## Using the Method of Characteristics to Solve Cauchy and Initial Value Problems

Definition: Cauchy problem. Assume three functionsand (boundary conditions) are given. The Cauchy problem is to find a solution u(x, y) of a PDE

which satisfies also

The curveis called a datum curve.

An Initial Value Problem, IVP is a special Cauchy problem with variablesand the datum curve

Method to solve Cauchy problems

Givenand Cauchy data (boundary conditions)

1. Parameterise the initial values:

2. Solve the characteristic equations

and

3. Findandin terms ofand(coordinate change)

4. Solve the compatibility condition

5. Calculateby using the coordinate change fromto

Example: Solvewith the initial condition

1. Parameterise the initial values:

2. Solve the characteristic equationsand

to get

3. Calculateandin terms ofandto get

4. Solve the compatibility condition to get

5. Express in terms of