(a)

(i) Compute the Fourier transform $\tilde{h}(k)$ of $h(x)=e^{-a|x|}$, where $a$ is a real positive constant.

(ii) Consider the boundary value problem

$$-\frac{d^{2} u}{d x^{2}}+\omega^{2} u=e^{-|x|} \quad \text { on }-\infty<x<\infty$$

with real constant $\omega \neq \pm 1$ and boundary condition $u(x) \rightarrow 0$ as $|x| \rightarrow \infty$.

Find the Fourier transform $\tilde{u}(k)$ of $u(x)$ and hence solve the boundary value problem. You should clearly state any properties of the Fourier transform that you use.

(b) Consider the wave equation

$$v_{t t}=v_{x x} \quad \text { on } \quad-\infty<x<\infty \text { and } t>0$$

with initial conditions

$$v(x, 0)=f(x) \quad v_{t}(x, 0)=g(x) .$$

Show that the Fourier transform $\tilde{v}(k, t)$ of the solution $v(x, t)$ with respect to the variable $x$ is given by

$$\tilde{v}(k, t)=\tilde{f}(k) \cos k t+\frac{\tilde{g}(k)}{k} \sin k t$$

where $\tilde{f}(k)$ and $\tilde{g}(k)$ are the Fourier transforms of the initial conditions. Starting from $\tilde{v}(k, t)$ derive d'Alembert's solution for the wave equation:

$$v(x, t)=\frac{1}{2}(f(x-t)+f(x+t))+\frac{1}{2} \int_{x-t}^{x+t} g(\xi) d \xi$$

You should state clearly any properties of the Fourier transform that you use.