Toronto Math Forum
APM346-2015F => APM346--Lectures => Web Bonus = Sept => Topic started by: Victor Ivrii on September 26, 2015, 12:43:13 PM
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http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter2/S2.3.P.html#problem-2.3.P.5 (http://www.math.toronto.edu/courses/apm346h1/20159/PDE-textbook/Chapter2/S2.3.P.html#problem-2.3.P.5) Problem 5
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Firstly, Let's use characteristic coordinates \begin{equation}\left\{\begin{aligned}&\xi=x+ct,\\&\eta=x-ct.\end{aligned}\right.\label{eq1}\end{equation} 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 \begin{equation}u=\phi(\xi)+\psi(\eta)\label{5}\end{equation} is the general solution to (2). Then \begin{equation}u|_{\xi=0}=\phi(0)+\psi(\eta)=g(t)\end{equation} \begin{equation}u|_{\eta=0}=\phi(\xi)+\psi(0)=h(t)\end{equation} Thus by (5)(6)(7) \begin{equation}u=g(t)+h(t)-\phi(0)-\psi(0)\end{equation} Since we have \begin{equation}u(0,0)=\phi(0)+\psi(0)\end{equation} \begin{equation}u(0,0)=h(0)=g(0)\end{equation} Then \begin{equation}u=g(t)+h(t)-g(0)\end{equation} solves Goursat problem.
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Good.
LaTex/ MathJax remark: avoid hard-coding. If you have equation which you want to refer to write
\begin{equation}
1+1=2
\label{myeq}
\end{equation}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 (https://en.wikibooks.org/wiki/LaTeX/Labels_and_Cross-referencing)