## 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 each there must be one initial value of given on this curve. We have used the coordinate to parameterise a characteristic curve and to 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 with as the integration constant. Similarly, if then the characteristic curves can be expressed as functions they are solutions of the ODE  