# 6 Well Posed PDE Problems     continued...

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.