A High Order Schema for the Numerical Solution of Ordinary Fractional Differential Equations

Cao, Junying; Xu, Chuanju

doi:10.1090/conm/586/11658

Cited by 24 publications

(41 citation statements)

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“…We also recall the property of hypergeometric functions related to transformations of variable (cf. [29, P. 390]): 9) and the Pfaff's formula on the linear transformation (cf. [2, (2.3.14)]): for integer n ≥ 0,…”

Section: Bessel Functionsmentioning

confidence: 99%

Rational Spectral Methods for PDEs Involving Fractional Laplacian in Unbounded Domains

Tang¹,

Wang²,

Yuan³

et al. 2020

SIAM J. Sci. Comput.

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Many PDEs involving fractional Laplacian are naturally set in unbounded domains with underlying solutions decay very slowly, subject to certain power laws. Their numerical solutions are under-explored. This paper aims at developing accurate spectral methods using rational basis (or modified mapped Gegenbauer functions) for such models in unbounded domains. The main building block of the spectral algorithms is the explicit representations for the Fourier transform and fractional Laplacian of the rational basis, derived from some useful integral identites related to modified Bessel functions. With these at our disposal, we can construct rational spectral-Galerkin and direct collocation schemes by pre-computing the associated fractional differentiation matrices. We obtain optimal error estimates of rational spectral approximation in the fractional Sobolev spaces, and analyze the optimal convergence of the proposed Galerkin scheme. We also provide ample numerical results to show that the rational method outperforms the Hermite function approach.

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Section: Bessel Functionsmentioning

confidence: 99%

Rational Spectral Methods for PDEs Involving Fractional Laplacian in Unbounded Domains

Tang¹,

Wang²,

Yuan³

et al. 2020

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

show abstract

“…We now here only mention a few part of related publications, such as the work by Yuste and Acedo [1], Langlands and Henry [2], Chen et al [3], Zhuang et al [4], Cao and Xu [5], Li and Ding [6], Sun and Wu [7]. For more results, readers can refer to the review article [8] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Some high-order difference schemes for the distributed-order differential equations

Gao

Sun

2015

Journal of Computational Physics

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“…Recently, some new works on the higher-order numerical approximation for the time-fractional derivatives can also be found. Cao and Xu [25] obtained a higher-order scheme for the numerical solution of the fractional ordinary differential equations starting from the equivalent integral form of the original differential equations. Gao and Sun [26] proposed a new L1-2 formula to approximate the Caputo time-fractional derivatives and investigated the applications of this formula into solving fractional differential equations.…”

Section: Introductionmentioning

confidence: 99%

Three-point combined compact difference schemes for time-fractional advection–diffusion equations with smooth solutions

Gao

Sun

2015

Journal of Computational Physics

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A High Order Schema for the Numerical Solution of Ordinary Fractional Differential Equations

Cited by 24 publications

References 0 publications

Rational Spectral Methods for PDEs Involving Fractional Laplacian in Unbounded Domains

Rational Spectral Methods for PDEs Involving Fractional Laplacian in Unbounded Domains

Some high-order difference schemes for the distributed-order differential equations

Three-point combined compact difference schemes for time-fractional advection–diffusion equations with smooth solutions

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