Method of characteristics

The method of characteristics is a method for solving quasi- linear partial differential equations of first order, ie, equations of the type

Where the discharges occur only linearly. The basic idea consists in the reduction of the problem to the solution of ordinary differential equations.

Idea

In order to convert the partial differential equation into a system of ordinary differential equations, the coordinates and two new coordinates and parameterized. First, the unknown function by derived ( applying the chain rule! )

A subsequent comparison of coefficients with the output equation yields a system of ordinary differential equations:

These three equations provide a solution to the partial differential equation in the new co-ordinates and on the other hand the transformation rule - the so-called characteristic lines or characteristic -. Thus, the solution found in the old coordinates are transformed back.

Example

Given a simple transport equation with initial condition:

With and constants. Derivation of coefficients and after comparison yields a system of ordinary differential equations:

Since the equations are decoupled from each other here completely, the solution is very simple:

From this it immediately and thus the solution of the transport equation in the old coordinate follows.