A finite difference method with non-uniform timesteps for fractional diffusion and diffusion-wave equations

Quintana-Murillo, J.; Yuste, S. B.

doi:10.1140/epjst/e2013-01979-7

Cited by 41 publications

(28 citation statements)

References 32 publications

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“…We should recall, at this point, that many authors have already considered similar problems, but most often adopting a fractional order only either in time or in space, see, e.g., [17][18][19][20]. When fluid flow through porous media is studied, usually, only fractional derivatives in time were considered.…”

Section: Statement Of the Problemmentioning

confidence: 99%

Identifying the Fractional Orders in Anomalous Diffusion Models from Real Data

Concezzi

Spigler

2018

Fractal Fract

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An attempt is made to identify the orders of the fractional derivatives in a simple anomalous diffusion model, starting from real data. We consider experimental data taken at the Columbus Air Force Base in Mississippi. Using as a model a one-dimensional fractional diffusion equation in both space and time, we fit the data by choosing several values of the fractional orders and computing the infinite-norm "errors", representing the discrepancy between the numerical solution to the model equation and the experimental data. Data were also filtered before being used, to see possible improvements. The minimal discrepancy is attained correspondingly to a fractional order in time around 0.6 and a fractional order in space near 2. These results may describe well the memory properties of the porous medium that can be observed.

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Section: Statement Of the Problemmentioning

confidence: 99%

Identifying the Fractional Orders in Anomalous Diffusion Models from Real Data

Concezzi

Spigler

2018

Fractal Fract

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“…This excellent approach in [21,22] allows to obtain a (relatively) high convergence order without the otherwise required very unnatural smoothness assumptions on the given solution. Other works for solving fractional differential equations with nonuniform meshes may be found in, for example, [12,19,24,25]. Motivated by the ideas in Diethelm [4] and Stynes et al [22], we will introduce a numerical method for solving (1.1) with the graded meshes and we prove that the optimal convergence order uniformly in t n for the proposed numerical method can be recovered when C 0 D α t y(t), α > 0 behaves as t σ , 0 < σ < 1.…”

Section: Introductionmentioning

confidence: 99%

Detailed error analysis for a fractional Adams method with graded meshes

2017

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We consider a fractional Adams method for solving the nonlinear fractional differential equation C 0 D α t y(t) = f (t, y(t)), α > 0, equipped with the initial conditions y (k) Here, α may be an arbitrary positive number and α denotes the smallest integer no less than α and the differential operator is the Caputo derivative. Under the assumption 2004) introduced a fractional Adams method with the uniform meshes t n = T (n/N), n = 0, 1, 2, . . . , N and proved that this method has the optimal convergence order uniformly in t n , that is, the optimal convergence order of this method cannot be obtained with the uniform meshes. However, it is well-known that for y ∈ C m [0, T ] for some m ∈ N and 0 < α < m, the Caputo fractional derivative C 0 D α t y(t) takes the form " C 0 D α t y(t) = ct α −α + smoother terms" (Diethelm et al. Numer. Algor. 36, 2004), which implies that C 0 D α t y behaves as t α −α which is not in C 2 [0, T ]. By using the graded meshes t n = T (n/N) r , n = 0, 1, 2, . . . , N with some suitable r > 1, we show that the optimal convergence order of this method can be recovered uniformly in t n even if C 0 D α t y behaves as

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“…(1) Use non-uniform meshes and approximate the solution near the singular point t = 0 by using finer meshes, see, [36], [39], [43], [56], [66]. (2) Use some nonpolynomial (or singular) basis functions or collocation spectral methods to capture the singularity of the solutions of (1.1), see [1], [5], [13], [14], [34], [27], [59], [63], [67].…”

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confidence: 99%

An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data

Ford

Yan

2017

FCAA

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In this paper, we shall review an approach by which we can seek higher order time discretisation schemes for solving time fractional partial differential equations with nonsmooth data. The low regularity of the solutions of time fractional partial differential equations implies standard time discretisation schemes only yield first order accuracy. To obtain higher order time discretisation schemes when the solutions of time fractional partial differential equations have low regularities, one may correct the starting steps of the standard time discretisation schemes to capture the singularities of the solutions. We will consider these corrections of some higher order time discretisation schemes obtained by using Lubich's fractional multistep methods, L1 scheme and its modification, discontinuous Galerkin methods, etc. Numerical examples are given to show that the theoretical results are consistent with the numerical results.

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A finite difference method with non-uniform timesteps for fractional diffusion and diffusion-wave equations

Cited by 41 publications

References 32 publications

Identifying the Fractional Orders in Anomalous Diffusion Models from Real Data

Identifying the Fractional Orders in Anomalous Diffusion Models from Real Data

Detailed error analysis for a fractional Adams method with graded meshes

An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data

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