The temperature $T(x, t)$ in a semi-infinite bar $(0 \leqslant x<\infty)$ satisfies the heat equation

$$\frac{\partial T}{\partial t}=\kappa \frac{\partial^{2} T}{\partial x^{2}}, \quad \text { for } x>0 \text { and } t>0$$

where $\kappa$ is a positive constant.

For $t<0$, the bar is at zero temperature. For $t \geqslant 0$, the temperature is subject to the boundary conditions

$$T(0, t)=a\left(1-e^{-b t}\right),$$

where $a$ and $b$ are positive constants, and $T(x, t) \rightarrow 0$ as $x \rightarrow \infty$.

(a) Show that the Laplace transform of $T(x, t)$ with respect to $t$ takes the form

$$\hat{T}(x, p)=\hat{f}(p) e^{-x \sqrt{p / \kappa}}$$

and find $\hat{f}(p)$. Hence write $\hat{T}(x, p)$ in terms of $a, b, \kappa, p$ and $x$.

(b) By performing the inverse Laplace transform using contour integration, show that for $t \geqslant 0$

$$T(x, t)=a\left[1-e^{-b t} \cos \left(\sqrt{\frac{b}{\kappa}} x\right)\right]+\frac{2 a b}{\pi} \mathcal{P} \int_{0}^{\infty} \frac{e^{-v^{2} t} \sin (x v / \sqrt{\kappa})}{v\left(v^{2}-b\right)} d v$$