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Mathematics 2015, 3(4), 1032-1044; doi:10.3390/math3041032

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

Department of Mathematics, Occidental College, Los Angeles, CA 90041, USA
Academic Editor: Indranil SenGupta
Received: 4 August 2015 / Accepted: 15 October 2015 / Published: 27 October 2015
View Full-Text   |   Download PDF [306 KB, uploaded 27 October 2015]   |  

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(N2) 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(N3) 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 logN) per iteration. Finally, some numerical results are shown to verify the accuracy and efficiency of the new method. View Full-Text
Keywords: circulant and toeplitz matrices; fast finite difference methods; fast fourier transform; fractional diffusion equations circulant and toeplitz matrices; fast finite difference methods; fast fourier transform; fractional diffusion equations
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Basu, T. A Fast O(N log N) Finite Difference Method for the One-Dimensional Space-Fractional Diffusion Equation. Mathematics 2015, 3, 1032-1044.

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