One-dimensional heat equation

Description

The one-dimensional hear equation is a differential equation involving three variables:

• An independent variable $t$ (the time parameter)
• An independent variable $x$ (the space parameter)
• A dependent variable $u$ (called the potential function or heat function)

The equation is:

$u_{t}=ku_{xx}$

Or in other words:

${\frac {\partial u}{\partial t}}=k{\frac {\partial ^{2}u}{(\partial x)^{2}}}$

Here, $k$ is a conductivity constant.

Interpretations

In terms of heat flow

The one-dimensional heat equation models the heat flow in a body, in the following sense. The amount of heat that flows into a point per unit time) depends on the way the heat is spatially distributed around that point.

For instance, if the temperature at both ends of a rod is being kept at different constant values, then at each point in between ,the temperature will attain a value that is somewhere in between the values at the ends. Note that in this case, the temperature is not constant at all points, but it varies linearly with position and hence the second derivative is 0. Thus, such a configuration is stable.

However, before all points in the rod attain these equilibrium temperatures, there will be a period where some points in the rod are at temperatures higher than that which is finally to be expected. In this case, the extent to which the temperature differs from a linear one is measured by $u_{xx}$, and it gets corrected at the same rate.

This is also an example of a diffusion equation -- it models a process whereby diffusion is occurring so as to equalize, or uniformize, a function across space as time passes.