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September 2018
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105 Reads
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13 Citations
International Journal of Computer Mathematics
In this paper, the high-dimensional Caputo-type parabolic equation with fractional Laplacian is studied by using the finite difference method. The convergence and error estimate of the established finite difference scheme are shown. And the illustrative examples are also displayed which support the theoretical analysis.
April 2017
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126 Reads
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34 Citations
Chaos Solitons & Fractals
In this paper, we present the finite difference method for Caputo-type parabolic equation with fractional Laplacian, where the time derivative is in the sense of Caputo with order in (0, 1) and the spatial derivative is the fractional Laplacian. The Caputo derivative is evaluated by the L1 approximation, and the fractional Laplacian with respect to the space variable is approximated by the Caffarelli–Silvestre extension. The difference schemes are provided together with the convergence and error estimates. Finally, numerical experiments are displayed to verify the theoretical results.
June 2016
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465 Reads
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103 Citations
Journal of Computational and Applied Mathematics
In this paper, a series of new high-order numerical approximations to αth (0<α<1) order Caputo derivative is constructed by using rth degree interpolation approximation for the integral function, where r≥4 is a positive integer. As a result, the new formulas can be viewed as the extensions of the existing jobs (Cao et al., 2015; Li et al., 2014), the convergence orders are O(τr+1-α), where τ is the time stepsize. Two test examples are given to demonstrate the efficiency of these schemes. Then we adopt the derived schemes to solve the Caputo type advection-diffusion equation with Dirichlet boundary conditions. The local truncation error of the derived difference scheme is O(τr+1-α+h2), where τ is the time stepsize, and h the space one. The stability and convergence of the proposed schemes for r=4 are also considered. Without loss of generality, we only display the numerical examples for r=4,5, which support the numerical algorithms.
... One of the approaches as mentioned in the literature is the Caffarelli-Silvestre extension which changes the fractional Laplace operator to an integer order differential operator [16,17]. Using this extension approach, Hu et al. [5,18] and Hu and Cheng [19] solved fractional diffusion equation using both finite difference and finite element methods respectively. These are the only works that focus on numerically approximating the extension problem. ...
September 2018
International Journal of Computer Mathematics
... The first one is based on the Caffarelli-Silvestre extension [8] or Dunford-Taylor formula [40], which transforms the nonlocal (−Δ) 2 to local elliptic operator. For example, by the Caffarelli-Silvestre extension, Nochetto et al. [29,30] made a systematical study on the finite element method for FDEs, Chen et al. [9,10] provided a multilevel methods for FDEs and derived a useful posteriori error estimator for adaptive methods, Hu et al. [20] gave a finite difference method for FDEs, Chen et al. [11] proposed a high accuracy and efficient spectral method for FDEs. Bonito et al. [5] used the Dunford-Taylor formula and sinc quadrature to develop a finite element method for FDEs. ...
April 2017
Chaos Solitons & Fractals
... Azin et al. [14] developed a hybrid numerical scheme based on Chebyshev cardinal functions and the modified Legendre functions to approximate the solution of (1) over a bounded time domain and an unbounded space domain. Li et al. [15] proposed a series of high-order numerical schemes on uniform mesh to solve Caputo-type advection-diffusion equation. The authors first constructed a series of high-order numerical algorithms to approximate the Caputo derivative and then derived a high-order finite difference scheme for solving Caputo-type advectiondiffusion equation. ...
June 2016
Journal of Computational and Applied Mathematics
