The Big Three Second-Order PDEs

Second-order PDEs pop up everywhere in science and engineering. The three most famous are:

• Heat Equation: Describes how heat diffuses through a medium.
• Wave Equation: Models vibrating strings, sound, and water waves.
• Laplace's Equation: Used in electrostatics, fluid flow, and more.

Their Forms

• Heat Equation: \[ \frac{\partial u}{\partial t} = D \frac{\partial^2 u}{\partial x^2} \]
• Wave Equation: \[ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \]
• Laplace's Equation: \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \]

Why Are They Important?

These equations help us predict how temperature, sound, or electric potential evolves in different environments.

Solution Techniques

• Separation of Variables
• Transform Methods
• Numerical Approaches

Each has its own quirks and applications.