6 Well Posed PDE Problems

In the previous sections we saw some examples of partial differential equations. We now consider some important issues regarding the formulation and solvability of PDE problems. A solution to a PDE can be described as simply a function that reduces that PDE to an identity on some region of the independent variables. In general, a PDE alone, without any auxiliary boundary or initial conditions, will either have an infinity of solutions, or have no solution. Thus, in formulating a PDE problem there are three components: (i) the PDE; (ii) the region of space-time on which the PDE is required to be satisfied; (iii) the auxiliary boundary and initial conditions to be met.

For a PDE based mathematical model of a physical system to give useful results, it is generally necessary to formulate that model as what mathematicians call a well posed PDE problem. A PDE problem is said to be well posed if

1.

a solution to the problem exists

2.

the solution is unique, and

3.

the solution depends continuously on the problem data.

If one of these conditions is not satisfied, the PDE problem is said to be ill-posed. In practice, the question of whether a PDE problem is well posed can be difficult to settle. Roughly speaking the following guidelines apply:

• The auxiliary conditions imposed must not be too many or a solution will not exist.
• The auxiliary conditions imposed must not be too few or the solution will not be unique.
• The kind of auxiliary conditions must be correctly matched to the type of the PDE or the solution will not depend continuously on the data.

• 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 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 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.

Laplace's equation on the rectangular region , subject to the Dirichlet boundary conditions

is well posed. For the case of these example boundary conditions, one can show that the unique solution to this BVP is . If any one of the four boundary conditions is deleted, then the problem becomes ill-posed, because is would then admit multiple solutions. If a second, independent Dirichlet condition were added on any part of the boundary, the problem would again be ill-posed, in this case due to lack of existence of a solution. More generally, if two, independent boundary conditions are imposed on any part of the boundary of the region, then the problem will fail to have a solution.

To illustrate that boundary value problems, not initial value problems, are the appropriate setting for elliptic PDE problems, we present the following example due to Hadamard. To view this problem as an initial value problem, one should think of y as a time variable. Consider the initial value problem

For and , it is clear that the corresponding solution to the above initial value problem is . For the case and , it is easy to verify that the corresponding solution is

Observe that the functions and are identical and that

uniformly in . Thus, we see that the data of the two problems, , and , , can be made arbitrarily close. But, if we compare the two solutions at , then we obtain

For positive, approaches infinity faster than , as goes to infinity. Therefore, we conclude that

illustrating that as the data for the two problems becomes more alike, the solutions become increasingly different. This is what is meant by failure of the solution to depend continuously on the problem data.

The following IBVP for the diffusion equation in one space variable is an example of a well posed parabolic PDE problem for .

One physical interpretation of this problem is that is the temperature at position and time in a one dimensional heat conducting medium (say a metal rod, for example) with thermal diffusivity . The initial condition, , specifies the temperature in the rod at the assigned time . The boundary conditions, and state that the ends of the rod are held at temperature zero for all time.

Simple problems such as this make excellent validation tools for the computational scientist. Since the exact solution to this IBVP can be shown (by separation of variables and Fourier series methods) to be

where

One can use this exact solution to test the results of a computer code.

The classic example of an ill-posed parabolic PDE problem is the "backward-in-time heat equation".

Here, if we think of as the temperature in a one dimensional heat conduction rod, the condition can be thought of as giving the temperature distribution at some specific time . The PDE problem calls for using this information, together with the heat balance equation and the boundary conditions to predict the temperature distribution at some earlier time, sat . It can be shown (see Schaum's Outline of PDE, solved problem 4.9) that if is not infinitely continuously differentiable, then no solution to the problem exists. If is infinitely continuously differentiable, then it is shown that the solution on does not depend continuously on the data, namely .

For second order hyperbolic PDE problems, the vibrating string is most frequently used as an example of a well posed problem. Think of as representing the vertical displacement at position and time of an ideal string which in static equilibrium occupies the horizontal line joining and . Then the following IBVP models the movement of the string subject to an initial displacement given by and an initial velocity given by .

A dramatic example of an ill-posed, second order hyperbolic PDE problem is given by the following BVP for the one dimensional wave equation. It can be shown that if is irrational, then the only solution of this BVP for the wave equation is u identically zero; whereas if is rational, the problem has infinitely many nontrivial solution. Thus the solution fails to depend continuously on the data - namely on the size of the region on which the problem is stated.