Show that for each $t>0$ and $x \in \mathbb{R}$ the function

$$K(x, t)=\frac{1}{\sqrt{4 \pi t}} \exp \left(-\frac{x^{2}}{4 t}\right)$$

satisfies the heat equation

$$\frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}$$

For $t>0$ and $x \in \mathbb{R}$ define the function $u=u(x, t)$ by the integral

$$u(x, t)=\int_{-\infty}^{\infty} K(x-y, t) f(y) d y$$

Show that $u$ satisfies the heat equation and $\lim _{t \rightarrow 0^{+}} u(x, t)=f(x)$. [Hint: You may find it helpful to consider the substitution $Y=(x-y) / \sqrt{4 t}$.]

Burgers' equation is

$$\frac{\partial w}{\partial t}+w \frac{\partial w}{\partial x}=\frac{\partial^{2} w}{\partial x^{2}}$$

By considering the transformation

$$w(x, t)=-2 \frac{1}{u} \frac{\partial u}{\partial x}$$

solve Burgers' equation with the initial condition $\lim _{t \rightarrow 0^{+}} w(x, t)=g(x)$.