The method of characteristics is a method that can be used to solve the initial value problem (IVP) for general first order PDEs. Consider the first order linear PDE
in two variables along with the initial condition . The goal of the method of characteristics, when applied to this equation, is to change coordinates from (x, t) to a new coordinate system in which the PDE becomes an ordinary differential equation (ODE) along certain curves in the x-t plane. Such curves, , along which the solution of the PDE reduces to an ODE, are called the characteristic curves or just the characteristics. The new variable s will vary, and the new variable will be constant along the characteristics. The variable will change along the initial curve in the x-t plane (along the line t = 0). How do we find the characteristic curves? Notice that if we choose
and
then we have
,
and along the characteristic curves, the PDE becomes the ODE
. (3)
Equations (2a) and (2b) will be referred to as the characteristic equations.
Here is the general strategy for applying the method of characteristics to a PDE of the form (1).
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