Using the Method of Characteristics to Solve Cauchy and Initial Value Problems
Definition: Cauchy problem. Assume three functions
and 
(boundary conditions) are given. The Cauchy problem is to find a solution u(x, y) of a PDE
which satisfies also 
The curve
is called a datum curve.
An Initial Value Problem, IVP is a special Cauchy problem with variables
and the datum curve
Method to solve Cauchy problems
Given
and Cauchy data (boundary conditions)
1. Parameterise the initial values:
2. Solve the characteristic equations
and
3. Find
and
in terms of
and
(coordinate change)
4. Solve the compatibility condition
5. Calculate
by using the coordinate change from
to
Example: Solve
with the initial condition 
1. Parameterise the initial values:
2. Solve the characteristic equations
and
to get
3. Calculate
and
in terms ofand
to get
4. Solve the compatibility condition 
to get
5. Express in terms of
