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Zbl 1242.65160
Gao, Guang-Hua; Sun, Zhi-Zhong; Zhang, Ya-Nan
A finite difference scheme for fractional sub-diffusion equations on an unbounded domain using artificial boundary conditions. (English)
[J] J. Comput. Phys. 231, No. 7, 2865-2879 (2012). ISSN 0021-9991

Summary: One-dimensional fractional anomalous sub-diffusion equations on an unbounded domain are considered. Beginning with the derivation of the exact artificial boundary conditions, the original problem on an unbounded domain is converted into mainly solving an initial-boundary value problem on a finite computational domain. The main contribution of our work, as compared with the previous work, lies in the reduction of fractional differential equations on an unbounded domain by using artificial boundary conditions and construction of the corresponding finite difference scheme with the help of the method of order reduction. The difficulty is the treatment of the Neumann condition on the artificial boundary, which involves the time-fractional derivative operator. The stability and convergence of the scheme are proven using the discrete energy method. Two numerical examples clarify the effectiveness and accuracy of the proposed method.

MSC 2000:: *65M06 Finite difference methods (IVP of PDE)
65M12 Stability and convergence of numerical methods (IVP of PDE)
35K05 Heat equation
35R11

Keywords: fractional differential equation; unbounded domain; finite difference scheme; stability; convergence; fractional anomalous sub-diffusion equations; artificial boundary conditions; initial-boundary value problem; method of order reduction; numerical examples

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