PDE notes: Method of Characteristic

1. Introduction

The method of characteristics is a general technique for solving first-order equations. The basic idea is to reduce the determination of explicit solutions to solving ODE.

Initial value problem in one space variable and time take the form

We solve (1) using the method of characteristics, which reduces the initial value problem to an initial value problem for a system of ODE. In this method, we depend on the observation that if is smooth curve, then along the curve, has rate of change

given by the chain rule. Comparing (3) with (1), we see that the expressions are identical if we set

and interpret as speed. The left-hand side of the PDE can also be interpreted as the derivative of , in the direction in space.

Now the PDE in (1) can be replaced by the ODE system,

These ODEs are called the characteristic equations. Take note that is constant along the curve . We solve (5) to get

where is an unknown constant. Equation (7) is the equation for the family of parallel characteristic curves of (1). We observe that at , we have , and hence this constant is the point where the characteristic curve starts. We call it the anchor point. If a point is given, we can always find the corresponding anchor point as

Now we solve the second characteristic ODE to get

where is an arbitrary constant.

Initial conditions for the ODE system are derived from the initial condition for the PDE problem (1). We already know that the initial condition for ODE (5) is . We write in place of , then the initial condition for ODE (6) is

Since u is constant along the characteristic curve,

Thus, given a point, there is a unique characteristic curve passing through , with anchor point at , and the general solution at is

Next we extend the method to allow for linear forcing terms. We try to solve the following PDE for on

Solution From the chain rule we have

Hence the characteristic equation becomes

The initial conditions are

The solution for the first characteristic equation is

Thus, the anchor point for the characteristic passing through the point is

The solution for the second characteristic equation is

and the solution along the characteristic is

For this kind of initial value problem, the method of characteristic is summarized as:

1. Rewrite the initial value problem (13) as a system of ODE consisting of the characteristic equations (16) and (17) with initial conditions (18) and (19).
2. Solve the ODE and initial condition for and , with the anchor point to get the solution along the characteristic.
3. Solve for as a function of . This effectively changes variables from to .
4. Write the solution .

We can extend our concepts to first-order equations with non-constant coefficients in the form

In applications, this PDE is usually accompanied by a side condition of the form

for , where is a curve of anchor points and is a given function. Suppose that is the solution to (24) subject to the side condition (25). We can think of as a two-dimensional surface in the space. Denote by the curve on the surface whose projection onto the plane is . The curve is called the initial curve. We parameterize the initial curve using the anchor points to get

Equation (24) states that the vector field is tangent to the solution surface , since the solution surface has normal

The solution surface can therefore be generated by integrating along the vector field, starting at each point of the curve .

Integral curves

of the vector field which starts from the initial curve satisfy the following vector ODE:

or, in component form, the system of ODEs:

The system of ODEs (31), (32), (33) are called the characteristic equations for the PDE (24). The solutions to the characteristic equations are called the characteristic curves for the PDE.

The procedure to solve the PDE (24) and (25) is as follows:

1. Solve the first two characteristic equations (31) and (32)to get and in terms of the characteristic variable and the anchor point :

   

2. Insert the solution from the previous step into equation (33) and solve the resulting equation for :

   

3. Apply the Inverse Function Theorem. In this step we solve the equations

   

   for

   

   The solution is guaranteed by the Inverse function theorem.

4. Write the solution for in terms of and to get the solution to the original PDE:

   

This procedure will work as long as the transformation in (36) and (37) is invertible. We can guarantee this locally by appealing to the Inverse Function Theorem.