Followed by this topic where we discuss the advection equation. We can extend the preceding notions to more complicated problems. First consider the linear initial value problem
where is a given continuous function. The left side of the PDE is a total derivative along the curves in the plane defined by the differential equation
Along these curves
or, in other words, is constant. Therefore
Again the PDE reduces to an ordinary differential equation (which was integrated to a constant) along a special family of curves, the characteristics defined by
The function gives the speed of these characteristic curves, which varies in spacetime.
Example. Consider the initial value problem
The characteristics are defined by the equation
which, on quadrature, gives
where is a constant. On these curves the PDE becomes
From the constancy of u along the characteristics, we have
This method, called the method of characteristics, can be extended to nonhomogeneous initial value problems. Let take an example when . We have the following pictures.
The above picture shows that how characteristic curves look like.
This picture shows the shape of solution in the spacetime.
When nonlinear terms are introduced into PDEs, the situation changes dramatically from the linear case discussed here and here. We will cover these interesting stuffs later on.