The Method of Characterisitcs Explained
To illustrate the method of characteristics consider the partial differential equation for
Obviously, this is in fact an ODE. The solution is found by integration with respect
to
and reads
where
is an arbitrary function of
is uniquely determined by fixing
along the line
or any curve in the
plane.
∂u
The initial value problem
and
for all
has the unique solution
To find this solution we identified a curve along which the PDE becomes an ODE – the lines
- where
is a constant, then solved the ODE for each
– for eachthere must be one initial value ofgiven on this curve. We have used the coordinateto parameterise a characteristic curve andto parameterise the different curves.
We can generalize this. Consider the general semilinear PDE
First choose a characteristic curve parameterised by
Definition: A characteristic curve is a curve in the
−plane, along which the PDE becomes an ODE. The characteristic curves, parameterised by
i.e.
are solutions of the characteristic equations
There is a one-parameter family of these curves which are parameterised by
If
then the characteristic curves can be expressed as functions y(x):
they are solutions of the ODE :
obtained from
withas the integration constant. Similarly, if
then the characteristic curves can be expressed as functions
they are solutions of the ODE
