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.

Full Set of Lecture Notes

The notes for this lecture are available here (24 pages).