Cauchy's method of characteristics

Cauchy had developed a method called method of characteristics which is based on geometric consideration. This method solves Eq.(), subjected to BC $u=g$ by converting the PDE Eq.() into an appropriate system of ODEs. Here is the big picture:

1. Let $u$ solves Eq.() and (). Fix a point $x \in \Omega.$
2. We would like to calculate $u(x)$ by finding some curve lying within $U$ which connects $x$ with a point $x_0 \in \Gamma,$ and along which one can compute $u.$
3. From the given BC $u=g$ on $\Gamma$, one know the value of $u$ at one end of this special curve i.e. at $x_0.$
4. The method of characteristic describe how to calculate $u$ all along the curve and thus in particular at $x.$

• Explore further: What can be the easiest form of such curve? The curve along, which $F$ is constant which automatically implies this constant value is nothing but value of $F$ at the boundary point $x_0.$ Read Sneddon, Chap. 2, Sec. 8 or Evans, Sec. 3.2.

• For solving non-linear PDEs like Eq.(), our main tool is Charpit's method. It is based on the concept of compatible systems of PDEs.