The Fourier equation, also known as the heat equation, is a fundamental partial differential equation that describes how heat distributes over time in a medium. It was developed by Joseph Fourier in the early 19th century. The standard form of the equation is shown here, where u represents temperature, t is time, alpha is the thermal diffusivity coefficient, and nabla-squared is the Laplace operator representing spatial derivatives. On the right, we can see how an initial temperature distribution evolves over time, gradually spreading out and decreasing in amplitude as heat diffuses through the medium.
The heat equation can be written in different forms depending on the dimensionality of the problem. In one dimension, such as heat flow along a rod, the equation uses the second derivative with respect to x. In two dimensions, like heat spreading across a flat surface, we add the second derivative with respect to y. For three-dimensional problems, we include all three spatial derivatives. These can all be written compactly using the Laplacian operator. The heat equation has numerous applications beyond thermal physics, including particle diffusion, financial modeling, and population dynamics. On the right, we can see a simulation of the one-dimensional heat equation showing how temperature evolves in a rod with fixed endpoints over time.
There are several methods to solve the heat equation. The separation of variables technique is fundamental, where we express the solution as a product of functions that depend on only one variable. For the heat equation, we write u(x,t) as X(x) multiplied by T(t). This leads to two ordinary differential equations that are easier to solve. The spatial components X(x) are typically sinusoidal functions that satisfy the boundary conditions, while the temporal components T(t) are exponential decay functions. Using Fourier series, we can represent the complete solution as an infinite sum of these component functions. Other solution methods include Fourier transforms, which are useful for unbounded domains, and numerical methods like finite difference schemes that approximate the derivatives with discrete values, essential for complex geometries or nonlinear problems.
The heat equation is often solved numerically using finite difference methods, which approximate the continuous derivatives with discrete differences. For the second spatial derivative, we use the central difference formula shown here, where u_{i,j} represents the temperature at position i and time step j. The time derivative is approximated using a forward difference. Combining these approximations gives us the explicit Forward-Time, Central-Space scheme, or FTCS. This method is simple to implement but has a stability condition: the ratio of time step to squared space step must be less than one-half. If this condition is violated, the numerical solution can become unstable and produce meaningless results. For better stability, implicit schemes like Crank-Nicolson or Backward Euler can be used, which are unconditionally stable but require solving a system of equations at each time step. On the right, we can see a simulation of heat diffusion on a 2D grid using the finite difference method, with a constant heat source at the bottom.
The heat equation was developed by Joseph Fourier in the early 19th century. Born in 1768, Fourier presented his first paper on heat theory in 1807 and later published his comprehensive work 'Analytical Theory of Heat' in 1822. His mathematical approach was initially controversial because it challenged the existing standards of mathematical rigor, but it eventually led to significant advances in mathematical analysis. Today, the heat equation has applications far beyond its original purpose. In engineering, it's used to design heat exchangers, cooling systems, and building insulation. In materials science, it helps model phase transformations and semiconductor processing. In biology and medicine, the equation describes processes like tumor growth and drug diffusion in tissues. The mathematical framework Fourier developed—particularly Fourier series and transforms—has become fundamental across many scientific and engineering disciplines, demonstrating how a focused study of heat conduction led to mathematical tools with universal applications.