Method of characteristics

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In mathematics, the method of characteristics is a technique for solving partial differential equations.

1 Characteristics of First-Order PDE
2 Example
3 Qualitative Analysis of Characteristics
4 External links
5 Bibliography

[edit] Characteristics of First-Order PDE

For a first-order PDE the method of characteristics discovers lines (called characteristic lines or characteristics) along which the PDE degenerates into an ordinary differential equation (ODE). Once the ODE is found it can be solved and transformed into a solution for the original PDE. Consider, as example, the advection equation (this example assumes familiarity with PDE notation, and solutions to basic ODEs).

[edit] Example

$a u_x + u_t = 0\,$

Where $a\,$ is constant and $u\,$ is a function over $x\,$ and $t\,$ . Our desire is to reform this linear first order PDE into an ODE along the appropriate curve; i.e. something of the form

$\frac{d}{ds}u(x(s), t(s)) = F(u, x(s), t(s))$ ,

where $(x(s),t(s))\,$ is a characteristic line. First, we find

$\frac{d}{ds}u(x(s), t(s)) = \frac{dx}{ds}u_x + \frac{dt}{ds}u_t$

by the chain rule. Now, notice if we set $\frac{dx}{ds} = a$ and $\frac{dt}{ds} = 1$ we get

$a u_x + u_t \,$

which happens to be the PDE we started with. Thus

$\frac{d}{ds}u = a u_x + u_t = 0$

So, along the characteristic line $(x(s), t(s))\,$ , the original PDE becomes the ODE $u_s = F(u, x(s), t(s)) = 0\,$ . Already we can make a very important observation: along the characteristics the solution is constant. Thus, $u(x_s, t_s) = u(x_0, 0)\,$ where $(x_s, t_s)\,$ and $(x_0, 0)\,$ lie on the same characteristic. But we are not done yet, the exact solution awaits. Now we have three ODEs to solve.

$\frac{dt}{ds} = 1$ , letting $t(0)=0\,$ we know $t=s\,$ ,
$\frac{dx}{ds} = a$ , letting $x(0)=x_0\,$ we know $x=as+x_0=at+x_0\,$ ,
$\frac{du}{ds} = 0$ , letting $u(0)=f(x_0)\,$ we know $u(x(t), t)=f(x_0)=f(x-at)\,$ .

So, we can conclude that the characteristic lines are straight lines (they could be curves, characteristic line is a bit of misnomer) with slope $a\,$ , and the value of $u\,$ remains constant along it.

[edit] Qualitative Analysis of Characteristics

Characteristics are also a powerful tool for gaining qualitative insights into PDE.

One can use the crossings of characteristics to find shockwaves. Intuitively, we can think of each characteristic line implying a solution to $u\,$ along itself. Thus, when two characteristics cross two solutions are implied. This causes shockwaves and the solution to $u\,$ becomes a multivalued function. Solving PDEs with this behavior is a very difficult problem and an active area of research.

Characteristics may fail to cover part of the domain of the PDE. This is called a rarefaction, and indicates the solution typically exists only in a weak, i.e. integral equation, sense.

The direction of the characteristic lines indicate the flow of values through the solution, as the example above demonstrates. This kind of knowledge is useful when solving PDEs numerically as it can indicate which finite difference scheme is best for the problem.

[edit] External links

More detailed information on the Method of Characteristics

[edit] Bibliography

Sarra, Scott The Method of Characteristics with applications to Conservation Laws, Journal of Online Mathematics and its Applications, 2003.
L.C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998. ISBN 0-8218-0772-2
A. D. Polyanin, V. F. Zaitsev, and A. Moussiaux, Handbook of First Order Partial Differential Equations, Taylor & Francis, London, 2002. ISBN 0-415-27267-X
A. D. Polyanin, Handbook of Linear Partial Differential Equations for Engineers and Scientists, Chapman & Hall/CRC Press, Boca Raton, 2002. ISBN 1-58488-299-9

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Category: Partial differential equations