008), pp. 204–226

    2008

  13. [20]

    W. Deng, B. Li, Z. Qian, and H. Wang , Time discretization of a tempered fractional Feynman–Kac equation with measure data, SIAM J. Numer. Anal., 56 (2018), pp. 3249–3275

    2018

  14. [21]

    Du , Nonlocal modeling, analysis, and computation , vol

    Q. Du , Nonlocal modeling, analysis, and computation , vol. 94, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, 2019. 24 C. SHENG, J. SHEN, T. TANG, L. WANG & H. YUAN

    2019

  15. [22]

    Duo and Y

    S. Duo and Y. Zhang , Computing the ground and first excited states of the fractional Schr¨ odinger equation in an infinite potential well , Commun. Comput. Phys., 18 (2015), pp. 321–350

    2015

  16. [23]

    , Finite difference methods for two and three dimensional fractional Laplacian with applications to solve the fractional reaction-diffusion equations, arXiv preprint arXiv:1804.02718, (2018)

    2018 arXiv

  17. [24]

    I. S. Gradshteyn and I. M. Ryzhik , Table of Integrals, Series, and Products , Elsevier/Academic Press, Amsterdam, eighth ed., 2015. Translated from the Russian, Translation edited and with a preface by Daniel Zwillinger and Victor Moll, Revised from the seventh edition [MR2360010]

    2015

  18. [25]

    Guo and Z

    B. Guo and Z. Wang , Modified Chebyshev rational spectral method for the whole line , in Proceedings of the fourth international conference on dynamical systems and differential equations, 2002, pp. 365–374

    2002

  19. [26]

    X. Guo, Y. Li, and H. Wang , A high order finite difference method for tempered fractional diffusion equations with applications to the CGMY model , SIAM J. Sci. Comput., 40 (2018), pp. A3322–A3343

    2018

  20. [27]

    Hatano and N

    Y. Hatano and N. Hatano, Dispersive transport of ions in column experiments: An explanation of long-tailed profiles, Water Resour. Res., 34 (1998), pp. 1027–1033

    1998

  21. [28]

    Hou and C

    D. Hou and C. Xu , A fractional spectral method with applications to some singular problems , Adv. Comput. Math., 43 (2017), pp. 911–944

    2017

  22. [29]

    Huang and A

    Y. Huang and A. Oberman, Numerical methods for the fractional Laplacian: a finite difference-quadrature approach , SIAM J. Numer. Anal., 52 (2014), pp. 3056–3084

    2014

  23. [30]

    B. Jin, R. Lazarov, and Z. Zhou , Error estimates for a semidiscrete finite element method for fractional order parabolic equations, SIAM J. Numer. Anal., 51 (2013), pp. 445–466

    2013

  24. [31]

    B. Jin, B. Li, and Z. Zhou, Numerical analysis of nonlinear subdiffusion equations, SIAM J. Numer. Anal., 56 (2018), pp. 1–23

    2018

  25. [32]

    Klein, C

    C. Klein, C. Sparber, and P. Markowich , Numerical study of fractional nonlinear Schr¨ odinger equations, Proc. Ser. A Math. Phys. Eng. Sci., 470 (2014), p. 20140364

    2014

  26. [33]

    Lischke, G

    A. Lischke, G. Pang, M. Gulian, F. Song, C. Glusa, X. Zheng, Z. Mao, W. Cai, M. M. Meerschaert, M. Ainsworth, et al., What is the fractional Laplacian? , arXiv preprint arXiv:1801.09767, (2018)

    2018 arXiv

  27. [34]

    Z. Mao, S. Chen, and J. Shen , Efficient and accurate spectral method using generalized Jacobi functions for solving Riesz fractional differential equations , Appl. Numer. Math., 106 (2016), pp. 165–181

    2016

  28. [35]

    Mao and J

    Z. Mao and J. Shen , Hermite spectral methods for fractional PDEs in unbounded domains , SIAM J. Sci. Comput., 39 (2017), pp. A1928–A1950

    2017

  29. [36]

    McCay and M

    B. McCay and M. N. L. Narasimhan, Theory of nonlocal electromagnetic fluids, Arch. Mech., 33 (1981), pp. 365–384

    1981

  30. [37]

    Metzler and J

    R. Metzler and J. Klafter , The random walk’s guide to anomalous diffusion: a fractional dynamics approach , Phys. Rep., 339 (2000), pp. 1–77

    2000

  31. [38]

    , The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics , J. Phys. A, 37 (2004), p. R161

    2004

  32. [39]

    E. W. Montroll and G. H. Weiss , Random walks on lattices. II , J. Math. Phys., 6 (1965), pp. 167–181

    1965

  33. [40]

    E. D. Nezza, G. Palatucci, and E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces , Bull. Sci. Math., 136 (2012), pp. 521–573

    2012

  34. [41]

    R. H. Nochetto, E. Ot ´arola, and A. J. Salgado , A PDE approach to fractional diffusion in general domains: a priori error analysis , Found. Comput. Math., 15 (2014), pp. 733–791

    2014

  35. [42]

    R. H. Nochetto, E. Otarola, and A. J. Salgado , A PDE approach to space-time fractional parabolic problems , SIAM J. Numer. Anal., 54 (2016), pp. 848–873

    2016

  36. [43]

    J. Shen, T. Tang, and L.-L. Wang, Spectral methods: algorithms, analysis and applications , vol. 41, Springer Science & Business Media, 2011

    2011

  37. [44]

    Shen and L

    J. Shen and L. Wang, Some recent advances on spectral methods for unbounded domains , Commun. Comput. Phys., 5 (2009), pp. 195–241

    2009

  38. [45]

    Shen, L.-L

    J. Shen, L.-L. Wang, and H. Yu, Approximations by orthonormal mapped Chebyshev functions for higher-dimensional problems in unbounded domains , J. Comput. Appl. Math., 265 (2014), pp. 264–275

    2014

  39. [46]

    Shlesinger, B

    M. Shlesinger, B. West, and J. Klafter , L´ evy dynamics of enhanced diffusion: Application to turbulence, Phys. Rev. Lett., 58 (1987), p. 1100

    1987

  40. [47]

    S. A. Silling , Reformulation of elasticity theory for discontinuities and long-range forces , J. Mech. Phys. Solids, 48 (2000), pp. 175–209

    2000

  41. [48]

    D. W. Sims, E. J. Southall, N. E. Humphries, G. C. Hays, C. J. Bradshaw, J. W. Pitchford, A. James, M. Z. Ahmed, A. S. Brierley, M. A. Hindell, et al. , Scaling laws of marine predator search behaviour , Nature, 451 (2008), p. 1098

    2008

  42. [49]

    Szeg¨o, Orthogonal polynomials, vol

    G. Szeg¨o, Orthogonal polynomials, vol. 23, American Mathematical Soc., 1939

    1939

  43. [51]

    T. Tang, H. Yuan, and T. Zhou , Hermite spectral collocation methods for fractional PDEs in unbounded domains , Commun. Comput. Phys., 24 (2018), pp. 1143–1168

    2018

  44. [52]

    Zhang, W

    Z. Zhang, W. Deng, and G. E. Karniadakis , A Riesz basis Galerkin method for the tempered fractional Laplacian , SIAM J. Numer. Anal., 56 (2018), pp. 3010–3039

    2018

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.