Going Beyond: Functions of Several Variables

PDEs are equations involving partial derivatives of functions with more than one variable.

Classic Examples

• The heat equation: \( \frac{\partial u}{\partial t} = D \frac{\partial^2 u}{\partial x^2} \)
• The wave equation: \( \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \)

Why Are PDEs Useful?

They describe complex systems, like heat flow, vibrations, and fluid movement.

Solving PDEs

Solutions often require advanced techniques and may only be possible for certain cases.