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# System prepended metadata

title: Weekly Report 2.2

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# Weekly Report 2.2
[toc]

## Method of separable variables
![](https://i.imgur.com/HQiyeE1.png)
![](https://i.imgur.com/BSGazYC.png)

然後討論 lamda>=<0 的情況
:::spoiler lamda>0
![](https://i.imgur.com/Ma2hNNo.png)
:::
:::spoiler lamda=0
![](https://i.imgur.com/EO98DuD.png)
:::

## Solutions for 2D Laplace's equations
Cartesian Coordinate System
![](https://i.imgur.com/HQiyeE1.png)
![](https://i.imgur.com/0C16UmQ.png) beta<0，和beta>0，式子長的一樣
![](https://i.imgur.com/EUAJ5iO.png)
所以 V 有 2X2 種型態
或是
![](https://i.imgur.com/72bsNqZ.png)
所以 V 有 2X2 種型態
所以
![](https://i.imgur.com/dHBNEuZ.png)
![](https://i.imgur.com/IMmITjt.png)
注意：要注意 possible complete solutions 不要寫成 invalid solutions

## Invalid Solutions in 2D Cartesian Coordinate
:::spoiler Picture
![](https://i.imgur.com/NJjoBGL.png)
:::
![](https://i.imgur.com/aWnQBBd.png)
因為前面，所以現在我們知道
![](https://i.imgur.com/TQhp2cz.png)
然而
![](https://i.imgur.com/eqox32c.png)

### Basic Problem: three boundary conditions
:::danger
每一種的推導過程相當類似，考試前強烈建議自己重新推推看。boundary conditions 可以當作特殊案例，當特殊案例常出現時，我們就可以幫它當作一種題型，要熟悉到可以看圖就可以反射想出結果。
:::


![](https://i.imgur.com/rWwquod.png)
:::spoiler 
![](https://i.imgur.com/dAg2FcE.png)
:::
![](https://i.imgur.com/P1AL7zc.png)
:::spoiler 
![](https://i.imgur.com/DG5icjR.png)
:::
![](https://i.imgur.com/34mr5IP.png)
:::spoiler 
![](https://i.imgur.com/BLAvJD9.png)
:::
![](https://i.imgur.com/X8ZONJJ.png)
:::spoiler 
![](https://i.imgur.com/e9eZQdd.png)
:::

### More than one non-zero boundary conditions

:::danger
這段同上，考試前推一下。要熟悉到可以看圖想結果。
:::
![](https://i.imgur.com/hbZGZCq.png)
:::spoiler 1. Detail
![](https://i.imgur.com/MvWAmiA.png)
:::
![](https://i.imgur.com/6S6jqDx.png)
:::spoiler 2. Detail
![](https://i.imgur.com/1QLCSue.png)
:::

### Hybrid boundary conditions
![](https://i.imgur.com/ruTPxJQ.png)
:::spoiler
![](https://i.imgur.com/9tJ52dg.png)
:::
### Division of Problem Domains
![](https://i.imgur.com/Ysnsq2v.png)
:::spoiler
![](https://i.imgur.com/aT5CMCm.png)
:::


## Solving Poisson’s Equations
:::danger
這段同上，考試前推一下。要熟悉到可以看圖想結果。
:::
![](https://i.imgur.com/2EdjrSa.png)
:::spoiler
![](https://i.imgur.com/2DDOwBb.png)
:::
![](https://i.imgur.com/2Rcq1i1.png)
:::spoiler
![](https://i.imgur.com/FzD2YxR.png)
:::
![](https://i.imgur.com/ItH3tiZ.png)
:::spoiler
![](https://i.imgur.com/9wtHXWW.png)
:::
### Check of boundary conditions
:::danger
這段同上，考試前推一下。要熟悉到可以看圖想結果。
:::
![](https://i.imgur.com/kPSQ8nz.png)
:::spoiler Boundary conditions for the Poisson solution
![](https://i.imgur.com/n4rZdLX.png)
:::
![](https://i.imgur.com/2jnr4Id.png)
:::spoiler
![](https://i.imgur.com/HRQE4Xf.png)
:::
![](https://i.imgur.com/WMFWHXS.png)
:::spoiler
![](https://i.imgur.com/frfbqY1.png)
:::

## Solution to the Poisson Equation with Homogeneous Boundary Conditions
![](https://i.imgur.com/7mCbaxo.png)
![](https://i.imgur.com/rZKLBM2.png)
![](https://i.imgur.com/q8BSFTZ.png)

### Solution to the Poisson Equation with Homogeneous Boundary Conditions
:::danger
這段應該要和 Laplace Equation 做比較
:::
![](https://i.imgur.com/dOYBtqY.png)
![](https://i.imgur.com/Y0rPh0p.png)

### Solution to the Laplace Equation with Non-Homogeneous Boundary Conditions
:::danger
這段應該要和 Poisson Equation 做比較
:::
![](https://i.imgur.com/F0pRoHM.png)
![](https://i.imgur.com/yBz2JjQ.png)

### Complete solution
![](https://i.imgur.com/aSnZBHm.png)


# 心得
這段感覺是過去的總和，不過相對沒有新鮮感，主要是數學的推導，很多的推導過程相當類似，考試前強烈建議自己重新推推看，而且可以把類似的部分拿出來做比較，透過對照找出不同之處可以讓我們對於該部分的印象特別加深。
對我來說 boundary conditions 可以把她當作特殊案例，當特殊案例常出現時，我們就可以幫它當作一種題型，自己做歸納，或是找出規律、記憶點來方便自己熟悉題型，直到可以看圖就可以反射想出結果。




