TT1 Problem 2

[Victor Ivrii]

Solve the wave equation with the following initial conditions
\begin{equation*}
\left\{\begin{aligned}
& u_{tt}- 4 u_{xx}= x t e^{-x^2} ,\qquad&& -\infty <x< \infty\\
&u (x,0) = e^{-x^2} ,\\
&u_t(x,0)= x e^{-x^2}\end{aligned}\right.
\end{equation*}

[Jessica Chen]

I don't know how to attach a pdf properly such that we can see its preview without download it.

You cannot. So scan to png or jpg. V.I.

[Biao Zhang]

To solve the question by using D'Alemeber formula:\\
\begin{multline*}
u(x,t)= \underbrace{\frac{1}{2}\bigl( G(x+ct)+G(x-ct) \bigr)}_1
+ \underbrace{\frac{1}{2c}\int_{x-ct} ^{x+ct} H(x')\,dx'}_2 +
\underbrace{\frac{1}{2c} \int_0^t \int _{x-c(t-t')} ^{x+c(t-t')} F(x',t')\, dx' dt'}_3
\end{multline*}
we have
\begin{align*}
c=2\\
G(x)=e^{-x^2}\\
H(x)=xe^{-x^2}\\
F(x,t)=xte^{-x^2}
\end{align*}
\begin{align*}
(1): \frac{1}{2}(e^{-(x+2t)^2}+e^{-(x-2t)^2})\\
(2): \frac{1}{4}\int_{x-2t} ^{x+2t} H(x')\,dx'\\
\hspace{1cm} =\frac{1}{8}(e^{-(x-2t)^2}-e^{-(x+2t)^2})
\end{align*}
combine (1) and (2) we have:
\begin{equation*}
\frac{3}{8}e^{-(x+2t)^2} +\frac{5}{8}e^{-(x-2t)^2}
\end{equation*}
now we need find (3):
\begin{align*}
\frac{1}{4} \int_0^t \int _{x-2(t-t')} ^{x+2(t-t')} x't'e^{-x'^2}\, dx' dt'\\
\frac{1}{8}  \int_0^t t'(e^{-(x+2t-2t')^2}-e^{-(x-2t+2t')^2})\, dt'
\end{align*}
This part i don't know how to integrate this part . but answer should be:
\begin{equation*}
\frac{3}{8}e^{-(x+2t)^2} +\frac{5}{8}e^{-(x-2t)^2}+\frac{1}{8}  \int_0^t t'(e^{-(x+2t-2t')^2}-e^{-(x-2t+2t')^2})\, dt'
\end{equation*}

[Victor Ivrii]

OK.

I graded this problem.

Remark. The last integral is non-elementary but changing variable we will get something like $\int_X^Y  (A+B s) e^{-s^2/2}\,ds$ where one part is integrable and another could be expressed through $\operatorname{erf}$. But this is optional. I have not punished those who did not the last step and marginally punished those who did incorrectly.

Consider the last term obtained by Biao (which is contribution of the RHE after we integrated by $x'$), but Biao erred with sign
\begin{equation}
\frac{1}{8}\int_0^t  s\Bigl( e^{-(x-2t+2s)^2}  -  e^{-(x+2t-2s)^2} \Bigr)\, ds=
F(x,t,2)-F(x,t,-2)
\label{a}
\end{equation}
with
\begin{equation}
F(x,t,k)= \frac{1}{8}\int_0^t  s e^{-(x-k t+ks)^2}   \, ds=
\frac{1}{8k^2}\int_{x-kt}^{x}  (x-kt+y)  e^{-y^2}\,dy
\label{b}
\end{equation}
where we made a substitution $y=x-kt+ks$.Then
\begin{equation*}
F(x,t,k)=
 \frac{1}{8k^2}\biggl[(x-kt) \int_{x-kt}^x   e^{-y^2}\,dy - \int_{x-kt}^x  y  e^{-y^2}\,dy\biggr];
\end{equation*}
 then (\ref{a}) equals
\begin{gather*}
 \frac{1}{32}
\biggl[(x-2t) \int_{x-2t}^x e^{-y^2}\,dy - (x+2t) \int_{x+2t}^x   e^{-y^2}\,dy
-\int_{x-2t}^x  y  e^{-y^2}\,dy + \int_{x+2t}^x  y  e^{-y^2}\,dy\biggr]\\[4pt]
=\frac{1}{32}\biggl[ (x-2t) \bigl( E(x)-E(x-2t)\bigr)- (x+2t) \bigl( E(x)-E(x+2t)\bigr) \biggr]+
\frac{1}{64}\biggl[ e^{-(x+2t)^2}-e^{-(x-2t)^2}\biggr]
\end{gather*}
with $E(z)=\int_0^z e^{-y^2}\,dy$ which could be expressed via $\operatorname{erf}$ easily.