Method of characteristics

From Free net encyclopedia

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

Contents

[edit]

Characteristics of First-Order PDE

For a first-order PDE, the method of characteristics discovers lines (called characterstic 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

<math>a u_x + u_t = 0\,</math>

Where <math>a\,</math> is constant, and <math>u\,</math> is a function over <math>x\,</math> and <math>t\,</math>. Our desire is to reform this linear first order PDE into a linear first order ODE, i.e. something of the form

<math> \frac{d}{ds}u(x(s), t(s)) = c(x(s), t(s))u </math>,

where <math>(x(s),t(s))\,</math> is a characteristic line. First lets see what happens when we find

<math>\frac{d}{ds}u(x(s), t(s)) = \frac{dx}{ds}u_x + \frac{dt}{ds}u_t</math>,

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

<math> a u_x + u_t \,</math>

which happens to be the PDE we started with. Thus

<math>\frac{d}{ds}u = a u_x + u_t = 0 </math>

So, along the characteristic line <math>(x(s), t(s))\,</math>, the derivative of <math>u\,</math> is constant. Already we can make a very important observation: along the characteristics the solution is constant. Thus, <math>u(x, t) = u(x_0, 0)\,</math> where <math>(x, t)\,</math> and <math>(x_0, 0)\,</math> lay on the same characterstic. But we are not done yet, the exact solution awaits. Now we have three ODEs to solve.

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

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

A solution is shown in the figure below as a surface plot and a contour plot. Notice, as we predicted, the solution is constant along the lines of slope <math>a\,</math>. This forces the wave along <math>x\,</math> as <math>t\,</math> advances.


[edit]

Qualititive Analysis of Characteristics

Characeristics 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 <math>u\,</math> along itself. Thus, when two characteristics cross two solutions are implied. This causes shockwaves and the solution to <math>u\,</math> 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 this problem.

[edit]

External links

[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
Retrieved from "http://www.netipedia.com/index.php/Method_of_characteristics"