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

Treena Basu
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Treena Basu: Department of Mathematics, Occidental College, Los Angeles, CA 90041, USA

Mathematics, 2015, vol. 3, issue 4, 1-13

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.

Keywords: circulant and toeplitz matrices; fast finite difference methods; fast fourier transform; fractional diffusion equations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2015
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