More specific guidelines can be stated for second order linear PDE problems.

- Well posed elliptic PDE problems usually take the form of a
boundary value problem (BVP) with the PDE required to hold
on the interior of some region and the solution required to
satisfy a single boundary condition (BC) at each point on
the boundary of the region. Typical boundary conditions are:
- Dirichlet BC - the solution value is specified on the boundary
- Neumann BC - the normal derivative of the solution is specified on the boundary
- Robin BC - a linear combination of the solution and its normal derivative is specified on the boundary.

- Well posed parabolic PDE problems usually
involve one or more spatial variables and a time variable as
well. Parabolic PDE models often arise in connection with
evolutionary systems in which the flux of some material
quantity is "down gradient" with respect to a field
variable. Typically, a well posed parabolic problem requires
the same boundary conditions on the spatial variables as in
the case of elliptic problems. In addition an initial
condition specifying the state of the system at time
**t = 0**is required. Thus, a well posed second order parabolic PDE problem usually takes the form of and initial boundary value problem (IBVP). - Well posed, second order, hyperbolic PDE
problems also require the same boundary conditions as
elliptic problems. Usually second order, hyperbolic PDE
model arise in connection with physical problems involving
wave motion, vibration or oscillation. In these problems,
two initial conditions at time
**t = 0**are required (one to describe the initial state of the system and another to describe the initial velocity).

A discussion of the well posedness of PDE problems involving systems of first order equations requires an understanding of the characteristic curves associated with such systems. Systems of first order equations are very important in the field of computational science, but are not dealt with here, since the remainder of this chapter focus on second order PDEs. To conclude this section, several examples of well posed and ill posed second order PDE problems are presented.