Numerical Methods for PDEs: Solving PDEs on a computer

Introduction

Looking at Nature

[math]\displaystyle{ u_x = \frac{\partial u}{\partial x} }[/math]

A second-order linear equation in two variables:

[math]\displaystyle{ Au_{xx} + Bu_{xy} + Cu_{yy} + Du_x + Eu_y + Fu = G }[/math]

Parabolic: [math]\displaystyle{ B^2 - 4AC = 0 }[/math]. This family of equations describe heat flow and diffusion processes.
Hyperbolic: [math]\displaystyle{ B^2 - 4AC \gt 0 }[/math]. Describe vibrating systems and wave motion.
Elliptic: [math]\displaystyle{ B^2 - 4AC \lt 0 }[/math]. Steady-state phenomena.

Taylor Series:

[math]\displaystyle{ f(x + h) = f(x) + \frac{f'(x)}{1!}h + \frac{f^{(2)}(x)}{2!}h^2 + \cdots + \frac{f^{(n)}(x)}{n!}h^n + R_n(x) }[/math]