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I am working on a homework problem from Guenther & Lee (section 5.3 #4 for anyone interested). I am asked to find a Green's function for

$u_t = au_{xx} + bu_x + cu, \quad 0\leq x \leq L, \quad t>0,\\ u(0,t) = u(L,t) = 0,\quad t>0,\\ u(x,0) = f(x),\quad 0\leq x \leq L.$

Previously we have used Green's Functions for solving the heat equation, but the derivation of the Green's function for the heat equation was given in terms of the Fourier Solution to the heat equation. So, should I find a Fourier integral solution first and then transform it into a Green's function? If so, how is this useful? It seems as though the Green's function is not helpful for solving the PDE then. I think that in general I am having difficulty understanding when to use Green's functions and how they are helpful, any insight is appreciated.

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