📂Partial Differential Equations

Partial Differential Equations

Definitions¹

Partial Differential Equations

For a natural number $k \in \mathbb{N}$ and an open set $U \subset \mathbb{R}^{n}$, the following expression is called a $k$-order partial differential equation.

$$ \begin{equation} F(D^{k}u(x), D^{k-1}u(x),\cdots,Du(x),u(x),x)=0\quad (x\in U) \end{equation} $$

Here, $D^{k}u$ is the multi-index notation. $F$ is given as follows, and the unknown $u$ is as follows.

$$ F : {\mathbb{R}}^{n^{k}}\times{\mathbb{R}}^{n^{k-1}}\times \cdots \times \mathbb{R}^{n}\times \mathbb{R}\times U \to \mathbb{R} \\ u : U \to \mathbb{R} $$

System of Partial Differential Equations

Given $\mathbf{F} : {\mathbb{R}}^{mn^{k}}\times{\mathbb{R}}^{mn^{k-1}}\times \cdots \times \mathbb{R}^{mn}\times \mathbb{R}^{m}\times U \to \mathbb{R}^{m}$ and the unknowns $\mathbf{u}:U \to \mathbb{R}^{m}$, $\mathbf{u}=(u^{1},\cdots,u^{m})$, the expression below

$$ \mathbf{F}(D^{k}\mathbf{u}(x),D^{k-1}\mathbf{u}(x),\cdots,D\mathbf{u}(x),\mathbf{u}(x),x)=\mathbf{0}\quad (x\in U) $$

is called a $k$-order system of partial differential equations.

Explanation

Partial differential equations are commonly abbreviated as PDEs. Solving a PDE means finding all $u$ that satisfy $(1)$, and such $u$ are called solutions.

Finding a solution means

1. Ideally, finding a simple and explicit solution,
2. When that’s not possible, determining the existence of a solution or other characteristics.

In most cases, $U, \Omega \subset \mathbb{R}^{n}$ in a partial differential equation represents an open set, and the variable $t$ always represents time, where $t\ge 0$. Also,

$$ Du=D_{x}u=(u_{x_{1}},\cdots,u_{x_{n}}) $$

represents the gradient of $u$. Here, $x=(x_{1},\cdots,x_{n})$.

Classification

Partial differential equations can be classified based on linearity as follows.

Linear

The partial differential equation $(1)$ is said to be linear if it satisfies the following equation for a given function $a_{\alpha}, f$.

$$ \sum _{| \alpha | \le k} a_{\alpha}(x) D^{\alpha} u = f(x) $$

If $f=0$, it is called a homogeneous linear PDE. If it’s not linear, it’s called non-linear. 2nd-order linear partial differential equations are further classified as:

• Hyperbolic PDE
• Parabolic PDE
• Elliptic PDE

Semilinear

A partial differential equation $(1)$ is called semilinear if it satisfies the following.

$$ \sum _{| \alpha | = k} a_{\alpha}(x) D^{\alpha} u + a_{0}\left( D^{k-1}u, \dots, Du, u, x \right) = 0 $$

In other words, a semilinear pde means a partial differential equation whose coefficients of the derivative of order $k$ (the highest order) depend only on $x$. For example,

• Reaction-diffusion equation $$ u_{t} - \Delta u = f(u) \qquad (\text{e.g. } f(u) = u^{2}) $$

Quasilinear

A partial differential equation $(1)$ is called quasilinear if it satisfies the following.

$$ \sum _{| \alpha | = k} a_{\alpha}(D^{k-1}u, \dots, Du, u, x)D^{\alpha} u + a_{0}\left( D^{k-1}u, \dots, Du, u, x \right) = 0 $$

Examples include

• Inviscid Burgers’ equation $$ u_{t} + uu_{xx} = 0 $$

Fully Non-linear

Non-linear equations that are not quasilinear are called fully nonlinear.