Heat Equation

Fourier's original application - watch heat diffuse

Fourier's Original Application

Joseph Fourier developed his series in the early 1800s while studying heat conduction. He wanted to understand how heat spreads through a metal bar over time - and discovered that the solution was beautifully expressed in terms of sine waves.

This was one of the most important discoveries in applied mathematics, showing how abstract periodic functions could solve real physical problems.

The Heat Equation

In one dimension, the heat equation is:

∂u/∂t = α ∂²u/∂x²

Where u(x,t) is temperature at position x and time t, and α is thermal diffusivity.

This equation says that the rate of temperature change at any point is proportional to how "curved" the temperature profile is there. Heat flows from hot regions (peaks) to cold regions (valleys).

The Fourier Solution

Fourier's brilliant insight was that each frequency mode decays independently. If the initial temperature profile is decomposed as:

u(x,0) = Σ bₙ sin(nπx/L)

Then the solution at time t is:

u(x,t) = Σ bₙ sin(nπx/L) e^(-αn²π²t/L²)

Each harmonic decays exponentially, with higher frequencies decaying faster (n² in the exponent). This is why sharp temperature variations smooth out quickly while broad temperature gradients persist longer.

Physical Intuition

High Frequencies Decay Fast

Sharp spikes in temperature (high frequency) smooth out almost instantly because they have steeper gradients for heat to flow down.

Low Frequencies Persist

Broad temperature variations (low frequency) change slowly. A warm room cools gradually compared to a hot spot.

Beyond Heat: Diffusion Everywhere

The same equation (called the diffusion equation) appears throughout science:

  • Chemical diffusion: How dye spreads in water
  • Finance: The Black-Scholes equation for options pricing
  • Biology: How chemicals spread in cells
  • Image processing: Gaussian blur is diffusion!

Historical Impact

Fourier's 1822 work "Théorie analytique de la chaleur" (The Analytical Theory of Heat) was initially controversial. Lagrange objected that discontinuous functions couldn't be represented by smooth sine waves.

It took decades for mathematicians to rigorously justify Fourier's methods, leading to the development of modern analysis and the concept of convergence. Fourier's practical engineering approach ultimately transformed mathematics.