Method of Characteristics
Summary
The method of characteristics was already employed to construct the solution of the (constant-coefficient) transport equation in Lecture 2. We now begin the general treatment of the method of characteristics for general first-order PDEs.
We write a first order PDE for an unknown function $u$ as
$$
F(Du,u,x)=0.
$$
Here $x\in U$, where $U \subset \mathbb{R}^{n}$ is open, and $u\colon \overline{U} \to \mathbb{R}$. $F$ is of the form
$$
F\colon \mathbb{R}^n \times \mathbb{R} \times \overline{U} \to \mathbb{R}.
$$
and construct solutions of
$$
\left\{
\begin{alignedat}{2}
F(Du,u,x) &= 0 &&\text{in}~U \\ u&=g\quad &&\text{on}~\Gamma
\end{alignedat}
\right.
$$
where $\Gamma \subseteq \partial U$, and $g\colon \Gamma \to \mathbb{R}$ is given. $F$ and $g$ are as smooth as we like and the boundary $\partial U$ is $C^1$. We give several examples before a theoretical treatment.
