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A stable explicitly solvable numerical method for the Riesz fractional advection–dispersion equations

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  • Zhang, Jingyuan

Abstract

In this paper, we present a finite difference scheme for solving the Riesz fractional advection-dispersion equations (RFADEs). The scheme is obtained by using asymmetric discretization technique and modify the shifted Grünwald approximation to fractional derivative. By calculating the unknowns in differential nodal-point sequences at the odd and even time-levels, the discrete solution of the scheme can be obtained explicitly. The computational cost for the scheme at each time step can be O(KlogK) by using the fast matrix-vector multiplication with the help of Toeplitz structure, where K is the number of unknowns. We prove that the scheme is solvable and unconditionally stable. We derive the error estimates in discrete l2-norm, which is optimal in some cases. Numerical examples are presented to verify our theoretical results.

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  • Zhang, Jingyuan, 2018. "A stable explicitly solvable numerical method for the Riesz fractional advection–dispersion equations," Applied Mathematics and Computation, Elsevier, vol. 332(C), pages 209-227.
    • Handle: RePEc:eee:apmaco:v:332:y:2018:i:c:p:209-227
    DOI: 10.1016/j.amc.2018.03.060
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    1. Mainardi, Francesco & Raberto, Marco & Gorenflo, Rudolf & Scalas, Enrico, 2000. "Fractional calculus and continuous-time finance II: the waiting-time distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 468-481.
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    4. Feng, L.B. & Zhuang, P. & Liu, F. & Turner, I., 2015. "Stability and convergence of a new finite volume method for a two-sided space-fractional diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 52-65.
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    1. Fahimi-khalilabad, Iraj & Irandoust-pakchin, Safar & Abdi-mazraeh, Somayeh, 2022. "High-order finite difference method based on linear barycentric rational interpolation for Caputo type sub-diffusion equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 199(C), pages 60-80.

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