The function $\theta(x, t)$ obeys the diffusion equation

$$\frac{\partial \theta}{\partial t}=D \frac{\partial^{2} \theta}{\partial x^{2}}$$

Verify that

$$\theta(x, t)=\frac{1}{\sqrt{t}} e^{-x^{2} / 4 D t}$$

is a solution of $(*)$, and by considering $\int_{-\infty}^{\infty} \theta(x, t) d x$, find the solution having the initial form $\theta(x, 0)=\delta(x)$ at $t=0$.

Find, in terms of the error function, the solution of $(*)$ having the initial form

$$\theta(x, 0)= \begin{cases}1, & |x| \leqslant 1 \\ 0, & |x|>1\end{cases}$$

Sketch a graph of this solution at various times $t \geqslant 0$.

[The error function is

$$\left.\operatorname{Erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-y^{2}} d y .\right]$$