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.
The kind of boundary condition
can vary from point to point on the boundary, but at any
given point only one BC can be specified. Physically a
Dirichlet BC usually corresponds to setting the value of a
field variable, such as temperature; a Neumann BC usually
specifies a flux condition on the boundary; and a Robin BC
typically represents a radiation condition. When the region
on which the PDE problem is posed is unbounded, one or more
of the above boundary conditions is usually replaced by a
growth condition that limits the behavior of the solution
"at infinity".
- 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.
