Ngô Quốc Anh

February 13, 2010

Linear First-Order Equations: Variable Coefficients

Filed under: PDEs — Ngô Quốc Anh @ 0:47

Followed by this topic where we discuss the advection equation. We can extend the preceding notions to more complicated problems. First consider the linear initial value problem

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

and

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

where $c=c(x,t )$ is a given continuous function. The left side of the PDE  is a total derivative along the curves in the $xt$ plane defined by the differential equation

$\displaystyle \frac{dx}{dt}=c(x,t)$.

Along these curves

$\displaystyle\frac{{du}}{{dt}} = {u_t} + {u_x}\frac{{dx}}{{dt}} = {u_t} + c(x,t){u_x} = 0$

or, in other words, $u$ is constant. Therefore

$\displaystyle u={\rm const.}$ on $\displaystyle \frac{dx}{dt}=c(x,t)$.

Again the PDE reduces to an ordinary differential equation (which was integrated to a constant) along a special family of curves, the characteristics defined by

$\displaystyle \frac{dx}{dt}=c(x,t)$.

The function $c=c(x,t)$ gives the speed of these characteristic curves, which varies in spacetime.

Example. Consider the initial value problem

$\displaystyle u_t -xt u_x=0, \quad x \in \mathbb R, t>0$

and

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

The characteristics are defined by the equation

$\displaystyle \frac{dx}{dt}=-xt$

$\displaystyle x = \xi {e^{ - \frac{{{t^2}}}{2}}}$

where $\xi$ is a constant. On these curves the PDE becomes

$\displaystyle {u_t} - xt{u_x} = {u_t} - \frac{{dx}}{{dt}}{u_x} = \frac{{du}}{{dt}} = 0$

or

$\displaystyle u={\rm const.}$ on $\displaystyle \frac{dx}{dt}=\xi {e^{ - \frac{{{t^2}}}{2}}}$.

From the constancy of u along the characteristics, we have

$\displaystyle u(x,t) = {u_0}\left( {x{e^{\frac{{{t^2}}}{2}}}} \right)$.

This method, called the method of characteristics, can be extended to nonhomogeneous initial value problems. Let take an example when $u_0(x)=\sin\left(\frac{\pi}{2}x\right)$. We have the following pictures.

The above picture shows that how characteristic curves look like.

This picture shows the shape of solution in the spacetime.

When nonlinear terms are introduced into PDEs, the situation changes dramatically from the linear case discussed here and here. We will cover these interesting stuffs later on.