I want to solve the the one-dimensional heat equation in an infinite rod with the initial condition that all the heat is initially concentrated at $x=0$, so $u(x,0)=\delta(x)$. The boundary condition is $u(\infty,t) = 0$
In the first 3 pages of the following paper, the one-dimensional heat equation in an infinite rod is solved by separation of variables.
It is specifically mentioned in the first line of the second page that $u \to 0$ ad $x \to \infty$
This does not seem to be consistent with the fact that $$X(x) = A \cos (\sqrt{\lambda}x) + B \sin (\sqrt{\lambda}x) \ne 0 $$
as $\ x \to \infty $
If I however apply my initial condition to the final result (equation 8 in the paper) then I obtain a Gaussian type function that does have the right boundary conditions: $u \to 0$ as $|x| \to \infty$.
Other threads on this forum have indicated that this problem should be solved with the Fourier transform method.
My question then becomes: How does this paper seem to get the right final answer through separation of variables, even with the boundary condition $u(\infty,t) = 0$? The solution in the paper for $X(x)$ does not satisfy the boundary condition, but the final solution does. It seems like this paper is cutting some corners in a mathematically less rigorous way to obtain the right answer, but I can't find out exactly where.