A fast algorithm for solving the space–time fractional diffusion equation. (15th March 2018)

• Record Type: Journal Article
• Authors: Duo, Siwei; Ju, Lili; Zhang, Yanzhi
• Is Part Of: Computers & mathematics with applications. Volume 75:issue 6(2018)
• Page Start: 1929
• Page End: 1941
• Publication Date: 2018-03-15
• Subjects: Fractional diffusion equation -- Spectral fractional Laplacian -- Caputo fractional derivative -- Matrix transfer method -- Discrete sine transform -- Matrix–vector product; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2017.04.008
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: In this paper, we propose a fast algorithm for efficient and accurate solution of the space–time fractional diffusion equations defined in a rectangular domain. The spatial discretization is done by using the central finite difference scheme and matrix transfer technique. Due to its nonlocality, numerical discretization of the spectral fractional Laplacian ( − Δ ) s α / 2 results in a large dense matrix. This causes considerable challenges not only for storing the matrix but also for computing matrix–vector products in practice. By utilizing the compact structure of the discrete system and the discrete sine transform, our algorithm avoids to store the large matrix from discretizing the nonlocal operator and also significantly reduces the computational costs. We then use the Laplace transform method for time integration of the semi-discretized system and a weighted trapezoidal method to numerically compute the convolutions needed in the resulting scheme. Various experiments are presented to demonstrate the efficiency and accuracy of our method.