Heat equation
The heat equation is a type of equation used in mathematics and physics to describe how heat spreads out in a particular area. The equation was first created by Joseph Fourier in 1822. People use the heat equation to model how heat spreads through different materials.
The heat equation is very important in the field of mathematics, and it is studied a lot. This equation is used in many areas of science and applied mathematics, such as probability theory and image analysis. Some solutions to the heat equation are called heat kernels, and they can provide important information about the area where the heat is spreading.
Scientists have found many different ways to use the heat equation. For example, it can be used to study random walks and Brownian motion in probability theory, and it can help to resolve pixelation in image analysis. In addition, variants of the heat equation are used in financial mathematics and quantum mechanics. Scientists have also found that solutions of the heat equation are helpful in understanding hydrodynamical shocks. People have been studying the heat equation for a long time, and it is a very important topic in mathematics and physics.
Heat Equation Media
Animated plot of the evolution of the temperature in a square metal plate as predicted by the heat equation. The height and redness indicate the temperature at each point. The initial state has a uniformly hot hoof-shaped region (red) surrounded by uniformly cold region (yellow). As time passes the heat diffuses into the cold region.
Solution of a 1D heat partial differential equation. The temperature (u) is initially distributed over a one-dimensional, one-unit-long interval (x = [0,1]) with insulated endpoints. The distribution approaches equilibrium over time.
The behavior of temperature when the sides of a 1D rod are at fixed temperatures (in this case, 0.8 and 0 with initial Gaussian distribution). The temperature approaches a linear function because that is the stable solution of the equation: wherever temperature has a nonzero second spatial derivative, the time derivative is nonzero as well.
Idealized physical setting for heat conduction in a rod with homogeneous boundary conditions.
Depicted is a numerical solution of the non-homogeneous heat equation. The equation has been solved with 0 initial and boundary conditions and a source term representing a stove top burner.
