| Abstract: |
| Fractional diffusion equations have been proven to accurately model anomalous diffusion processes in nature. However, numerical schemes applied to space-fractional diffusion equations result in dense or full coefficient matrices with computational complexity and storage capacity of $O(N^3)$ per time step and $O(N^2)$ respectively, which is increasingly problematic for larger N. This talk seeks to provide a more efficient and robust algorithm for numerically approximating a second-order accurate numerical solution to the discretized one-dimensional two-sided space-fractional diffusion equation that requires only $O(N \log N)$ computational work per time step and $O(N)$ memory by utilizing the Crank-Nicolson scheme and studying the structure of the resulting coefficient matrix. A fast iterative scheme is used to solve the resulting system of equations. Numerical results are shown to illustrate the second-order accuracy and efficiency of the new method. |
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