📂Partial Differential Equations

Notation for Nonlinear First-Order Partial Differential Equations

Notation¹

A nonlinear first-order partial differential equation is denoted as follows.

$$ \begin{equation} F(Du, u, x) = F(p, z, x) = 0 \label{eq1} \end{equation} $$

• $\Omega \subset \mathbb{R}^{n}$ is an open set
• $x\in \Omega$
• $F : \mathbb{R}^n \times \mathbb{R}^n \times \bar{ \Omega } \to \mathbb{R}$ is the given function
• $u : \bar{ \Omega } \to \mathbb{R}$ is the variable of $F$

Description

Solving a nonlinear first-order partial differential equation $F$ means finding a variable $u$ that satisfies $F=0$ for the given $F$. Here, let $x$ be a variable that encompasses both time and space.

$$ x=(x_{1}, \dots, x_{n}=t) $$

The function $F$ is denoted as follows.

$$ F=F(p, z, x)=F(p_{1}, \dots, p_{n}, z, x_{1}, \cdots, x_{n}) $$

• $p=Du(x) \in\mathbb{R}^n$
• $z=u(x)\in \mathbb{R}$
• $x\in \bar{ \Omega }$

And the function $F$ is assumed to be sufficiently smooth to be differentiable. This is generally the case, so it’s not particularly a strong condition. Then, the gradient of $F$ with respect to each variable is as follows.

$$ \begin{cases} D_{p} F=(F_{p_{1}},\ \cdots,\ F_{p_{n}}) \\ D_{z}F=Fz \\ D_{x}F=(F_{x_{1}},\ \cdots,\ F_{x_{n}} )\end{cases} $$

The Clairaut’s equation is represented using this notation as follows.

$$ F(Du,\ u,\ x)=xDu+f(Du) $$

Boundary Value Problem

Differential equations $\eqref{eq1}$ are commonly given along with boundary conditions. In such cases, they are denoted as follows.

$$ \begin{align*} F(Du,\ u,\ x)&=0 && \text{in } \mathbb{\Omega} \\ u&=g && \text{on } \Gamma \end{align*} $$ In this case, $\Gamma \subset \partial \Omega, g : \Gamma \to \mathbb{R}$.