Heat Equation

The Heat Equation in one spatial variable is a partial differential equation that describes the distribution of heat (or variations in temperature) in a given region over time. It is expressed as

\[u_t = \alpha u_{xx}\]

Here:

  • $u = u(x,t)$ represents the temperature at position $x$ and time $t$.

  • $u_t$ is the partial derivative of $u$ with respect to time $t$, representing the rate of change of temperature over time

  • $u_{xx}$ is the second partial derivative of $u$ with respect to position $x$, representing the curvature of the temperature profile in space.

  • $\alpha$ is a positive constant known as the thermal diffusivity of the material, which characterizes how fast heat diffuses through the material.

Using alternative notation, this can be written

\[\frac{\partial u}{\partial t}(x,t) = \alpha \frac{\partial^2 u}{\partial x^2}(x,t).\]

Notice that $\frac{\partial^2 u}{\partial x^2}$ is the Laplacian in one dimension. In higher dimensions, the heat equation can be adjusted to account for the additional heat flow contribution along the other axes by the simple modification

\[\frac{\partial u}{\partial t} = \alpha \nabla^2 u,\]

where $\nabla^2 u$ is the Laplacian.

We can add a term for heat generated by sources, $P(x,t)$ to get

\[\frac{\partial u}{\partial t}(x,t) = \alpha \frac{\partial^2 u}{\partial x^2}(x,t) + P(x,t). \tag{4}\]

Heat Flow Through a Wire

Here, we will give a model for the flow of heat through a thin, insulated wire whose ends are kept at a constant temperature of $0^\circ c$ and whose initial temperature distribution is to be specified.

Let’s assume the wire is stretched along the $x$-axis, starting at $x=0$ and extending to $x=L$.

Equation (4) above governs the flow of heat in the wire. We have two other constraints. First, we are keeping the ends of the wire at $0^\circ c.$ Thus, we require that

\[u(0, t) = u(L, t) = 0 \tag{5}\]

for all $t$. These are called boundary conditions. Second, we must be given the initial temperature distribution $f(x)$. That is, we require:

\[u(x,0) = f(x), \quad 0 < x < L. \tag{6}\]

Equation (6) is referred to as the initial condition on $u$.

Combining equations (4), (5), and (6), we have the following mathematical model for the heat flow in a uniform wire without internal sources $(P = 0)$ whose ends are kept at the constant temperature $0^\circ c$:

\[\frac{\partial u}{\partial t}(x,t) = \alpha \frac{\partial^2 u}{\partial x^2}(x,t), \quad 0 < x < L, \quad t > 0, \tag{7}\] \[u(0, t) = u(L, t) = 0, \quad t>0, \tag{8}\] \[u(x,0) = f(x), \quad 0 < x < L. \tag{9}\]

This model is an example of an initial-boundary value problem. Finding a function $u(x,t)$ that satisfies all three of these conditions lets us know the temperature at any position and time.

One more thing to note. When the temperature reaches a steady state and $u$ does not depend on time, and there are no sources, then $\partial u / \partial t = 0$ and the temperature satisfies Laplace’s equation

\[\nabla^2 u = 0.\]

To see how to solve the heat equation, see Separation of Variables.