The Use of Finite Difference/Element Approaches for Solving the Time-Fractional Subdiffusion Equation

Zeng, Fanhai; Li, Changpin; Liu, Fawang; Turner, Ian

doi:10.1137/130910865

Cited by 275 publications

(151 citation statements)

References 55 publications

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“…There are two predominant approaches for approximating the fractional derivative: one approach is by using Lubich's convolution quadrature [27]- [29] and another approach is by using the L1 scheme (or Diethelm's finite difference method). For the recent developments for solving fractional ordinary (or partial ) differential equations by using the Lubich's convolution quadrature method, readers may refer to e.g., [39], [11], [3], [42], [6], [44], [46], [47], [22], [21], [19], etc.…”

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confidence: 99%

An Analysis of the Modified L1 Scheme for Time-Fractional Partial Differential Equations with Nonsmooth Data

Yan¹,

Khan²,

Ford³

2018

SIAM J. Numer. Anal.

139

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Abstract. We introduce a modified L1 scheme for solving time fractional partial differential equations and obtain error estimates for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Jin et al. (2016, An analysis of the L1 scheme for the subdiffusion equation with nonsmooth data, IMA J. of Numer. Anal.,36, established an O(k) convergence rate for the L1 scheme for smooth and nonsmooth initial data for the homogeneous problem, where k denotes the time step size. We show that the modified L1 scheme has convergence rate O(k 2−α ), 0 < α < 1 for smooth and nonsmooth initial data in both homogeneous and inhomogeneous cases. Numerical examples are given to show that the numerical results are consistent with the theoretical results.

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confidence: 99%

An Analysis of the Modified L1 Scheme for Time-Fractional Partial Differential Equations with Nonsmooth Data

Yan¹,

Khan²,

Ford³

2018

SIAM J. Numer. Anal.

139

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“…Recent advances in the fractional calculus concern the fractional derivative modeling in applied science, see [2,9,38], the theory of fractional differential equations, see [21], numerical approaches for the fractional differential equations, see [26,55] and the references therein. Another hot issue is the theory of hemivariational inequalities which is based on properties of the Clarke generalized gradient, defined for locally Lipschitz functions.…”

Section: Introductionmentioning

confidence: 99%

A class of fractional differential hemivariational inequalities with application to contact problem

Zeng

Liu

Migórski

2018

Z. Angew. Math. Phys.

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Abstract. In this paper, we study a class of generalized differential hemivariational inequalities of parabolic type involving the time fractional order derivative operator in Banach spaces. We use the Rothe method combined with surjectivity of multivalued pseudomonotone operators and properties of the Clarke generalized gradient to establish existence of solution to the abstract inequality. As an illustrative application, a frictional quasistatic contact problem for viscoelastic materials with adhesion is investigated, in which the friction and contact conditions are described by the Clarke generalized gradient of nonconvex and nonsmooth functionals, and the constitutive relation is modeled by the fractional Kelvin-Voigt law.Mathematics Subject Classification. 35L15, 35L86, 35L87, 74Hxx, 74M10.

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“…The corresponding fractional Fick's law has been proposed [4]. Recently, a modified fractional Fick's law has been used to describe processes that become less anomalous as time progresses by the inclusion of a secondary fractional time derivative acting on a diffusion operator [5], Till now, various kinds of anomalous diffusion equations have been studied numerically, see [7,8,9,10,11,12,13,14,15,16,17] and many the references cited therein. However, it seems that only a few numerical studies are available for the two-term subdiffusions of the above form.…”

Section: Introductionmentioning

confidence: 99%

High‐order compact difference schemes for the modified anomalous subdiffusion equation

Ding

2015

Numerical Methods Partial

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In this paper, two kinds of high-order compact finite difference schemes for second-order derivative are developed. Then a second-order numerical scheme for a Riemann-Liouvile derivative is established based on a fractional centered difference operator. We apply these methods to a fractional anomalous subdiffusion equation to construct two kinds of novel numerical schemes. The solvability, stability and convergence analysis of these difference schemes are studied by using Fourier method. The convergence orders of these numerical schemes are O(τ 2 + h 6 ) and O(τ 2 + h 8 ), respectively. Finally, numerical experiments are displayed which are in line with the theoretical analysis.

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The Use of Finite Difference/Element Approaches for Solving the Time-Fractional Subdiffusion Equation

Cited by 275 publications

References 55 publications

An Analysis of the Modified L1 Scheme for Time-Fractional Partial Differential Equations with Nonsmooth Data

An Analysis of the Modified L1 Scheme for Time-Fractional Partial Differential Equations with Nonsmooth Data

A class of fractional differential hemivariational inequalities with application to contact problem

High‐order compact difference schemes for the modified anomalous subdiffusion equation

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