分數階微分方程能夠描述許多物理現象；此外，數值解法有助於研究者理解其演變。在本論文中，我們專注於數值方法，以應對一個眾所周知的相變模型，即Allen-Cahn方程，將時間中的普通微分算子替換為Caputo分數微分算子。這個方程也可以被表示為具有非線性源項的時間分數二階双曲型方程。我們採用L1向後差分(BDF)方法，分別使用1和2的階數離散化時間分數算子，並應用有限元方法離散化空間域。由於Allen-Cahn方程涉及非線性源項，我們考慮使用凸分裂法處理它。

論文的第一部分討論了L1-BDF2方法的誤差估計。我們擴展了Bangti Jin開發的分析方法，以分析L1-BDF2方案的局部截斷誤差和收斂速度。此外，我們提供了數值驗證結果以支持我們的理論發現。結果表明，L1-BDF2方法具有與L1-BDF1方法相同的截斷誤差階數。

在論文的第二部分，我們應用L1-BDF2方案與凸分裂法來解時間分數Allen-Cahn方程。我們的目標是研究方程沿著相應的自由能的耗散性質。我們提出了兩個新的耗散輔助泛函，這些泛函在分數階趨近於1時恢復為自由能。此外，我們表明L1方案與線性和非線性凸分裂方法都保持離散最大值原則，並展示了相對於輔助泛函的離散耗散定律。在論文結尾，我們提供了數值結果來驗證我們的理論結論。

A fractional order differential equation can characterize many physical phenomena; moreover, the numerical solutions facilitate researchers in understanding its evolution. In this dissertation, we focus on numerical methods to tackle a well-known phase transition model, the Allen-Cahn equation, by replacing the ordinary differential operator in time with the Caputo fractional differential operator. The equation can alternatively be represented as a time-fractional second-order hyperbolic equation with a nonlinear source term. We adopt the L1 backward difference formulation (BDF) with orders 1 and 2 to discretize the time fractional operator and apply the finite element method to discretize the spatial domain.
Since the Allen-Cahn equation involves a nonlinear source term, we consider the convex splitting method to deal with it.
The first part of the dissertation discusses the error estimation of the L1-BDF2 method. We extend the analysis approach that Bangti Jin developed to analyze the local truncation error and the convergence rate of the L1-BDF2 scheme. Additionally, we provide numerical validation results to support our theoretical findings. Consequently, the analysis shows that the L1-BDF2 method possesses the same order of truncation error as the L1-BDF1 method.
In the second part of the dissertation, we apply the L1-BDF2 scheme with the convex splitting method to solve the time-fractional Allen-Cahn equation. We aim to investigate the dissipation property of the corresponding free energy along the equation. We propose two new dissipative auxiliary functionals that recover the free energy as the fractional order tends to 1. Additionally, we show that the L1 scheme with both linear and nonlinear convex splitting methods preserves the discrete maximum principle and presents the discrete dissipation law regarding the auxiliary functional. At the end of the thesis, we provide numerical results to validate our theoretical claims.

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