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The Method of Characteristics & Conservation Laws - The Method of Characteristics

Scott A. Sarra

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

first order pde   (1)

in two variables along with the initial condition {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, characteristic 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

characteristic ODE   (2a)


characteristic ODE   (2b)

then we have


and along the characteristic curves, the PDE becomes the ODE


Equations (2a) and (2b) will be referred to as the characteristic equations.

General Strategy

Here is the general strategy for applying the method of characteristics to a PDE of the form (1).

  • Step 1. Solve the two characteristic equations, (2a) and (2b). Find the constants of integration by setting (these will be points along the t = 0 axis in the x-t plane) and t(0)=0. We now have the transformation from to , and .
  • Step 2. Solve the ODE (3) with initial condition , where are the initial points on the characteristic curves along the t = 0 axis in the x-t plane.
  • Step 3. We now have a solution . Solve for s and in terms of x and t (using the results of Step 1), and substitute these values in to get the solution to the original PDE as .

Scott A. Sarra, "The Method of Characteristics & Conservation Laws - The Method of Characteristics," Loci (September 2004)


Journal of Online Mathematics and its Applications