As we know, solution to

\begin{align}

&u_t -u_{xx}=0, \label{A}\\

&u|_{t=0}=\delta(x)\label{B}

\end{align}

is

\begin{equation}

\frac{1}{\sqrt{4\pi t}}e^{-x^2/4t}

\label{C}

\end{equation}

where initial condition currently is understood as $u(x,t)\to 0$ as $t\to +0$, $x\ne 0$; $u(0,t)\to \infty$ as $t\to +0$ and $\int_{-\infty}^\infty u(x,t)\,dx=1$.

**a.** Consider 1D "radioactive cloud" problem:

\begin{align}

&u_t + v u_x-u_{xx}+\beta u=0, \label{D}\\

&u|_{t=0}=\delta(x)\label{E},

\end{align}

where $v$ is a wind velocity, $\beta$ shows the speed of "dropping on the ground".

*Hint:* Reduce to (\ref{A})--(\ref{B}) by $u= ve^{-\beta t}$ and $x=y+vt$, use (\ref{C}) for $v$ and then write down $u(x,t)$.

**b.** Find "contamination level" at $x$

\begin{equation}

D(x)=\beta \int _0^\infty u(x,t)\,dt.

\label{F}

\end{equation}

*Hint:* by change of variables $t= y^2$ with appropriate $z$ reduce to calculation of

\begin{equation}

\int \exp(-ay^2-by^{-2})\,dy

\label{G}

\end{equation}

and calculate it using f.e.

https://www.wolframalpha.com/ with input

`int_0^infty exp (-ay^2-b/y^2)dy`

(you may need to do it few times)

**c.** Later