Date: 28.1.1999
Janne Viljas, 44468K

Session 2, small projects:
The method of characteristics

First-order quasi-linear PDE

The method of characteristics, in its simplest form, concerns equations of the form 
a(x,t,f) f_x(x,t) + b(x,t,f) f_t(x,t) = c(x,t,f)                     (1)
which is a general quasi-linear first-order partial differential equation in two variables.
I will mainly consider the linear case 
a(x,t) f_x(x,t) + b(x,t) f_t(x,t) + c(x,t) f(x,t) = d(x,t)           (2)
or even its further simplification with c = 0 and/or d = 0.

The method

To simplify things, take first as a starting point the linear, homogeneous case with c = 0, d = 0 
a(x,t) f_x(x,t) + b(x,t) f_t(x,t) = 0                                (3)
Consider a curve in the xt-plane, parametrized with p 
x=x(p)
t=t(p)
Substitute these for x and t in the unknown function f(x,t) and define a function F of p by 
F(p) := f(x(p),t(p))
Differentiate this with respect to p and use the chain rule to get 
dF       dx       dt
-- = f_x -- + f_t --
dp       dp       dp
Comparing this with the equation (3) we see that on curves (x(p),t(p)) which satisfy 
dx
-- = a(x,t)
dp
                                                                     (4)
dt
-- = b(x,t)
dp
the derivative dF(p)/dp = 0, i.e. F and thus the solution f(x,t) has a constant value.
The family of curves defined by (4) are called the characteristics of equation (3).
Getting rid of the parameter we obtain for them the necessary condition 
dt   b(x,t)
-- = ------                                                          (5)
dx   a(x,t)
Summing up, the solution of equation (3) is that

f(x,t) = const. on the characteristics given by (5).

Depending on the equation (1-3) and initial conditions imposed on it, the solution may be restricted to just some part of the (x,t)-space. Note that no two characteristics can ever cross each other, if the solution is to be well-behaving. A crossover leads inevitably to a discontinuity. Many real-world "physical" solutions require this possibility, however! I'll try to clear these things out with some examples.

It would not be very hard to derive the prescription for the general equation (1), but I'll just quote the result. (See for example [1].) The characteristic curves of (1) are now defined as a family of curves (x(p),t(p),u(p)) which go along the solution surface so that they satisfy the equations 

dx
-- = a(x,t,f)
dp

dt   
-- = b(x,t,f)                                                        (6)
dp

du
-- = c(x,t,f)
dp
The characteristics are the projections of these curves to the xt-plane. The whole solution surface can be generated by solving the characteristics starting from some initial value curve at p=0 
x(s,0)=x_0(s)
t(s,0)=t_0(s)
u(s,0)=u_0(s)
with s another parameter. That's probably all that needs to be known for the purposes of this project.

Examples

The method of characteristics finds applications as an analytical method in many areas of physics. Some conservation laws result in quasi-linear first-order PDE's. In particular I should mention fluid dynamics, where the elementary investigation of high speed flow of air and surface waves in channels makes use of it. The formation of shock waves and hydraulic jumps (bores) get simple interpretations in this view (discontinuity in solution). These are, however, the subject of another project. I still reserve the right to take examples directly from this field, but for now I only refer the reader to the book [2]. Some of the following examples are taken from [3] and include solutions of some exercises with MATLAB. The PDEtools package of Maple may also be used.

Actually, I'll put the examples on a different page.

References

[1] Ronald B. Guenther and John W. Lee, Partial Differential Equations of Mathematical Physics and Integral Equations, Dover, New York (1988)
[2] A.R. Paterson, A first course in fluid dynamics, Cambridge University Press, Cambridge (1983)
[3] Jeffrey Cooper, Partial Differential Equations with MATLAB, Birkhäuser (1998)