Partial Differential Equations on Unbounded Domains
In our study of PDEs we noted the differences among the types of equations: parabolic, hyperbolic, and elliptic. Those classifications dictate the types of initial and boundary conditions that should be imposed to obtain a well-posed problem. There is yet another division that makes one method of solution preferable over another, namely, the nature and extent of the spatial domain. Spatial domains may be bounded, like a bounded interval, or unbounded, like the entire set of real numbers. It is a matter of preference which type of domain is studied first. It seems that boundaries in a problem, which require boundary conditions, make a problem more difficult. Therefore, we first investigate problems defined on unbounded domains.