I'll add my typed solutions.

(a) The ordinary heat equation is: \begin{equation}

u_t = ku_{xx} \end{equation}

Now consider $U(x,t) = u(x+ct,t)$. The partial derivatives needed for the heat equation are given by: \begin{equation}

\begin{cases}

U_t = cu_x + u_t \\

U_x = u_x \\

U_{xx} = u_{xx} \end{cases} \end{equation}

Now let's plug these into the heat equation: \begin{equation}

U_t = kU_{xx} \longrightarrow (u_t + cu_x) = k(u_{xx}) \longrightarrow u_t + cu_x = ku_{xx} \end{equation}

Therefore the heat equation with a convection term is obtained from the ordinary heat equation with a change of variables.

(b) Now we use the change of variabes $u(x,t) = U(x-ct,t)$ *(Note: Thank you Rong Wei for pointing out the fact that this should be c here, as I was unable to solve the problem otherwise!)*. Now the partial derivatives are given by: \begin{equation} \begin{cases}

u_t = -cU_x + U_t \\

u_x = U_x \\

u_{xx} = U_{xx} \end{cases} \end{equation}

Let's plug this into the heat equation with a convective term: \begin{equation}

u_t + cu_x = ku_{xx} \longrightarrow (-cU_x + U_t) + c(U_x) = k(U_{xx}) \longrightarrow -cU_x + cU_x + U_t = kU_{xx} \longrightarrow U_t = kU_{xx}

\end{equation}

So with this change of variables, the equation reduces to the familiar heat equation.

We can then use the usual formula to arrive at the solution: \begin{equation}

u(x,t) = \int_0^{\infty}G(x-ct,y,t)g(y)dy = \frac{1}{2\sqrt{\pi{}kt}}\int_0^{\infty}e^{-(x-ct-y)^2/(4kt)}g(y)dy \end{equation}

(c) I don't think we can use the method of continuation directly to solve IVBP with Dirichlet or Neumann boundary conditions as $x>0$ for the heat equation with a convection term on ${x>0, t>0}$. This is because we used a change of variables to define the problem.

(d)We have \begin{equation}

u(x,t) = v(x,t)e^{\alpha{}x + \beta{}t} \end{equation}

The partial derivatives are given by: \begin{equation} \begin{cases}

u_t = v_te^{\alpha{}x + \beta{}t} + \beta{}ve^{\alpha{}x + \beta{}t} \\

u_x = v_xe^{\alpha{}x + \beta{}t} + \alpha{}ve^{\alpha{}x + \beta{}t} \\

u_{xx} = v_{xx}e^{\alpha{}x + \beta{}t} + 2\alpha{}v_xe^{\alpha{}x + \beta{}t} + \alpha{}^2ve^{\alpha{}x + \beta{}t} \end{cases} \end{equation}

Let's plug them in to the heat equation with a convection term. \begin{equation}

v_te^{\alpha{}x + \beta{}t} + \beta{}ve^{\alpha{}x + \beta{}t} + cv_xe^{\alpha{}x + \beta{}t} + c\alpha{}ve^{\alpha{}x + \beta{}t} = ke^{\alpha{}x + \beta{}t}(v_{xx} + 2\alpha{}v_x + \alpha{}^2v) \end{equation}

Gathering like terms, we arrive at: \begin{equation}

v_t + v(\beta{} + c\alpha{} - k\alpha{}^2) + v_x(c-2k\alpha{}) = kv_{xx} \end{equation}

For this to reduce to the heat equation, we want the coefficients in front of $v$ and $v_x$ to be zero. So: \begin{equation} \begin{cases}

\beta{} + c\alpha{} - k\alpha{}^2 \\

c-2k\alpha{} \end{cases} \end{equation}

Solving first for $\alpha{}$, we get that $\alpha{} = \frac{c}{2k}$. We can then plug this into the equation for $\beta{}$ and solve to get that $\beta{} = \frac{-c^2}{4k}$.

(e) ~~I will work on this part in a little while. So far I think I have gotten all of the same solutions as Rong Wei.~~ Added my solution below.

For the case of the half-line and Dirichlet boundary condition, we will have the solution: \begin{equation}

u(x,t) = \frac{e^{\alpha{}x + \beta{}t}}{2\sqrt{\pi{}kt}}\int_0^{\infty}[e^{-(x-y)^2/4kt} - e^{-(x+y)^2/4kt}]g(y)dy \end{equation}

In the case of Neumann boundary conditions, we cannot use a similar method.