Trinh @ Bath

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vpde_lecture22 [2020/03/24 14:45]
trinh [The 1D heat equation with zero Dirichlet conditions]
vpde_lecture22 [2020/03/26 13:42] (current)
trinh
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 ====== MA20223 Lecture 22 ====== ====== MA20223 Lecture 22 ======
  
-//This lecture is about terminology but it is important not to be bogged down by terminology. You will practice by doing!// +//This lecture is about terminology but it is important not to be bogged down by terminology. You will practice by doing! This lecture covered part of **Chapter 15**.// 
 + 
 + 
 +<html> 
 +<iframe width="560" height="315" src="https://www.youtube.com/embed/30UzA62y_XE" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe> 
 +</html>
  
 ===== An example ===== ===== An example =====
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 ==== The 1D heat equation with zero Dirichlet conditions ====  ==== The 1D heat equation with zero Dirichlet conditions ==== 
  
-Now let's return to the study of the heat equation with zero Dirichlet conditions stated at the top of this note. In the video, we will go through the procedure from start to end in solving the BVP. In the end, the solution will be given by a Fourier series,+Now let's return to the study of the heat equation with zero Dirichlet conditions stated at the top of this note. In the video, we will go through the procedure from start to end in solving the BVP. In the end, the solution will be given by a Fourier sine series,
  
 $$ $$
-u(x, t) \sim \frac{a_0}{2} + \sum_1^\infty \left[a_n \cos\left(\frac{n\pi x}{L}\right) +u(x, t) \sim \sum_1^\infty \left[b_n \sin\left(\frac{n\pi x}{L}\right)\right]. 
-b_n \sin\left(\frac{n\pi x}{L}\right)\right]. +
 $$ $$
  
-where $a_n$ and $b_n$ are the usual Fourier coefficients defined with the function $f(x)$ on $[0, L]$. +where $b_n$ will be the Fourier sine coefficients of the odd $2\pi$ extension of $f(x)$ on $[0, \pi]$.  
 + 
 +See the video for details of the calculation; we were not able to get to the end of the calculation by the lecture's end, but it will be simple for us to finish it all up in [[vpde_lecture23|MA20223 Lecture 23]]