Consider the initial-boundary value problem

$$\begin{gathered} \frac{\partial u}{\partial t}=\frac{\partial^{2} u}{\partial x^{2}}, \quad 0<x<\infty, \quad t>0 \\ u(x, 0)=x e^{-x}, \quad 0 \leqslant x<\infty \\ u(0, t)=\sin t, \quad t \geqslant 0 \end{gathered}$$

where $u$ vanishes sufficiently fast for all $t$ as $x \rightarrow \infty$.

(i) Express the solution as an integral (which you should not evaluate) in the complex $k$-plane

(ii) Explain how to use appropriate contour deformation so that the relevant integrand decays exponentially as $|k| \rightarrow \infty$.