The Method of Characteristics
When we write (for example) ut we mean the partial derivative@u means.
The simple linear PDE ut+cux = 0, where c is a constant, has general solution u(x, t) = f(x—ct). This can be verified directly: by the chain rule
u = f 0(x — ct) @ (x — ct) = f 0(x — ct)
and ut = f 0(x — ct) (x — ct) = —cf 0(x — ct)
and so for any di↵erential function f, u = f(x — ct) satisfies ut + cux = 0. The initial value problem in which u(x, 0) = u0(x) hence has solution u(x, t) = u0(x—ct). This is usually thought of as describing wave propagation in which every point on the initial disturbance translates at the same constant speed c so that the wave has the same shape at all times. If c > 0 the wave moves left to right or if c < 0 the wave moves right to left as t increases.
Figure 1.1: Linear wave propagation for c > 0. The red curve shows wave profile at t = 0 and the others show it at successively later times.
Considring the less simple, nonlinear PDE ut + uux = 0 in a similar way we can guess an implicit description of the solution u = f(x—ut). Note that u appears twice in this equation and in general one cannot use it to obtain an explicit expression for u in terms of t and x. However it can be solved locally and it can be shown that u satisfying u = f(x — ut) satisfies ut + uux = 0. We give a wave interpretation of this implicit solution: each point on the initial wave profile moves not at the same speed, but at speed u determined by its amplitude. In particular, as t increases, the points that have positive amplitude move to the right, those with negative amplitude move left and those with zero amplitude do not move. More general, the same argument applies to solutions of ut + c(u)ux = 0; for u such that c(u) is positive/negative/zero, the wave profile moves right/moves left/is stationary as t increases.
Figure 1.2: Nonlinear wave propagation. The red curve shows wave profile at t = 0 and the others show it at successively later times.
1.2 Quasilinear first-order PDEs
The first order PDE a(t, x, u)ut + b(t, x, u)ux = c(t, x, u), (1.1)where a, b, c are functions of t, x and u (but not the derivatives of u) is said to be quasilinear. From now on we will not indicate the dependence of a, b, c on t, x, u. For this equation to be linear, we would need to have a and b independent of u and c has to be linear in u.A solution of this PDE u = f(t, x) defines the surface F(t, x, u) = f(t, x) — u = 0 in (x, t, u)-space. Now recall two facts from multivariable calculus (e.g. Maths 2A).
Fact 1 The gradient vector
Fact 2 For a parametric curvet = t(⌧), x = x(⌧), u = u(⌧) in (t, x, u)-space, the vectordt , dx, du is tangential to the curve for each ⌧ . Stating this another way, any solution ofthe system of equations a curve is (t, x, u)-space which has tangent vector (a, b, c) at each point.
When a, b and c are the coefficients in a quasilinear first order PDE (1.1), (1.2) are called the Monge equations associated with that PDE.
Since u = f is a solution of the quasilinear PDE (1.1), we have aft + bfx — c = (a, b, c) • (ft, fx, —1) = 0. This tells us that the tangent vector (a, b, c) is perpendicular to a normal (ft, fx, —1) of the solution surface. Thus the solutions of the Monge equations are curves lying in a solution surface.
Like any (locally smooth) 3D surface, the solution surface may be described parametrically in terms of two parameters. In the present context, parameter ⌧ , say, is taken to change along solution curves (for the Monge equations) and the other σ, say, labels distinct curves. The projection of each of these curves onto the (x, t)-plane is called a characteristic curve or simply a characteristic. Often, initial or boundary conditions are attached to the PDE (e.g. the solution is given on the line t = 0 for an initial value problem, or on x = 0 for a boundary value problem) and will always use ⌧ = 0 to designate the point where the solution curves of the Monge equations intersect this initial data curve.
Example 1.1. Find the characteristics forut + cux = 0,where c is a constant. Hence find its general solution and the solution such that u(x, 0) = u0(x). So These have solutions t = ⌧ + t0, x = c⌧ + x0, u = u0, where t0, x0 and u0 are constant on a given characteristic, but may be di↵erent for di↵erent characteristics. Initial conditions are given on the data curve t = 0 (i.e. u is specified at t = 0 for all x) and we choose t0 = 0 so that this is also the curve ⌧ = 0. In a boundary value problem (solution specified at x = 0 for all t) we would choose x0 = 0 instead.
We then choose to write x0 = σ and then u0 = F(σ) since the u0 depends on the choice of characteristic. So we get the parametric description of the solution surfaces
Figure 1.3: Characteristics of ut + cux = 0 The general solution u = u(x, t) is then given in parametric form by u(x, t) = F(σ), where x = σ + c⌧, t = ⌧, and, in this case, we may eliminate the parameters to give an explicit formula for u = u(x, t); u(x, t) = F(x — ct). That is, the general solution describes the unchanging propagation of the initial solution profile with velocity c, as described in §1.1.
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