Heat and Wave equation - Green's function versus Fourier series?

Question

I am learning how to solve the heat and wave equation in bounded domains in 1 and 2D as well as in $\mathbb{R}$ and $\mathbb{R}^2$. In the latter case I have learned the representation formulas i.e. heat kernel/Green's function and d'Alembert's formula. I do not understand why the bounded domain case (where I am using Fourier series and separation of variables) is so different from the solutions in the whole space. I have tried solving the heat equation on $\Omega=[a,b] \subset \mathbb{R}$ for $a<b$ and taking the limit as $a\rightarrow -\infty$ and $b\rightarrow \infty$ but I am not getting anywhere. Can someone explain to me why we need to different methodologies for bounded domain case and the whole space for both the heat and wave equation?