Web bonus problem -- Week 2

[Jingxuan Zhang]

Differentiating Ioana's $(A)$ and equating it with his(her?) $(B)$, we have the symbolic system
$$\left( \begin{array}{cc|c}
1 & -1 & 2x\\
1+x& 1-x&3x^{2}\\
\end{array} \right)

\implies

\left( \begin{array}{c}
\varphi'(X)\\
\psi'(Y)
\end{array} \right)
=
\left( \begin{array}{c}
X\\
-Y
\end{array} \right)
\text{ where X, Y are the arguments of $\varphi, \psi$ resp. }
\implies

\left( \begin{array}{c}
\varphi(X)\\
\psi(Y)\end{array} \right)
=
\left( \begin{array}{c}
X^{2}/2+C_{1}\\
-Y^{2}/2+C_{2}
\end{array} \right)

\implies
u=(x+t)^{2}/2 - (x-t)^{2}/2 + const.
$$

Fixed now. For uniqueness we impose that the characteristics intersect the initial data, that is, precisely when
$$x^{2}-2x+2C, x^{2}+2x-2C$$
both have solution. This happens whenever
$$-t-1/2\leq x\leq t+1/2.$$

[Victor Ivrii]

Jingxuan
Misprint at the very end, correct it. However $\phi'(X)=X$ implies $\phi(x)=X^2/2+c$, etc. So you need to take it into account and find it from (B).

Just for fun, simplify

We need also answer the question, where this solution is uniquely determined.

[Ziyuan Wang]

Differentiating Ioana's $(A)$ and equating it with his(her?) $(B)$, we have the symbolic system
$$\left( \begin{array}{cc|c}
1 & -1 & 2x\\
1+x& 1-x&3x^{2}\\
\end{array} \right)

\implies

\left( \begin{array}{c}
\varphi'(X)\\
\psi'(Y)
\end{array} \right)
=
\left( \begin{array}{c}
X\\
-Y
\end{array} \right)
\text{ where X, Y are the arguments of $\varphi, \psi$ resp. }
\implies

\left( \begin{array}{c}
\varphi(X)\\
\psi(Y)\end{array} \right)
=
\left( \begin{array}{c}
X^{2}/2+C_{1}\\
-Y^{2}/2+C_{2}
\end{array} \right)

\implies
u=(x+t)^{2}/2 - (x-t)^{2}/2 + Cx+D.
$$

The last two terms in the final step are empirical, and I urgently seek a theoretical account for it.

u=(x+t)²/2-(x-t)²/2, where u is define on t>=0 and x is any real number.

[Ioana Nedelcu]

Simplifying further, we get $$ u = 2xt + C $$ but using the second initial condition implies C = 0 so the solution is $ u(x,t) = 2xt $ (thanks for going over this in class!)