Well Posed Initial-Value Problem

For a proper authoritative definition of ``well posed'' in the field of finite difference schemes, see, e.g., [495]. The definition we will use here is less general in that it excludes amplitude growth from initial conditions which is faster than polynomial in time.

We will say that an initial-value problem is well posed if the linear system defined by the PDE, together with any bounded initial conditions is marginally stable.

As discussed in [460], a system is defined to be stable when its response to bounded initial conditions approaches zero as time goes to infinity. If the response fails to approach zero but does not exponentially grow over time (the lossless case), it is called marginally stable.

In the literature on finite difference schemes, lossless systems are classified as stable [495]. However, in this book series, lossless systems are not considered stable, but only marginally stable.

When marginally stable systems are allowed, it is necessary to accommodate polynomial growth with respect to time. As is well known in linear systems theory, repeated poles can yield polynomial growth [460]. A very simple example is the ordinary differential equation (ODE)

which, given the initial condition , has solutions