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

作者：Zhao, Meng; Wang, Hong; Cheng, Aijie^*

来源：Journal of Scientific Computing, 2018, 74(2): 1009-1033.

DOI：10.1007/s10915-017-0478-8

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

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.

• 出版日期2018-2
• 单位山东大学