Math 553 Partial Differential Equations

Professor Richard S. Laugesen

Section 1.1
Cauchy Problem for Quasilinear Equations

(see also JOHN pp 1-17; Z & T chp 3; EPSTEIN pp 28-35; CARRIER & PEARSON chp 6)

The heart of solving first order PDEs is the method of characteristics. It is a way of representing a surface (the graph of the solution of the differential equation) as a family of paths, and then using the PDE to generate a system of ODEs for finding these paths. The last step in the method is an inversion process where one changes from the parametric representation of the surface to a representation as a function of the original independent variables (if this is possible).

The important steps that are involved in using the method of characteristics are:

• parametrizing the data for the problem (i.e. the value of u(x,y) along some curve, often the statement of an initial condition)
• solving the system of ODEs and applying the data as initial conditions (this determines the characteristics of the problem, and hence a parametric representation of the solution surface)
• inverting the parametrization to get back to a representation of the solution in terms of x and y variables rather than parameters

As you do this type of calculation, you need to ask yourself about the region where the solution is defined. There are generally three things you need to look at in this regard:

• there may be restrictions imposed through the statement of the data, e.g., if the initial data is u(x,0) = 1 + x² for 1 < x < 2 (so that Gamma is a short piece of curve, rather than being infinite in extent) then the restriction 1 < s < 2 should be carried through to the end to see how it affects where the solution is defined.
• the theoretical justification (a theorem!) for the method of characteristics requires that the coefficients in the PDE (and the functions giving the data) be continously differentiable. The region of existence must be chosen so that these assumptions are true. So if the coefficients have, for example, points of discontinuity, these points need to be taken into account in determining the region of existence.
• finally, the parametrization must be inverted. Only in a region where this is possible will the solution exist. Perhaps you can accomplish this inversion explicitly, but in any case inversion is theoretically possible (at least locally) provided the change-of-variables Jacobian matrix d(x,y)/d(s,t) is nonsingular, which is the meaning of the determinant condition (8) on p. 16.

These three considerations basically tell you the region in the xy-plane where the solution is valid.

Here are some related questions that you should resolve for yourself as you read:

• how general is the method of characteristics? i.e. can any quasilinear 1st order PDE be treated?
• in what way does the method simplify for semilinear equations?

You should also have thought about how to apply the method of characteristics to problems in 3, 4, ..., n independent variables.

There are a good number of problems in MCOWEN that you can do for skill building: p 23 no 4, 5, 6, 7 (each has several parts). In addition JOHN p 18 no 1 has several good problems that can be solved using characteristics.