A partial differential equation, or PDE, is a mathematical equation that involves partial derivatives of a function with respect to multiple independent variables. Unlike ordinary differential equations which involve functions of a single variable, PDEs describe how quantities change across multiple dimensions such as space and time. A classic example is the heat equation, which describes how temperature distributes over space and time.
Partial differential equations are classified in several ways. By order, we distinguish first-order PDEs where the highest derivative is first order, and second-order PDEs where the highest derivative is second order. By linearity, we have linear PDEs where coefficients are constants or functions of independent variables only, and nonlinear PDEs where coefficients depend on the solution itself. Common examples include the heat equation, wave equation, and transport equation.
Let's visualize the heat equation in action. We start with an initial temperature distribution shaped like a sine wave. The boundaries are kept at zero temperature. As time progresses, we can see how the temperature diffuses and the peak gradually decreases while spreading out. This demonstrates the fundamental property of heat diffusion - hot spots cool down and spread their energy to surrounding areas.
Now let's examine the wave equation, which governs oscillatory phenomena like vibrating strings and sound waves. Unlike the heat equation where energy dissipates, the wave equation preserves energy as the wave propagates. We can see how the sinusoidal wave travels from left to right while maintaining its shape and amplitude. This demonstrates the fundamental property of wave motion - energy transport without material transport.
Partial differential equations are ubiquitous in science and engineering. Maxwell's equations describe electromagnetic phenomena and form the foundation of modern electronics. The Schrödinger equation governs quantum mechanical systems. Navier-Stokes equations model fluid flow, crucial for aerodynamics and weather prediction. In finance, the Black-Scholes equation helps price options. PDEs provide the mathematical framework for understanding and predicting complex phenomena across virtually every field of science and technology.