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作者(中文):曾仲麟
作者(外文):Tseng, Chung-Lin
論文名稱(中文):分數階擴散方程的一個快速解法
論文名稱(外文):An Efficient Solver for Fractional Diffusion Equations
指導教授(中文):王偉成
指導教授(外文):Wang, Wei-Cheng
口試委員(中文):林文偉
朱家杰
劉晉良
張德健
楊肅煜
口試委員(外文):Lin, Wen-Wei
Chu, Chia-Chieh
Liu, Jinn-Liang
Chang, Der-Chen
Yang, Suh-Yuh
學位類別:博士
校院名稱:國立清華大學
系所名稱:數學系
學號:100021802
出版年(民國):108
畢業學年度:107
語文別:英文
論文頁數:30
中文關鍵詞:分數階微分預處理方程解法迭代
外文關鍵詞:fractionaldifferentialpreconditionequationsdirectiterative
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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.
Contents
1 Introduction. . . . . .. . . . . .. . . . . .. . . . . .. . . . .5
2 De nitions of Fractional Derivative and Fractional Laplacian . . .6
2.1 Fractional derivative in 1D . . . . . . . . . . . . . . . . . .7
2.2 Fractional Laplacian in Rn . . . . . . . . . . . . . . . . . . 8
2.3 Fractional Laplacian on bounded domains . . . . . . . . . . . 9
3 Discretization and Precondition. . . . . .. . . . . .. . . . . . 9
3.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Precondition . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 Time Dependent Problem . . . . . . . . . . . . . . . . . . . 12
4 Numerical Results. . . . . .. . . . . .. . . . . .. . . . . .. 14
4.1 Generalized Eigenvalue Problem . . . . . . . . . . . . . . . 14
4.2 Time Dependent Problems . . . . . . . . . . . . . . . . . . . 24
5 Conclusion . . . . . .. . . . . .. . . . . .. . . . . .. . . . .26
Reference. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 27
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