### Author Topic: Web bonus problem : Week 3 (#1)  (Read 3106 times)

#### Victor Ivrii

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##### Web bonus problem : Week 3 (#1)
« on: September 26, 2015, 12:43:13 PM »

#### Yiqi Shi

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##### Re: Web bonus problem : Week 3 (#1)
« Reply #1 on: September 27, 2015, 02:45:49 PM »
Firstly, Let's use characteristic coordinates \left\{\begin{aligned}&\xi=x+ct,\\&\eta=x-ct.\end{aligned}\right.\label{eq1} When using the characteristic coordinates, the problem become: \begin{gather}u_{\xi\eta}=0\qquad \text{as }\xi>0,\eta>0,\\[3pt]u|_{\xi=0}=g(t)\qquad \text{as }t<0 ,\\[3pt]u|_{\eta=0}=h(t)\qquad \text{as } t>0.\end{gather} Then by 2.4.1 equation(4), we have $$u=\phi(\xi)+\psi(\eta)\label{5}$$ is the general solution to (2). Then $$u|_{\xi=0}=\phi(0)+\psi(\eta)=g(t)$$ $$u|_{\eta=0}=\phi(\xi)+\psi(0)=h(t)$$ Thus by (5)(6)(7) $$u=g(t)+h(t)-\phi(0)-\psi(0)$$ Since we have $$u(0,0)=\phi(0)+\psi(0)$$ $$u(0,0)=h(0)=g(0)$$ Then $$u=g(t)+h(t)-g(0)$$ solves Goursat problem.

#### Victor Ivrii

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##### Re: Web bonus problem : Week 3 (#1)
« Reply #2 on: September 27, 2015, 03:48:53 PM »
Good.

LaTex/ MathJax remark: avoid hard-coding. If you have equation which you want to refer to write
Code: [Select]
1+1=2\label{myeq}Then to refer to it write (\ref{myeq}).

The same label should not be defined more than once (but referred any number of times) and (a lesser mistake) the given environment (equation, multline, line in align or gather) should have just no more than one label. See more
https://en.wikibooks.org/wiki/LaTeX/Labels_and_Cross-referencing