Differential Equations
Study of equations involving derivatives and their applications.
Advanced Topics
Partial Differential Equations (PDEs)
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.
Examples
Modeling how heat spreads through a metal rod.
Describing the shape of a vibrating guitar string.
In a Nutshell
PDEs describe changes in systems with multiple variables like temperature or waves.