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(Sorry for the messy title, trying to include the keypoints of the problem.)

I am new to the theory on how to solve this kind of PDE problem which is presented below; I am unsure on which method to use, I am currently thinking of making Fourier transform and take it from there. Now, the full problem description is

[The] temperature inside a quantum wire is given by the solution to the following PDE:

$\left\{\begin{array}{ll}u_{t}(x,t)-vu_{x}(x,t)-\alpha u_{xx}(x,t)=Ae^{-x^2/L^2}\delta(t-t_0)\quad (-\infty<x<\infty,\quad t>0)\\u(x,0)=u_0e^{-x^2/L^2}\quad (-\infty<x<\infty)\end{array}\right.$

where $v>0$, $\alpha>0$, $A>0$, $L>0$, $t_0>0$ and $u_0>0$ are constants.

Calculate the Greenfunction for the system above! All integrals must be calculated and the Greenfunction must be defined appropriately.

Edit (forgot the question)

Now, my question is if I may Fourier transform this directly to obtain the Green's function, or do I have to homogenize the equation first? (More generally, how do I continue from here?) I know the solution can be written as a series of Green's functions, but the assignment forces me to make the complete derivation of the Green function, so I cannot guess the solution.

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Sketch: For simplicity, let us just consider \begin{align} u_t-u_x-u_{xx}= e^{-x^2}\delta(t-t_0). \end{align} Observe, if you set $v(t, x) = e^{-t/4+x/2}u(t, x)$, then we see that \begin{align} v_t-v_{xx} = e^{-t/4+x/2}(u_t-u_x-u_{xx}) = e^{-t/4+x/2-x^2}\delta(t-t_0). \end{align}

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