# Solving first order PDEs with Characteristics

Use the method of characteristics to solve the PDE system

- 25 November, 2012 -
- Reading Diary -
- Tags : numerical analysis
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We begin by parameterizing as functions of such that

This gives the system of ODE

Unfortunately, we don’t know what is; but we do have the parameterization parameter , so we can solve the system backwards in time.

which is equivalent to the second order ODE

which has the solution

Now, we know that , so we want to find the time such that , or , then

and

Let’s quickly verify that this is indeed the solution

Since this is fully linear, the solution is unique and the characteristics intersect.