(a) Find the Laplace transform of

$$y(t)=\frac{e^{-a^{2} / 4 t}}{\sqrt{\pi t}}$$

for $a \in \mathbb{R}, a \neq 0$.

[You may use without proof that

$$\left.\int_{0}^{\infty} \exp \left(-c^{2} x^{2}-\frac{c^{2}}{x^{2}}\right) d x=\frac{\sqrt{\pi}}{2|c|} e^{-2 c^{2}} .\right]$$

(b) By using the Laplace transform, show that the solution to

$$\begin{aligned} \frac{\partial^{2} u}{\partial x^{2}} &=\frac{\partial u}{\partial t} \quad-\infty<x<\infty, \quad t>0 \\ u(x, 0) &=f(x) \\ u(x, t) \quad \text { bounded, } \end{aligned}$$

can be written as

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

for some $K(|x-\xi|, t)$ to be determined.

[You may use without proof that a particular solution to

$$y^{\prime \prime}(x)-s y(x)+f(x)=0$$

is given by

$$\left.y(x)=\frac{e^{-\sqrt{s} x}}{2 \sqrt{s}} \int_{0}^{x} e^{\sqrt{s} \xi} f(\xi) d \xi-\frac{e^{\sqrt{s} x}}{2 \sqrt{s}} \int_{0}^{x} e^{-\sqrt{s} \xi} f(\xi) d \xi .\right]$$