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.
Highlights
The history of fractional calculus is almost as long as integer calculus, it is only in the last few decades that it has gained much importance
This paper develops a fast solution method for the implicit finite difference scheme Equation (6) for the one-dimensional space-fractional diffusion equation developed by Meerschaert and Tadjeran in [15,16]
The fast method consists of carefully analyzing the structure of the coefficient matrix resulting from the finite difference method, delicately decomposing the coefficient matrix into a combination of sparse and structured dense matrices and applying an iterative scheme, in this case the conjugate gradient squared method
Summary
The history of fractional calculus is almost as long as integer calculus, it is only in the last few decades that it has gained much importance. They utilize this decomposition and applied an operator splitting technique to the one-dimensional space-fractional diffusion equation to develop a fast operator-splitting finite difference method for the space-fractional diffusion equation in one space dimension This method has a computational work account of O(N log N ) per iteration and has a memory need of O(N log N ) per time step, due to the use of the banded coefficient matrix. The non-local nature of the fractional derivatives results in a full coefficient matrix of the system This is in contrast to numerical methods for second-order diffusion equations which usually generate banded coefficient matrices of O(N ) nonzero entries and can be solved by fast solution methods such as multigrid methods, domain decomposition methods, and wavelet methods. The development of fast and robust numerical methods with efficient storage for the space-fractional diffusion equation is crucial for the applications of fractional diffusion equations
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