#### Date of Award

1-1-2012

#### Document Type

Campus Access Dissertation

#### Department

Mathematics

#### First Advisor

Hong Wang

#### Abstract

Fractional diffusion equations model phenomena exhibiting anomalous diffusion that can not be modeled accurately by the second order diffusion equations. Because of the non-local property of fractional differential operators, the numerical methods have full coefficient matrices which require storage of O(N^2) and computational cost of O(N^3), where N is the number of grid points.

Together we develop a fast finite difference method for the one-dimensional space fractional diffusion equation, which only requires storage of O(N) and computational cost of O(N log^2N), while retaining the same accuracy and approximation property as the regular finite difference method. Numerical experiments are presented to show the utility of the method.

For example, with 1024 computational nodes, the new scheme developed for the one-dimensional problem has about 40 times of CPU reduction than the standard scheme.

For the two-dimensional space fractional diffusion equation we devise a fast iterative scheme, which only requires storage of O(N) and computational cost of O(N log N) while retaining the same order accuracy as the regular finite difference method.

Our preliminary numerical example runs for two-dimensional model problem of intermediate size seem to indicate the observations: to achieve the same accuracy, the new method uses less than one thousandth of CPU time and one thousandth memory than the standard method does. This demonstrates the utility of the method.

#### Recommended Citation

Basu, T.(2012). *Fast Solution Methods For Fractional Diffusion Equations and Their Applications.* (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/etd/1585