Use of the numerical solution on finite interval to describe the behavior of PDE on $\Bbb R$

Question

When we want to simulate the solution of some one dimensional PDE $$ \mathcal{L}(u) = f,\quad \text{on } \mathbb{R} $$ on the real line, why do we use the solution in finite interval $[a,b]$ to describe the behavior of the PDE on $\Bbb R$?

Answer 1

One could ask the same question for an ordinary differential equation $\dot y=f(y,t)$. Most of the numerical methods proceed by iterating some algorithm in time. Thus, they provide only finite-time solutions (which are already very useful). A few numerical methods provide long-time solutions, such as the harmonic balance method (HBM). However, speaking of the HBM, all the transient information around $t=0$ is lost with this method. Depending on the purpose of the study, one may be interested in the short-time behavior, or rather in the long-time behavior.