Superconvergence Points for the Spectral Interpolation of Riesz Fractional Derivatives

Deng, Biwei; Zhang, Zhimin; Zhao, Xuan

doi:10.1007/s10915-019-01054-6

Cited by 9 publications

(9 citation statements)

References 39 publications

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“…The above estimate indicates the exponential convergence under the assumption that the source term f is smooth. The results in (Mao et al 2016) extend the studies on spectral methods for solving one-sided fractional equations found in (Chen, Shen and Wang 2016b, Zayernouri and Karniadakis 2014, Zayernouri, Ainsworth and Karniadakis 2015b; see also (Mao and Karniadakis 2018, Samiee, Zayernouri and Meerschaert 2019a, Samiee, Zayernouri and Meerschaert 2019b for general two-sided fractional equations, (Zayernouri, Ainsworth and Karniadakis 2015a) for tempered fractional diffusion equations, and (Deng, Zhang and Zhao 2019) for the study of superconvergence points.…”

Section: Spectral-galerkin Methods In Bounded Domainssupporting

confidence: 70%

Numerical methods for nonlocal and fractional models

D'Elia,

Du,

Glusa

et al. 2020

Preprint

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Partial differential equations (PDEs) are used, with huge success, to model phenomena arising across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDE models fail to adequately model observed phenomena or are not the best available model for that purpose. On the other hand, in many situations, nonlocal models that account for interaction occurring at a distance have been shown to more faithfully and effectively model observed phenomena that involve possible singularities and other anomalies. In this article, we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis, and specific concrete examples. We then provide extensive discussions about numerical methods, including finite element, finite difference, and spectral methods, for determining approximate solutions of the nonlocal models considered. In that discussion, we pay particular attention to a special class of nonlocal models that are the most widely studied in the literature, namely those involving fractional derivatives. The article ends with brief considerations of several modeling and algorithmic extensions which serve to show the wide applicability of nonlocal modeling.

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Section: Spectral-galerkin Methods In Bounded Domainssupporting

confidence: 70%

Numerical methods for nonlocal and fractional models

D'Elia,

Du,

Glusa

et al. 2020

Preprint

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“…The framework of the proof is the same as Theorem 4.4. The detail is referred to the Theorem 3.1 in [6].…”

Section: )mentioning

confidence: 99%

“…For example, the superconvergence of integer-order derivative of various Jacobi-Gauss-type spectral interpolations were considered in [27,28,33,34] respectively. Zhang, Zhao and Deng generalized the study to Riemann-Liouville and Riesz fractional derivatives with the order of 0 < α < 1 in [6,36]. However, as pointed out in [6], Lagrange-type interpolations fail to converge when applying to left Riemann-Liouville fractional derivative with α > 1.…”

mentioning

confidence: 99%

“…Zhang, Zhao and Deng generalized the study to Riemann-Liouville and Riesz fractional derivatives with the order of 0 < α < 1 in [6,36]. However, as pointed out in [6], Lagrange-type interpolations fail to converge when applying to left Riemann-Liouville fractional derivative with α > 1. More specifically, the error −1 D α x (u − u N ) will not converge at x = −1, where u N is the Legendre-Lobatto or Left-Radau interpolant of u(x).…”

mentioning

confidence: 99%

“…Consequently, the Hermite interpolation makes the singularity at x = ±1 to be under control comparing with Lagrange-type interpolations. While the theory of convergence and superconvergence for Lagrange-type interpolations is well understood (see, [6,27,33,34,36]), the study on Hermete interpolation has so far received little attention.…”

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confidence: 99%

See 2 more Smart Citations

Superconvergence points of integer and fractional derivatives of special Hermite interpolations and its applications in solving FDEs

Deng

Zhang

2019

ESAIM: M2AN

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In this paper, we study convergence and superconvergence theory of integer and fractional derivatives of the one-point and the two-point Hermite interpolations. When considering the integer-order derivative, exponential decay of the error is proved, and superconvergence points are located, at which the convergence rates are O(N −2 ) and O(N −1.5 ), respectively, better than the global rate for the one-point and two-point interpolations. Here N represents the degree of interpolation polynomial. It is proved that the α-th fractional derivative of (u − u N ) with k < α < k + 1, is bounded by its (k + 1)-th derivative. Furthermore, the corresponding superconvergence points are predicted for fractional derivatives, and an eigenvalue method is proposed to calculate the superconvergence points for the Riemann-Liouville fractional derivative. In the application of the knowledge of superconvergence points to solve FDEs, we discover that a modified collocation method makes numerical solutions much more accurate than the traditional collocation method.

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Recent Development Strategies for Bismuth‐Driven Materials in Sustainable Energy Systems and Environmental Restoration

Adhikari

Mandal

Kim

2022

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Bismuth(Bi)‐based materials have gained considerable attention in recent decades for use in a diverse range of sustainable energy and environmental applications due to their low toxicity and eco‐friendliness. Bi materials are widely employed in electrochemical energy storage and conversion devices, exhibiting excellent catalytic and non‐catalytic performance, as well as CO2/N2 reduction and water treatment systems. A variety of Bi materials, including its oxides, chalcogenides, oxyhalides, bismuthates, and other composites, have been developed for understanding their physicochemical properties. In this review, a comprehensive overview of the properties of individual Bi material systems and their use in a range of applications is provided. This review highlights the implementation of novel strategies to modify Bi materials based on morphological and facet control, doping/defect inclusion, and composite/heterojunction formation. The factors affecting the development of different classes of Bi materials and how their control differs between individual Bi compounds are also described. In particular, the development process for these material systems, their mass production, and related challenges are considered. Thus, the key components in Bi compounds are compared in terms of their properties, design, and applications. Finally, the future potential and challenges associated with Bi complexes are presented as a pathway for new innovations.

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Superconvergence Points for the Spectral Interpolation of Riesz Fractional Derivatives

Cited by 9 publications

References 39 publications

Numerical methods for nonlocal and fractional models

Numerical methods for nonlocal and fractional models

Superconvergence points of integer and fractional derivatives of special Hermite interpolations and its applications in solving FDEs

Recent Development Strategies for Bismuth‐Driven Materials in Sustainable Energy Systems and Environmental Restoration

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