Notes
1.1 Cauchy Problem for Quasilinear Equations
(see also JOHN pp 1-17; 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 solution of the differential equation) as a family of lines, and then using the PDE to generate a system of ODEs for finding these lines. The last step in the method is an inversion process where one changes from the parametric representation by a collection of lines to a representation as a function (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 line, 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 values 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, i.e. the first example we did in class required that u(0,y) = 1 + y2 for 1 < y < 2. The inequality here needs to 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.
These three steps basically define the region in the x vs y where the solution is valid.
The main theory that lies behind the method of characteristics (we will cover this in class) gives conditions on the data which ensure that the inversion is possible. Hence these conditions help with the third step above, unless you can analyse conditions for the inversion explicitly.
An alternate solution thechnique is Lagrange's method. It is useful because it delivers a general solution, i.e. one containing an arbitrary function. However, it may not capture all solutions, and it's usefulness is predicated to some extent on ingenuity in finding functions f(x,y,z) that are constant along characteristics.
Related questions that you should resolve for yourself as you read:
- how general is the method of characteristics? i.e. can any 1st order PDE be treated?
- how general is Lagrange's method?
- what are semilinear and quasilinear equations, and why are they singled out for special treatment?
You should also have thought about how to apply the ideas above to problems in 3, 4, etc. independent variables.
1.2 Weak Solutions for Quasilinear Equations
The principal goal of this section is to introduce the idea of a "weak" solution. This can mean a variety of things, but in this first exposure it means a solution of an integrated form of the PDE. This is significant because, generally speaking, an integrated form of a PDE involves fewer derivates of the unknown u, and hence can be satisfied with a "less smooth" function u.
To keep things simple, the discussion in this section is centered around equations that are what the author refers to a "conservation laws". There has been a good deal of research on such equations in the past 30 years, and it is generally known that this research has been concerned with (potentially) non-smooth solutions. Moreover, the prototypical conservation law in continuum mechanics arises in the theory of gas dynamics (u there usually represents the gas density or a component of the fluid velocity) and one is interested in this case with shock wave ("waves" that exibit discontinuities in u) and acceleration waves ("waves" in which u is continuous but its first derivatives are not continuous). Bottom line: here is a physical applications area where discontinuous solutions are the norm rather than the exception.
In reading MCOWEN and following my lectures you should come to realize that what we are considering here is the idea of "patching together" solutions which are smooth in their regions of definitions but can "fit together" only in a discontinuous way. And in studying such a problem there are four steps:
- determine a point at which the data for the problem is not smooth
- find the smooth solution ul to the left of the non-smooth point (using the method of charateristics)
- find the smooth solution ur to the right of the non-smooth point (using the method of charateristics)
- find the curve where these two solutions intersect (see the text) so that the integrated form of the equation is satisfied. OR, fill the region between them with a "fan". This last step depends on whether the characteristics in the two smooth regions are either "approaching" each other or "departing" from each other.
In a more general problem there may be multiple regions to worry about, especially if there are multiple points where the initial data is not smooth. As time (i.e. y) proceeds, some of the lines along which discontinuities exit may disappear, thereby giving rise to smoother (though not fully classical) solutions!
One of the things you should ask yourself in this section is: how general is the discussion? For example, can the same type of discussion be done if (say) the function G(u) also depended explicitly on x? Etc.
For practice try the following problems: MCOWEN p 28 no 2,3,4,5,6,8,9 and JOHN p 19 no 4,6
1.4 Concluding Remarks on First-Order Equations
This last section is a direct lead in to the classification theorems in Chapter 2 for second order equations. Our main focus here is on semilinear 1st order equations (since in Chapter 2 we consider only semilinear 2nd order problems). And the tact that we take is to ask: under what conditions on a curve in the x y plane can we compute the solution and all its derivatives along that curve. The point is this. If we can compute all these derivatives, then we can form a Taylor series that will serve as a candidate for the solution, and, with good luck (to be justified in Chapter 2) we will be able to show this series defines a smooth solution of our differential equation.
What we find is that the curve must not be a characteristic to do this. Turned around, we conclude that if a solution of a 1st order semilinear equations is going to be non-smooth as one crosses a curve, that curve must be a characteristic!
One last thing we did here was to introduce changes of variables. Specifically, since characteristics are so special to a 1st order equation, there may be some simplifications that occur if we switch to a new set of variables in which the characteristics become straight lines. Indeed, we saw that by doing this the PDE became a one parameter family of ODEs and hence was amenable to solution using methods from ordinary differential equations.