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).
For Tuesday read Section 1.1 and give problem 4 on page 23 of MCOWEN a try. Bring questions about your attempts to class and we will use these to guide the discussion.
The important steps that are involved in using the method of characteristics are:
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:
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.
Related questions that you should resolve for yourself as you read:
You should also have thought about how to apply the ideas above to problems in 3, 4, etc. 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 usuing characteristics
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:
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
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.
