Journal
COMPUTERS & MATHEMATICS WITH APPLICATIONS
Volume 73, Issue 6, Pages 1155-1171Publisher
PERGAMON-ELSEVIER SCIENCE LTD
DOI: 10.1016/j.camwa.2016.06.007
Keywords
Finite difference method; Riemann-Liouville fractional derivative; Fractional diffusion equation; Crank-Nicolson scheme; Variable coefficients; Fast Bi-CGSTAB algorithm
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Funding
- Australian Research Council [LP0348653]
- National Natural Science Foundation of China [11301040, 11226166]
- State Scholarship Fund from China Scholarship Council
- Australian Research Council [LP0348653] Funding Source: Australian Research Council
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In this paper, we consider a type of fractional diffusion equation (FDE) with variable coefficients on a finite domain. Firstly, we utilize a second-order scheme to approximate the Riemann-Liouville fractional derivative and present the finite difference scheme. Specifically, we discuss the Crank-Nicolson scheme and solve it in matrix form. Secondly, we prove the stability and convergence of the scheme and conclude that the scheme is unconditionally stable and convergent with the second -order accuracy of theta(tau(2) + h(2)). Furthermore, we develop a fast accurate iterative method for the Crank-Nicolson scheme, which only requires storage of theta(m) and computational cost of theta(m log m) while retaining the same accuracy and approximation property as Gauss elimination, where m = 1/h is the partition number in space direction. Finally, several numerical examples are given to show the effectiveness of the numerical method, and the results are in excellent agreement with the theoretical analysis. (C) 2016 Elsevier Ltd. All rights reserved.
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