Question

PDE: How do you get the characteristic curve from \( u_t+(x+t)u_x=t \)?

Answer 1

Well, you'd first assume that \(t\) and \(x\) are functions of some parameter \(s\). In that case,
\[ \frac{du}{ds} = \frac{dt}{ds} \frac{\partial u}{\partial t} + \frac{dx}{ds} \frac{\partial u}{\partial x} = t\]
where we identify
\[ \frac{dt}{ds} = 1\]
and
\[ \frac{dx}{ds} = x + t\]

From here we can just let \[t = s\]
and simply write x as a function of t. The second equation becomes

\[ \frac{dx}{dt} = x + t \]
or
\[\frac{dx}{dt} - x = t\]

The solution to this equation is

\[x(t) = e^t - (t+1)\]

Which is the characteristic curve that you're looking for.

Answer 2

Whoops, that should be
\[ x(t) = c\cdot e^t - (t+1) \] where c is an arbitrary constant.

Answer 3

For the first step when choosing \(x \) and \(t \) as a function of \(s \), is this similar to finding the directional derivative?
\( D{ \widehat {<1, x+t>} } u = \cfrac{t}{\sqrt{1+(x+t)^2} } \)

Answer 4

Choosing \(x\) and \(t\) as functions of \(s\) defines a set of curves in the \((x,t)\) plane. Solving the resulting differential equation (which is \( \frac{du}{dt} = t\)) will tell you how the solution evolves along that curve. Then the problem is essentially solved. This is the method of characteristics, and is what I assumed you were asking about.

I'd have to think about your question some more, but that's not normally part of the process I use when I use this technique.

Answer 5

Oh wait, no, of course. That's the idea. The question then is how to find the curves that follow that derivative field, which is where the rest of the technique comes in.

Answer 6

I'm trying to get an intuitive handle on applying the method of characteristics. It's starting to make sense, but there are still hazy spots in my understanding. You explanation helps.

Answer 7

I think about it this way - the essential purpose of the method of characteristics is to transform a partial differential equation into a collection of ordinary differential equations, which are obviously much easier to solve.

In the process, we create a family of curves that cover the (x,t) plane. By solving the ordinary differential equations, we find out how the solution evolves as you travel along each curve.

So at the end of the day, rather than think about a point in the plane by asking "what is x and what is t", we're really asking "which curve are we on, and how far along it are we?"