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A Fast O ( N log N ) Finite Difference Method for the One-Dimensional Space-Fractional Diffusion Equation

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  • Treena Basu

    (Department of Mathematics, Occidental College, Los Angeles, CA 90041, USA)

Abstract

This paper proposes an approach for the space-fractional diffusion equation in one dimension. Since fractional differential operators are non-local, two main difficulties arise after discretization and solving using Gaussian elimination: how to handle the memory requirement of O( N 2 ) for storing the dense or even full matrices that arise from application of numerical methods and how to manage the significant computational work count of O( N 3 ) per time step, where N is the number of spatial grid points. In this paper, a fast iterative finite difference method is developed, which has a memory requirement of O( N ) and a computational cost of O( N log N ) per iteration. Finally, some numerical results are shown to verify the accuracy and efficiency of the new method.

Suggested Citation

  • Treena Basu, 2015. "A Fast O ( N log N ) Finite Difference Method for the One-Dimensional Space-Fractional Diffusion Equation," Mathematics, MDPI, vol. 3(4), pages 1-13, October.
    • Handle: RePEc:gam:jmathe:v:3:y:2015:i:4:p:1032-1044:d:57869
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    References listed on IDEAS

    as
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    2. James W. Kirchner & Xiahong Feng & Colin Neal, 2000. "Fractal stream chemistry and its implications for contaminant transport in catchments," Nature, Nature, vol. 403(6769), pages 524-527, February.
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