1.1 PDEs Basics

PDEs have more than on independent variable $x, y . .$
The dependent variable is an unknown function $u (x, y, .)$
Its derivatives are denoted as $\frac{\partial u}{\partial x} = u_{x}$

A PDE is an identity that relates the independent variables, dependent variable $u$ , and its partial derivatives of $u$ .

F (x, y, u (x, y), u_{x} (x, u), u_{y} (x, y)) = F (x, y, u, u_{x}, u_{y}) = 0

The order of a PDE is the highest derivative that appears.

The solution is $u (x, y, \dots)$ that satisfies the equation.

Homogenous Equation:

L u = 0

Where $L$ is a linear operation
Ex: $u_{x} + y u_{y} = 0$ where $L = \frac{\partial}{\partial x} + \frac{y \partial}{\partial y}$

Inhomogenous Equation:

L u = g, g \neq 0

Ex: $(\cos x y^{2}) u_{x} - y^{2} u_{y} = \tan (x^{2} + y^{2})$

The advantage of a linear equation $L u = 0$ is that if $u_{1}, u_{2}, \dots, u_{n}$ are solutions then so is:

\sum_{j = 1}^{n} c_{j} u_{j} (x), (c_{j} are constants)

Examples

$u_{x x} = 0$

⟹ u_{x} = f (y) ⟹ u = f (y) x + g (y)

$u_{x x} + u = 0$

⟹ u = f (y) \cos (x) + g (y) \sin x

from ODEs

$u_{x y} = 0$

⟹ u_{y} (x, y) = f (y) ⟹ u = F (y) + g (x)