Partial Differential Equations
Heat, Waves, and how continuous space dictates motion
1. Building Intuition
While Ordinary Differential Equations (ODEs) deal with functions of a single variable (like time), Partial Differential Equations (PDEs) deal with functions of multiple variables (like time and space). They capture how ripples travel across a pond, how heat spreads through an engine, or how quantum probability distributions evolve.
Notice how the sharp edges of the initial temperature profile quickly smooth out as the heat diffuses into the cooler surroundings over time.
2. The Mathematics
The three classic PDEs found in engineering are the Heat Equation, Wave Equation, and Laplace Equation.
PDEs are generally solved by finding basis functions via Separation of Variables and applying Orthogonality — exactly why Fourier Series were explicitly developed!
3. Applications
Thermodynamics
Predicting heat dissipation in microprocessors and heat exchangers.
Acoustics
Simulating sound pressure waves travelling through concert halls.
Structural Mechanics
Modelling the vibrating membrane of a drum head (2D wave equation).
Quantum Mechanics
The Schrödinger equation is a PDE describing the wave function of a quantum system over time.