在這篇文章中，我們討論對時間項偏微分採取整數階與分數階方式的擴散方程在不同時間點上數值解的總和，並討論加入Allen-Cahn方程的非線性項。在時間項分數階擴散方程中，我們比較兩種離散分數階的方式，其一為Grünwald-Letnikov，另一種為Backward Difference Formula-1，兩者均用有限差分法離散空間。在時間分數階Allen-Cahn方程中，

我們觀察每個時間點上數值解的總和；在週期邊界下不論整數階或是分數階擴散方程的數值解總和會不變。對於隨機初始值下Allen-Cahn方程，在同初始值下觀察到不同分數階在當前時間與下個時間點的數值總和差隨時間改變，在時間-數值總和差的圖中，取出不同分數階數值總和差絕對值最大值的點，不同分數階的點會靠近一條直線。且分數階越靠近1，數值總和差的絕對值會越大，同時發生的時間點也越往後。

同時我們也設計兩個實驗。第一個實驗觀察的目標是在不同時間分數階下擴散面積的變化。小的分數階擴散面積在前期時間較大到中期趨緩，較大的分數階的擴散面積在中期時間便會超過較小的分數擴散面積。第二個實驗我們測試一行進波問題，我們得到關於不同分數階下行進波的變化的結論(1)分數階越小則行進波的高度越低。(2)以Backward Difference Formula-1離散式模擬波的前進速度相較於Grünwald-Letnikov離散式來得快。

In this article, we investigate the numerical summation of solutions at different time points for the first-order or fractional-order diffusion equations in time, with or without the nonlinear term. In the time-fractional diffusion equation, we compare two discrete fractional-order methods: the Grünwald-Letnikov or the backward difference formula-1 method in time and standard finite difference in space.

We study the summation of numerical solutions at each time point. Under the same periodic boundary condition, the first-order or fractional-order diffusion equation, the summation of numerical solutions is conservative in time. We add the nonlinear term, we observe that the summation of numerical solutions is not conservative over time. In addition, we extract the time at the maximal absolute difference in numerical summation for each fractional order. These maximal absolute difference grows linearly with the different fractional orders. Moreover, for the larger fractional order, the maximal absolute difference in numerical summation becomes larger and happens later.

We design two experiments. In the first experiment, we observe the variation in the diffusion area under different fractional time orders. Smaller fractional order leads to a larger diffusion area during the early to middle stages. However, larger fractional order result in a larger diffusion areas after the middle stages. In the second experiment, we put a traveling wave in the initial condition configuration. We conclude that under different fractional orders, (1) smaller fractional order leads to lower wave height, and (2) the backward difference formula-1 discrete scheme simulates the wave's propagation speed faster than the Grünwald-Letnikov discrete scheme.

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