這段時間，分數階微分算子作為能夠發展準確描述複雜異常系統的數學模型的重要工具而備受關注。由於分數階微分算子是非局部的，相應的線性系統具有稠密，結構化的扥波力茲矩陣。 許多研究致力於為這種線性系統開發穩健且有效的解法。在這論文中，我們提出了一種基於新預處理器的分數階擴散方程的數值方法，使用這方法解分數階擴散方程我們可以使用直接法和迭代法，每個時間步長具有總O（N logN）運算量。數值結果表明新的方法相較現有方法是有競爭力的。

The fractional order differential operators have attracted considerable attention recently as an essential tool for developing more sophisticated mathematical models that can accurately describe complex anomalous systems. Since the fractional order differential operators are nonlocal, the corresponding linear system involves are dense, structured Toeplitz matrix. Many research activities are devoted to developing robust and efficient solvers for such linear systems. In this thesis, we propose a numerical method for the fractional diffusion equations based on a new preconditioner that
can be used to develop direct and iterative solvers for fractional diffusion equations with total O(N log N) operations per time step. Numerical results suggest the new method is a competitive alternative to existing methods.

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