Ngô Quốc Anh

February 11, 2010

Linear First-Order Equations: Characteristic curve

Filed under: Nghiên Cứu Khoa Học, PDEs — Ngô Quốc Anh @ 0:20

We are going to discuss several topics involving hyperbolic PDEs. We will focus on the so-called characteristic curve. We start with the following first-order equations

\displaystyle u_t+c(x,t) u_x=0, \quad x \in \mathbb R, t>0

with the init data

\displaystyle u(x,0)=u_0(x), \quad x\in \mathbb R.

Advection Equation. In this section, we consider the case when c is a constant. We show that the initial value problem for the advection equation, namely

\displaystyle u_t+cu_x=0, \quad x \in \mathbb R, t>0

\displaystyle u(x,0)=u_0(x), \quad x\in \mathbb R

has a unique solution given by u(x, t ) = u_0(x - c t ), which is a right traveling wave of speed c.

The way to see it is to make a change of coordinates to

\displaystyle x' = ct-x, \quad t'=t+cx.

Replacing all x and t derivatives by x' and t' derivatives. By the chain rule

\displaystyle {u_x} = \frac{{\partial u}}{{\partial x}} = \frac{{\partial u}}{{\partial x'}}\frac{{\partial x'}}{{\partial x}} + \frac{{\partial u}}{{\partial t'}}\frac{{\partial t'}}{{\partial x}} = -\frac{{\partial u}}{{\partial x'}} + c\frac{{\partial u}}{{\partial t'}}

and

\displaystyle {u_t} = \frac{{\partial u}}{{\partial t}} = \frac{{\partial u}}{{\partial x'}}\frac{{\partial x'}}{{\partial t}} + \frac{{\partial u}}{{\partial t'}}\frac{{\partial t'}}{{\partial t}} = c\frac{{\partial u}}{{\partial x'}} +\frac{{\partial u}}{{\partial t'}}.

Hence

\displaystyle {u_x} + c{u_t} = ({c^2} + 1)\frac{{\partial u}}{{\partial t'}}.

The equation now takes the form u_{t'}=0 in the new (primed) variable. Thus the solution is

\displaystyle u(x,t) = f(x') = f(ct - x)

with f an arbitrary function of one variable. The initial condition (at t=0) tells us that f(-x)=u_0(x). Thus

\displaystyle u(x,t) = {u_0}(x - ct).

Characteristic curves. The above solutions suggests that u is constant on the curve x-ct=\xi where \xi is constant. The totality of all the curves x-ct=\xi, where \xi is constant, is called the set of characteristic curves for this problem.

The constant \xi acts as a parameter and distinguishes the curves. A graph of the set of characteristic curves on a spacetime diagram (i.e., in the xt plane) is called the characteristic diagram for the problem.

The characteristic curves, or just characteristics. are curves in spacetime along which signals propagate. In the present case. the signal is the constancy of u that is carried along the characteristics. Further-and this will turn out to be a defining feature-the PDE reduces to an ordinary differential equation along the characteristics.

We end this section by proving an example.

The equation is u_t+2u_x=0 with the initial data \sin(2\pi x). Thus the general solution is \sin(2\pi(x-2t)).

This picture shows that in the case when coefficient c is constant, the characteristic curves are parallel. This is not true in the case when c is dependent of either x or t. This circumstance is discussed here.

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