Journal of Scientific Computing

, Volume 74, Issue 2, pp 1009–1033 | Cite as

A Fast Finite Difference Method for Three-Dimensional Time-Dependent Space-Fractional Diffusion Equations with Fractional Derivative Boundary Conditions



We develop a fast finite difference method for time-dependent variable-coefficient space-fractional diffusion equations with fractional derivative boundary-value conditions in three dimensional spaces. Fractional differential operators appear in both of the equation and the boundary conditions. Because of the nonlocal nature of the fractional Neumann boundary operator, the internal and boundary nodes are strongly coupled together in the coupled linear system. The stability and convergence of the finite difference method are discussed. For the implementation, the development of the fast method is based upon a careful analysis and delicate decomposition of the structure of the coefficient matrix. The fast method has approximately linear computational complexity per Krylov subspace iteration and an optimal-order memory requirement. Numerical results are presented to show the utility of the method.


Anomalous diffusion Finite difference method Fractional derivative boundary condition Space-fractional diffusion equation Stability and convergence analysis Maximum-minimum principle Toeplitz matrix 



This work was supported in part by the OSD/ARO MURI Grant W911NF-15-1-0562, by the National Natural Science Foundation of China under Grants 11471194, 91630207 and 11571115, by the National Science Foundation under Grant DMS-1620194, by the National Science and technology major projects of China under Grants 2011ZX05052 and 2011ZX05011-004, by Natural Science Foundation of Shandong Province of China under Grant ZR2011AM015, by Taishan Scholars Program of Shandong Province of China, and by the China Scholarship Council (File No. 201606220127). The authors would like to express their sincere thanks to the referees for their very helpful comments and suggestions, which greatly improved the quality of this paper.


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© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.School of MathematicsShandong UniversityJinanChina
  2. 2.Department of MathematicsUniversity of South CarolinaColumbiaUSA

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